












































































































































































CIVIL ENGINEER’S POCKET-BOOK. 


MR. TRAUTWINE’S ENGINEERING WOR 


The Field Practice of Laying out Circular Curves for Rail¬ 
roads. By John C. Trautwine, Civil Engineer. Thirteenth 
edition, 12mo, gilt edge.f 

A Method of Calculating the Contents of Excavations and 

Embankments. By John C. Trautwine, Civil Engineer. 
Ninth Edition, 8vo, cloth. 

The Civil Engineer’s Pocket-Book of Mensuration, Trigo¬ 
nometry, Surveying, Hydraulics, Hydrostatics, Instruments and 
their adjustments, Strength of Materials, Masonry, Principles 
of Wooden and Iron Roof and^Bridge Trusses, Stone Bridges 
and Culverts, Trestles, Pillars, ‘ Suspension Bridges, Dams, 
Railroads, Turnouts, Turning Platforms, Water Stations, Cost, 
of Earthwork, Foundations, Retaining Walls, etc., etc., etc. 

In addition to which the elucidation of certain important 
Principles of Construction is made in a more simple manner 
than heretofore. By John C. Trautwine, C. E. 12mo, 832 
pages, illustrated with about VOO wood-cuts. Morocco, flap, 
gilt edge. Fourteenth edition, thirty-fifth thousand ... 8 

Any of the above books will be sent to any part of the Li 
States or Canada on receipt of list price. 

Send money in Registered Letter, P. 0. Order, or Check. 

JOHN WILEY & SONS, 

Scientific Publishers, 
v.15 Astor Place, New York ( 
* 

\ 



V 









THE 




CIVIL ENGINEER’S POCKET-BOOK, 

/ 


OF 


lensuration, Trigonometry, Surveying, Hydraulics, Hydrostatics, 
Instruments and their Adjustments, Strength of Materials, 
Masonry, Principles of Wooden and Iron Roof and 
Bridge Trusses, Stone Bridges and Culverts, 

^ Trestles, Pillars, Suspension Bridges, Dams, 

Bailroads, Turnouts, Turning-Plat¬ 
forms, Water Stations, Cost of 
Earthwork, Foundations, 

Eetaining Walls, 

Etc., Etc., Etc. 

N ADDITION TO WHICH THE ELUCIDATION OF CERTAIN 
IMPORTANT PRINCIPLES OF CONSTRUCTION IS 
MADE IN A MORE SIMPLE MANNER 
THAN HERETOFORE. 





OF EXCAVATIONS AND 


COPIOUSLY ILLUSTRATED. 


dFouv*—^ ^ " — • 



NEW YORK: 

JOHN WILEY & SONS, 


15 ASTOR PLACE. 
LONDON: E. & F. N. SPON. 
1889 . 






Entered, according to Act of Congress, In ibe year I8S2* by 
JOHN C. TRACT WINE, 

in tne Office of the Librarian of Congress at Washington. 


Copyright by John C. T'p.autwtne, Jr., 18S9. 


i 




[ 

1 


WM. RUTTER A CO., 

BOOK MANUFACTURERS, 
SEYENTH A CHEPRY isTS., PHILA. 



Press of Jus. B. Rodgers Printing Co., Phila., Pa. 










THE AUTHOE 


Idiates tljis “mctl; 


TO THE MEMORY OF HIS FRIEND, 

THE LATE 

BENJAMIN H. LATROBE, Esq. ? 

CIVIL ENGINEER. 


V 


No pains have been spared to maintain the position of 
this as the foremost Civil Engineer’s Pocket-Book, not only 
in the United States, but in the English language. 

JOHN WILEY & SONS, 

Scientific Publishers, 

15 Astor Place, New York City. 



PREFACE TO FIRST EDITION. 


QHOULD experts in engineering complain that they do not find 
^ anything of interest in this volume, the writer would merely 
remind them that it was not his intention that they should. The 
book has been prepared for young members of the profession; and 
one of the leading objects has been to elucidate in plain English, a 
few important elementary principles which the savants have envel¬ 
oped in such a haze of mystery as to render pursuit hopeless to any 
but a confirmed mathematician. 

Comparatively few engineers are good mathematicians; and in the 
writer’s opinion, it is fortunate that such is the case; for nature rarely 
combines high mathematical talent, with that practical tact, and 
observation of outward things, so essential to a successful engineer. 

There have been, it is true, brilliant exceptions; but they are very 
rare. But few even of those who have been tolerable mathematicians 
when young, can, as they advance in years, and become engaged in 
business, spare the time necessary for retaining such accomplish¬ 
ments. 

Nearly all the scientific principles which constitute the founda¬ 
tion of civil engineering are susceptible of complete and satisfactory 
explanation to any person who really possesses only so much element¬ 
ary knowledge of arithmetic and natural philosophy as is supposed 
to be taught to boys of twelve or fourteen in our public schools.* 


* Let two little boys weigh each other on a platform scale. Then when they 
balance each other on their board see-saw, let them see (and measure for themselves) 
that the lighter one is farther from the fence-rail on which their board is placed, in 
the same proportion as the heavier boy outweighs the lighter one. They will then 
have learned the grand principle of the lever. Then let them measure and see that 
the light one see-saws farther than the heavy one, in the same proportion; and they 
will have acquired the principle of virtual velocities. Explain to them that equality 
i(f moments means nothing more than that when they seat themselves at their metis- 






Vlll 


PREFACE. ! 

The little that is beyond this, might safely be intrusted to the 
savants. Let them work out the results, and give them to the engi¬ 
neer in intelligible language. We could afford to take their words for 
it, because such things are their specialty ; and because we know that 
they are the best qualified to investigate them. On the same princi¬ 
ple we intrust our lives to our physician, or to the captain of the 
vessel at sea. Medicine and seamanship are their respective special¬ 
ties. 

If there is any point in which the writer may hope to meet the 
approbation of proficients, it is in the accuracy of the tables. The 
pains taken in this respect have been very great. Most of the tables 
have been entirely recalculated expressly for this book; and one of 
the results has been the detection of a great many errors in those in 
common use. He trusts that none will be found exceeding one, or 
sometimes two, in the last figure of any table in which great accuracy 
is required. There are many errors to that amount, especially where 


ured distances on their see saw, they balance each other. Let them see that the weight 
of the heavy hoy, when multiplied by his distance in feet from the fence-rail 
amounts to just as much as the weight of the light one when multiplied by his dis¬ 
tance, and that each of these amounts is in foot-pounds. Explain to them that the 
lighter boy, because he swings faster than the other, has greater kinetic energy, not¬ 
withstanding his slighter weight, and will therefore bump harder against the ground. 
The boys may then go in to dinner, and probably puzzle their big lout of a brother 
who has just passed through college with high honors. They will not forget what 
they have learned, for they learned it as play, without any ear-pulling, spanking, t 
or keeping in. Let their bats and balls, their marbles, their swings, &c., once 
become their philosophical apparatus, and children may be taught ( really taught) 
many of the most important principles of engineering before they can read or 
write. 

It. is the ignorance of these principles, so easily taught even to children, that con-, 
stitutes what is popularly called “The Practical Engineer;” which, in the great j 
majority of cases, means simply an ignoramus, who blunders along without knowing 
any other reason for what he does, than that he has seen it done so before. And it 
is this same ignorance that causes employers to prefer this practical man to one who 
is conversant with principles. They, themselves, were spanked, kept in, &e, when 
boys, because they could not master leverage, equality of moments, and virtual velo¬ 
cities, enveloped in x’s, p’s, Greek letters, square-roots, cube-roots, &c, and they 
naturally set down any man as a fool who could. They turn up their noses at science, 
not dreaming that the word means simply, knowing why. And it must be confessed 
that they are not altogether without reason ; for the savants appear to prepare their 
books with the express object of preventing purchasers, (they have but few readers,} 
from learning why. 






PREFACE. 


IX 


the recalculation was very tedious, and where, consequently, interpo 
lation was resorted to. They are too small to be of practical import¬ 
ance. He knows, however, the almost impossibility of avoiding larger 
errors entirely; and will be glad to be informed of any that may be 
detected, except the final ones alluded to, that they may be corrected 
in case another edition should be called for. Tables which are abso¬ 
lutely reliable, possess an intrinsic value that is not to be measured by 
monev alone. With this consideration the volume has been made a 
trifle larger than would otherwise have been necessary, in order to 
admit the stereotyped sines and tangents from his book on railroad 
curves. These have been so thoroughly compared with standards 
prepared independently of each other, that the writer believes them 
to be absolutely correct. 

In order to reduce the volume to pocket-size, smaller type has been 
used than would otherwise have been desirable. 

Many abbreviations of common words in frequent use have been 
introduced, such as abut, cen, diag, hor, vert, pres, &c, instead ot 
abutment, center, diagonal, horizontal, vertical, pressure, &c. They 
can in no case lead to doubt; while they appreciably reduce the 
thickness of the volume. 

Where prices have been added, they are placed in footnotes. They 
are intended merely to give an approximate or comparative idea of 
value; for constant fluctuations prevent anything farther. 

The addresses of a few manufacturing establishments have also 
been inserted in notes, in the belief that they might at times be found 
convenient. They have been given without the knowledge of the 
proprietors. 

The writer is frequently asked to name good elementary books on 
civil engineering; but regrets to say that there are very few such in 
our language. “ Civil Engineering,” by Prof. Mahan of West Point; 
“Roads and Railroads,” by the late Prof. Gillespie; and the “Manual 
for Railroad Engineers,” by George L. Vose, Civ. Eng, and Professor 
of Civil Engineering in Bowdoin College, Brunswick, Maine, are 


X 


PREFACE. 


the best. The last, published by Lee & Shepard, Boston, 1873, is the : 
most complete work of its class with which the writer is acquainted. 

Many of Weale’s series are excellent. Some few of them are 
behind the times; but it is to be hoped that this may be rectified in 
future editions. Among pocket-books, Haswell, Hamilton’s Useful] 
Information, Henck, Molesworth, Nystrom, Weale, &c, abound in 
valuable matter. 

The writer does not include Kankine, Moseley, and Weisbach,i 
because, although their books are the productions of master-minds, 
and exhibit a profundity of knowledge beyond the reach of ordinary 
men, yet their language also is so profound that very few engineers 
can read them. The writer himself, having long since forgotten the 
little higher mathematics he once knew, cannot. To him they are but 
little more than striking instances of how completely the most simple 
facts may be buried out of sight under heaps of mathematical rubbish. 

Where the word “ ton ” is used in this volume, it always means 
2240 lbs, because that is its meaning in U. S. law. 

John C. Trautwink. 


Philadelphia, September, 1876. 







PREFACE TO THE TWENTY-SECOND THOUSAND. 


S INCE the appearance of its last edition (the twentieth thousand) in 
1883, the “ Pocket-Book ” has been thoroughly revised, and many 
important additions and other alterations have been made. These 
t necessitated considerable change in the places of the former matter, 
and it was deemed best to turn this necessity to advantage, and to make 
a thorough re-arrangement, putting all of the articles, as for as possi¬ 
ble, in a rational order, so that the reader might, in many cases, be able 
to readily find a desired subject without the use of the index, which, 
notwithstanding, has been greatly enlarged. All of the articles for¬ 
merly in the appendix, and such of those in the glossary as seemed out 
of place there, have been transferred to their proper places in the body 
of the work. 

In making the re-arrangement, the rule has been to proceed from the 
abstract to the concrete, from the theoretical to the practical, from the 
general to the particular. All this will appear from a glance at the new 
and very full table of contents on pages xxiii to xxxii. Thus, beginning 
with Arithmetic, we proceed to Mensuration of lines, angles, surfaces, 
solids, the simple, in each case, preceding the complex. Next come the 
various branches of Surveying, with surveying and engineering instru¬ 
ments. Then Sound, Heat, Air and Water, Hydrostatics, Hydraulics, 
Water Supply, and Mechanics. Next, under Materials and their 
Properties, we have Specific Gravity, Weight, Weights and Measures, 
Weights, Dimensions, Prices, etc, of many manufactured and other arti¬ 
cles, and Strength of Materials. This last leads naturally to strengths 
.of beams and trusses, after which follow Suspension Bridges. Proceed¬ 
ing next to subjects more closely connected with actual construction, 
and in which theory plays a less important part, we come to Well 
Boring, Dredging, Foundations, and Stonework, the latter including 
-j Quarrying and Masonry, Bricks, Mortar, Cements and Concretes, Re¬ 
taining Walls, and Arches of brick and stone. Then follows the arti¬ 
cle on Railroads, under which head have been grouped, in order, 
Railroad Construction (including Grades, Curves, Earthwork, Tunnels, 
Trestles, Roadway, and Turnouts), Railroad Equipment (including 
Turntables, Water Stations, &c), Railroad Rolling-Stock and Operation, 
nd Railroad Statistics. By the transfer to the body of the book of the 
arious articles in the appendix, the glossary comes to its proper place 
the end of the volume, just before the index. 

It is believed that this thorough re-arrangement will, in many cases, 
1 iable the reader to find a subject more readily by the aid of the table 

xi 




Xll 


PREFACE TO THE TWENTY-SECOND THOUSAND. 


of contents than b)'- that of the index; or, indeed, as already suggested, 
without the aid of either, and by the natural place of the subject in the 
order of arrangement. 

The following are some of the principal changes of subject matter 
in this edition. (The former pages of revised articles are giveu in 
parentheses.) On page 34, a table of Fractions reduced to exact deci¬ 
mals has been added. On pages 128 to 140, are two new Tables of 
Circumferences and Areas of Circles, to not less than four places of deci¬ 
mals. These tables have been calculated independently by two persons, 
expressly for this work, and the two results compared after the casting 
of the plates. They are therefore believed to be correct.* The rules 
for finding chords, radii, Ac, of Circular Arcs, p 141 (16, 17) have been 
greatly extended, re-modeled, re-illustrated, and systematized. Under , 
Thermometers, pj> 213, Ac (309, Ac), the rules for conversion of Fahren¬ 
heit to Centigrade readings, and vice versa , have been put into more con¬ 
venient shape, and others added for the conversion of both of these 
scales to that of Reaumur and vice versa. In consonance with this 
change, and to make this subject complete, three new Conversion Tables 
have been substituted for the one formerly given. The article on Flota- . 
tion, Ac, p 235 (635) has been corrected and amplified. The opening 
articles in Hydraulics, pp 237, Ac (535, Ac), in which are explained the 
divisions of the total head and their several offices, have been re-written, 
and it is believed they will be found to give a fuller and more satisfac¬ 


tory explanation than before, of the various phenomena connected with 


this subject. On p 254 will be found a new formula for the discharge 
through a pipe of varying diameter. The remarks on City Water-Pipes, 
Valves and Fire-Hydrants, pp 293 to 305 (572 to 577) have been re-writ¬ 
ten, enlarged, and modernized. New forms of Pipe-Joint, Stop-Valve, j 
and Fire-Hydrant, and several modern appliances, are described and 


illustrated ; and recent data are given for the Cost of Pipe and Laying* 


And as nearly accurate as their number of decimal places permits, except that if 
1 be added to the final decimal of the circumferences and areas corresponding to the 
following diameters, their error in excess will be a little less tbau their present error 
in deficiency. ( Here followed, in the twenty-second thousand, a list of these diameter 
The greatest error in these is now less than 1 in the Jinal decimal. It will then be 
less than as in all the othe; s. ( These trifling inaccuracies were corrected in the twenty- 
fifth thousand). 1 










PREFACE TO THE TWENTY-SECOND THOUSAND. xiii 




On p 364 (587) the Velocities, &c, of Falling Bodies are tabulated in con¬ 
venient form. The remarks on Centrifugal Force, p 366 (494), have 
been revised, and the explanation made clearer. On p 396 will be found a 
concise but full account of the arrangement of the new Standard Rail¬ 
way Time. The various articles on pp 398 to 433 (357 to 382) in which 
* are given the Dimensions, Weights, Prices, Ac, of various manufactured 
( articles in common use, have been thoroughly overhauled, and corrected 
up to the present time. Among these may be noted especially Iron 
^ Pipes, p 405 (364) to which Boiler Tubes have been added; Bolts, Nuts, 
and Washers, pp 406 to 408 (374 to 376); a new list of Wire Gauges, giv¬ 
ing the new British Standard Gauge, p 410 (368); Wire Ropes, p 413 
(380); Tin Plates, p 419 (379); and Window-Glass, p432 (514). 

On p 435 (174) is a new table of crushing loads of American woods, 
deduced from the last census. The former table was based upon Hodg- 
kinson’s experiments, made chiefly with foreign woods. The coefficients 
for transverse strengths of timber, p 493 (185), have also been changed, 
where necessary, to correspond with results of the census experiments. 

On p 449 (234) is a revised and complete list of Phoenix Segment Col¬ 
umns, illustrated. A number of changes have been made in Rivets 
and Riveting, pp 468, Ac (653, Ac). 

Under Transverse Strength, pp 478, Ac (183, Ac), have been grouped 
the several articles on Moments of Rupture, of Inertia, and of Resist¬ 
ance, Open and Closed Beams, Shearing of Beams, Ac, which were for¬ 
merly scattered through widely separated parts of the book (183, Ac, 
, 194, Ac, 217, Ac, 642 to 650). These have been carefully revised and 
amended, and arranged into a systematic whole, from which the student 
1 can now obtain a clear and correct notion of this somewhat troublesome 
subject. On pp. 521 to 523 (211 to 213) are new rules and tables for I 
and channel beams; and. on pp 525 to 527 (373) similar rules and tables 
for angle and T iron. These have been carefully compiled from the 
latest tables of the Pencoyd Works, which were selected on account of 
the large range of sizes made by those works, and the completeness of 
the data obtainable in regard to them. On p 524 (304) are illustrations 
of recent bridges of I beams in use on the Penna R R. The article on 
^nearingof (or vertical strains in) Beams, p 532 (642), is entirely re¬ 
written and re-illustrated. A new article on Riveted Girders is given, pp 
537 to 546 (214 to 217), in which much fuller rules, used in the best prac¬ 
tice of to-day, are substituted for those of Fairbairn ; and illustrations 
from the present standards of the Penna R R are given in place of 
those from the Charing Cross Railway of a quarter of a century ago. 

In Trusses, the remarks on Counter-bracing, pp564 to 570 (275 to 281), 
the treatment of the Fink Roof Truss, pp 574, Ac (264, Ac), and that 
of the Braced Arch, p 592 (274), have been corrected and simplified. 
Increased dimensions have been given in the tables of Wooden Howe 







XIV PREFACE TO THE TWENTY-SECOND THOUSAND. 




and Pratt Trusses, pp 595 and 596 (284 and 285), to provide for the raoder 
weights of such loads as would probably be placed upon such structure! 

Considerable changes are made in Suspension Bridges, pp 615 to 62 
(588 to 597). The formulae for the strains in the main chains have bee 
re-arranged, and those for the strains in the back-stays and on the pier 
have been much extended and illustrated, and made to cover the groun 
more fully. The description of the Pierce Well-borer, p 626 (636), ha 
been corrected, other boring tools described and illustrated, and a ne^j 
article on Artesian Well-Drilling added. The Nasmyth Steam-Ilan 
mer Pile-Driver is described on p 642. The articles on Stonewori < 
pp 651 to 668 (310 to 313) have been greatly enlarged by the add . 
tion of new articles on Machine Rock-Drills, Air Compressors, an 
Modern Explosives. In the first named, sectional views and descri] 
tions of several of the more prominent modern drills are given. 

Under Railroads, pp 722 to 818 (409 to 428, &c), will be found coi 
lected most, or all, of the former matter of the book, relating exclu 
sively to that subject, and much that is new. A Table of Curves, wit 
Radii, <tc, in metres , has been added, on p 728, for the convenience o 
those who have occasion to work by the metric system. Under Earth 
work, pp 747, &c (418 to 428, 435 to 441), original rules and table 
have been added for estimating the cost of moving earth by means o 
the modern wheeled and drag scrapers, and that of moving earth am 
rock by cars and locomotives. In this connection, also, the operatioi 
of the modern steam excavators, or land dredges, is described, and dat. 
given concerning their capacity and their cost of operation. 

The article on Trestles, p 755, &c (307, <fec), is greatly enlarged h; 
descriptions of the new Portage, Kinzua, and other recent iron tres 
tics. The former articles (390 to 40S) on Rail joints and Turnout 
have been replaced by new, enlarged, and re-illustrated articles, pi 
7C3 to 789, based upon present practice. Under Rail-joints the prin, 
cipal place has been given to the fish and angle plates now so gener 
ally used, after which follow the modern Fisher and Gibbon joints. 
The older forms, including most of those illustrated in former editions i 
are briefly referred to for purposes of comparison. Under Turnout#! 
the lengthy details of the stub-switch stand, &c, are omitted, and {hi! 
point and Wharton switches are given greater prominence. Severn 
important improvements having been made in the Wharton since out] 
last edition (in which a new description of that switch was given),a] 
further revision became necessary, and the new article is accompanies] 
by a new cut, showing the switch in its latest form. Under Frogs !| 
prominence is given to the Steel-Rail Frog, which has now supplanted 
the older forms on first-class lines; the Mansfield Elastic Frog isde 
scribed and illustrated; and an explanation of the working of the! 
Spring-Rail Frog is added, with original cuts. The directions for laying! 
out turnouts from curves have been simplified. 





PREFACE TO THE TWENTY-SECOND THOUSAND. XV 


Under Turntables, the description of the Sellers table has been re¬ 
written and brought into line with recent improvements; and several 
modern forms of plate and cast-iron turntables are described and illus¬ 
trated. The remarks on Track-Tanks, p 802 (434), have been extended, 
and a cut added, showing a cross-section of the Penna R R standard 
tank. 

The scattered data (411 to 413) on Locomotives and Cars, and the 
Railroad Statistics formerly given (409, 410) have made room for tabu¬ 
lated modern data, pp 805 to 818, on these subjects, much fuller than 
those which they have replaced, and covering both standard and narrow- 
gauge roads. 

The following are among the more important of the minor changes. 

Compare 


New page with Old page 

55 Exterior Angles.62 

56 Definition of Complement and Supplement . . .62 

110 Table of Polygons.15 

112 Trigonometry. Case 2.39 

146 to 148 Circular Segments. 22, 24, 25 

149 &c Ellipse. 25, 26, 630 

159 Cylindric Ungulas. 31, 32, 630 

220 Records of Rain-falls.. . . 518 

232 Rankine’s Formula for Thickness of Wall at Base . . 531 

246 Remarks Introductory to Table 2.541 

248 Art 3 ... 542 

293 Weights of Cast-Iron Pipes.364 

332 Polygon of Forces.470 

348 Center of Gravity (foot).* . . 442 

380 Specific Gravity. Introduction.383 

381 &e Heading of Table. 384 &c 

384 Foot-Note. 

425 Creosote, &c . . . . . . . . 359 to 362 

433 Paper.151, 152 

436 Crushing Loads of Masonry and of Ice (foot-notes) . .175 

438, 439 Iron Pillars. 222, 223, 234 

504 to 506 Deflections, &c, of Beams .... 196 to 198 

572 First paragraph.262 

573 Foot-Note . . . ..263 

710 Brick Arches, Penna R R, Phila. 

803 Track Scales.409 


A number of minor additions have been made to the glossary. 

Most of the new matter is in nonpareil, the larger of the two types 
heretofore used. Boldfaced type has been freely used ; but only for 
the purpose of guiding the reader rapidly to a desired division of a 
subject. For emphasis , italics have been employed. 

Illustrations which were lacking in clearness or neatness have been 
re touched and re-lettered, or replaced with new and better cuts. The 
new matter is very freely illustrated. 

New rules have been put in the shape of formuke, and many of the 










Xvi PREFACE TO THE TWENTY-SECOND THOUSAND. 

old rules have been re-cast into the same form. In doing this, th< 
terms of the formulas have, except where this would be exceedingh 
cumbersome, been written out in full, as in former editions, so that tin 
reader is not compelled to look back over a number of pages to fin< 
the meaning of arbitrary symbols, and to tax his mind with remember 
ing them when found. It is believed that the formula? will be found a 
least as easy to use as the rules. Take, for example, the second formul; 
in Art 10, p 262 (556). In its former shape it read : 

“ To find the time reqd to fill the reservoir, m, from any level, c, above the top o! 
the opening, to any upper level, d." (This should have read “ to the level a of tin 
upper reservoir.”) 


“Rule. First find the area in sq ft of a hor section of the reservoir in , which i 
supposed to he of uniform section throughout its depth. Mult together this area 
the constant number 2, and the sq rt of the vert height, a c, in ft. Call the prod p 
Mult together the area of the opening n, in sq ft; the coeffof contraction (usnall; 
about .62, whether the disch be into the air or under water); and the constant 8.02 
Call the prod y. Div p by y. The quot will be the reqd time in secs.” 

It is safe to say that any one, capable of using the above rule, can a.' 
readily use the following formula, in which it now appears 


Seconds required 

to raise level in m from 
c to a 


l height a c v hor area of ., 0 
_in ft m in sq ft ^ z 

are :?o S 7<r g x .62^ 


and, moreover, the formula has the great advantages of showing the 
whole operation at a glance, of making its 'principle more apparent 
and of being much more convenient for reference. The formula might 
have been slightly condensed by giving the product of .62 X 8.03, in 
stead of the factors; but it was preferred to give the latter because they 
show the principle of the formula. The coefficient .62 had already been 
explained. The article referred to has also been otherwise improved. 

lhe addition of new matter, and a number of blank spaces necessarily 
left in making the re-arrangement, have increased the number of pages 
about one-fifth. 

The new index is in stricter alphabetical order than that of form,el 
editions, and contains more than twice as many entries, although much j 
repetition has been avoided by the free use of cross-references, without 
which this part of the work might have been indefinitely extended. 

I lie selection of articles of manufacture or merchandise for illustra¬ 
tion, has been guided by no other consideration than their fitness for the 

purpose, and the courtesy of the parties representing them, in supply¬ 
ing information. 

The writer gratefully acknowledges the kindness of those who have 
assisted in furnishing and arranging data. J. C T Jr i 

Philadelphia, January, 1885. 











PREFACE TO THE TWENTY-FIFTH THOUSAND. 


For tliis edition, the articles on Flow of Water in Channels, pp 
268 etc., Friction, pp 370 etc., and Timber Preservation, pp 425 etc., 
have been re-written. In the first named, Kutter’s formula is given, 
with tables to facilitate its use, and instructions for preparing a dia¬ 
gram from which its results may be taken by inspection. Under 
Friction are given the results of recent researches, including the now 
famous experiments of Capt. Douglas Galton on brake friction. The 
new article on Timber Preservation embodies, besides other matter, 
results recently published by a Committee of the American Society 
of Civil Engineers. 

Prices and descriptions of manufactured articles etc., have been re¬ 
vised to date. 

A number of changes have been made in the articles on Force in 
Rigid Bodies, and Trusses, in order to further simplify the treatment 
of those subjects. These changes include the enlargement, and the 
re-arrangement in more convenient form, of the articles on Gravity 
and Falling Bodies, Descent on Inclined Planes, Angular Velocity, 
Moment of Inertia, Radius of Gyration and Centrifugal Force. 

On p 230 a, the subject of Distribution of Pressure in Plane Sur¬ 
faces is explained more fully than in earlier editions. The empir¬ 
ical formula, p 243, for flow of water in pipes, has been so modified 
as to give more closely approximate results. An article on the Fa¬ 
tigue of Materials has been added, p 435. The rules for Strengths 
of Pillars, pp 439, etc., and those for Limited Deflections of Beams, 
pp 510 etc., have been simplified. On p 678 is given a summary of 
the results of Mr. Eliot C. Clarke’s recent experiments on the 
Strength etc. of Cements. Many other minor improvements have 
been made. 

Advantage has been taken of these changes to still further extend 
the substitution of the larger for the smaller type. 

J. C. T. Jr. 

Philadelphia, April, 1886. 

xvii 



PREFACE TO THE TWENTY-SEVENTH THOUSAND. 


The present edition contains revised formulae for thicknesses of cyl¬ 
inders under internal pressure, p. 232, a revised table of thicknesses 
for cast-iron pipe, p. 233, additional tables of coefficient “c” by 
Kutter’s formula, pp. 275 etc., and a new and fuller table of values 
of foreign coins, p. 386. Air. Pegrain’s suggested uniform loading 
for railroad bridges, to be substituted for the usual wheel loads in 
specifications, is given on p. 546. The formulae for approximate 
weights of truss bridges, p. 605, have been revised, and the table 
of dimensions and weights of large locomotives, p. 807, has been 
made to include some still heavier engines now in use. For minor 
changes see pp. 154, 155, 156, 159, 192, 358, 368, 397, 432 etc. 
About thirty pages more of the smaller type have been replaced 
by the larger. 

J. C. T., Jr. 

Philadelphia, March, 1S87. 


PREFACE TO THE THIRTY-SECOND THOUSAND, THIRTEENTH EDITION. 


The principal changes made in preparing for this edition are 
those in the articles on Mechanics and on Strength of Materials. 
The introductory remarks on both subjects (pp. 306 to 319 and pp. 
434, etc.), the rules for Parallel Forces and Center of Gravity (pp. 
347 to 351 h) and the remarks on Deflections of Beams (pp. 504 to 
505 b) have been re-written and greatly extended; and lists of Boiled 
Steel I Beams (p. 523 c) and of Separators for Beams (pp. 523 6 
and d ), have been added. 

A number of the rules under Mensuration have been put into 
better shape (See Arcs, pp. 141, etc.; Sectors, etc., p. 146, and Cones, 
etc., etc., pp. 160 to 162) ; several additions have been made to the 
article on Turnouts (pp. 770, etc.) ; and prices, etc., have been 
revised to date. A number of less important changes will also be 
noticed. 

„ J. C. T., Jr. 

Philadelphia, June, 1888 . 
xviii 







CONTENTS. 


> 


In many cases, a subject may be found more quickly by means of this table of 
contents, than by the index, page 833. See also Glossary, page 819. 


MATHEMATICS. 



Arithmetic. 


PAGE 


fractions. 

Addition; subtraction; multiplica¬ 
tion; division; greatest common 
divisor; to reduce to lowest 

terms. 33 

Conversion into decimals... 34 

Decimals. 

Addition ; subtraction ; multiplica¬ 
tion; division. 35 

Duodecimals. 35 

Simple Proportion. 35 

Compound Proportion. Ratio. 35 

Arithmetical Progression. 36 

Geometrical Progression. 36 

Permutation. 36 

Combination. 36 

Alligation. 36 

Equation of Payments. 37 

Simple Interest. 37 

Compound Interest. 37 

^Discount. 37 

Commission, Brokerage. 37 

Insurance. 37 

Fellowship, Partnership. 37 

Logarithms. 

Table of. 38 

To find roots by. 39 

Hyperbolic or Naperian. 39 

Roots and Powers. Square and Cube. 

Tables of. 40 

Powers of large numbers. To find. 48 
Roots of large numbers. To find.. 52 
Roots of decimals. To find. 53 


Geometry, Mensuration, 
and Trigonometry. 


Liues, Figures, and Solids defined.... 54 


Tines. 
Lines. To divide. 


54 


Angles. 

PAGE 


Angles. 

Defined ; Different kinds of.. 54 

Interior and exterior. 55 

Right angles. To draw. 55 

Paralie! lines. To draw. 56 

Angles. To draw; To bisect. 56 

Angles in a circle. 56 

Complement and Supplement. 56 

Angles in a parallelogram. 57 

Minutes and Seconds in Decimals 
of a Degree. 

Table of. 57 

Angles. To measure by a 2-ft rule, 

&c. 58 

Sines, tangents, &c. 

Defined. 59 

Table of. 60 

Chords. Table of. 105 


Surfaces. 

Polygons. 

Regular. Tables, Ac, of. 110 

Triangles. 

Different kinds of; properties of... 110 

Area of. To find. 110,111 

Side of. To find; having the area 

and angles. Ill 

Right-angled. Properties of... 111,112 

Angles and sides of. To find. 112 

Trigonometrical Problems. 112 

Parallelograms. 

Properties of; To find area of, &c... 119 
To draw a square on a given line.. 119 

Trapezoids; Trapeziums. ... 120 

Hexagons; Octagons, &c. To draw.. 121 
Polygons. 

Regular. To draw. 121 

To reduce to a triangle of equal 

area. 121 

To reduce a large figure to a smaller 

one. 122 

To reduce the scale of a map. v . 122 

Irregular figures. To measure. 122 


xxiii 
























































XXIV 


CONTENTS. 


PA6E 


Circles. 

Properties of.... 123 

Radius, Diameter, Circumf, Area, 

&c. To find. 123 

Center of a given circle. To find.. 123 
To draw, through three points, &e. 123 

Tangents to. To draw. 124 

Tables of. 

Diameters in units, eighths, six¬ 
teenths, &e. 125 

Diameters in units and tenths.... 128 
Diameters in units and twelfths; 

as in feet and inches. 134 

Arcs. Circular. 

Radius, Rise, Chord, Length, and 

Ordinates of. To find. 141 

of large radius. To draw.1415 

in frequent use. Tables of.. 142 

Tables of lengths of, &c . 143 

Circular Sectors, Rings, Zones, and 

Lunes. 146 

Circular Segments. 

Area of. To find. 146 

Area of. Table of.. 147 

Ellipse. 

Properties of; Ordinates and Cir¬ 
cumference of. To find. 149 

Elliptic Arcs. Table of lengths 


Area of. To find. 150 

To draw; To draw tangents to, &c. 150 

Oval or false ellipse. To draw. 151 

Cjnia Recta, Cynia Reversa, Ogee. 

To draw. 151 

Parabola. Common or conic. 

Properties of. 152 

Parabolic curve. Length of. 

To find. 152 

Table of. 153 

Area of. To find. 152 

Parabolic zone or frustum. To find 

area of.. 152 

To draw. To draw a tangent to... 153 

Cycloid.... 154 


Solids. 

Regular bodies. Tetraedron, Ilexae- 


dron, &c . 154 

Parallelopipeds. 

Properties of; Solidities of. To 

. 155 

Prisms. 

Solidity of. To find. 155 

Frustum. Solidity of. To find. 155 

Surface of. To find. 156 

Cylinders. 

Solidity of: Surface of. To find... 156 
Contents of, in cubic feet and in 

U S gallons. 157 

Wells. Masonry in walls of. 158 

Cylindric Ungnlas. 

Solidity of; Surface of. To find... 159 
Circular Rings. 

Solidity of; Surface of. To find... 159 


„ PAG 8 

Pyramids and Cones. 

Properties of; Solidities of: Sur- f 

faces of. 160 

Frustums of. Solidities of; Sur- 

faces of.160a 

Prismords. 

Solidity of.160i 


Wedges. 


! 


Defined; Solidity of. 161 

Spheres. 

Properties of. 162 

Solidity of; Surface of. To find... 162 , 

Solidity of; Surface of. Table of.. 163 

Segments and Zones of. 166 

Spherical Shell.. 166 

Spheroid or Ellipsoid. 166 

Paraboloid or Parabolic Conoid. 

Defined ; Solidity of. 167 

Frustum of. Solidity and surface 

„ of . 167 

Circular Spindle. 

Defined; Solidity of; Surface of.... 167 

Middle zone of. Solidity of.. 167 


SURVEYING, LEVELING, 
Ac. 


Surveying. 


Land Surveying. 

Method of Procedure; Tests of 
accuracy, Distribution of error, 

&c . 168 - 

Sloping ground. Table of allow¬ 
ances for. 

Chains and pins.’ 

Meridian. To find by means of 

North Star. 

Sines of polar distances of North 

Star. 

Secants of North Latitudes.. 

Elongations of North Star. 

Times of. 

Traverse Table. 

Surveying Instruments. 

Transit. 

Description. . 

Adjustments.. 

Variation Vernier. 

Cross hairs. To replace. 

Bubble-glass. To replace. 

Theodolite. Adjustments of. 

Pocket Sextant. Description and 

Adjustments. 

Compass. 

Adjustments. 

Variation.. 


176 

176 


176 

177 
177 . 


( 

177 


) 

17? f 


180 


188 

191 

K3 

193 

193 

193 


194 


195 

196 


Leveling. 


Contour Lines. 197 

Leveling Instruments. 

Y Level. 

Description. 201. 

Adjustments. 202 



































































CONTENTS. 


XXV 


PAGE 


Forms for Level Note-books. 204 

Hand Level. Description; Adjust¬ 
ment.••... 205 

Builder’s Plumb Level. Adjust¬ 
ment. 206 

Clinometer or Slope Instrument. 

Adjustment. 206 


Leveling by the barometer; by the 
boiling point. 


NATURAL PHENOMENA, 
FORCES AND SUBSTANCES. 

Sound. 

Sound. Velocity of; Distance trav¬ 
eled by. 211 


Heat. 

| Heat. Expansion of Solids by ; Es¬ 
timated heat of fires in common 


Thermometers. 

Conversion of readings by the three 
principal scales. Buies and tables 
for. 


Air and Water. 

Air. 

Extent; Composition; Weight; 

Quantity breathed. 215 

Temperature; Conduction of heat; 

Expansion by heat. 215 

Pressure in diving bells &e; Dew¬ 
point. 215 

Greatest recorded heat and cold.... 215 

Wind. Velocity and force of.. 216 

Water 

Composition; Weight at different 

temperatures. 217 

Weight of Sea Water. 217 

Ice... 217 

Compressibility of Water. 217 

Effects of fresh and salt water on 

metals, .. 217 

Tides. 219 

Rain. , _ 

Annual fall; Greatest recorded 
falls; Falls in different climates. 220 
One inch of depth. Equivalents 

of.... 221 

Snow. Weight of, &c. 221 

Evaporation, Filtration, Leakage. 22- 


II yd r os tat I cs. 

Hydrostatics defined. 

Pressure of water. 

In general.. 

Against surfaces under varying 
conditions.'•... 


222 

222 

223 


PAG* 


To divide surfaces into portions 

sustaining equal pressures. 227 

Transmission of. 227 

Center of. 227 

To find, under various conditions. 228 

Walls for resisting. 229 

Distribution of pressure in plane 

surfaces. 231a 

Cylinders for resisting, 
lteuleaux’s formula for thick¬ 
ness. 232 

Practical considerations. 233 

Thickness for riveted iron cylin¬ 
ders. 233 

Thickness lor cast-iron pipes. 233 

Thickness for lead pipes. 234 

Valves must lie closed slowly. 234 

Buoyancy, Flotation, Metacenter. 

General laws. 234 

Equilibrium of floating bodies. 235 

Stability of structures diminished 

by upward pressure of water. 236 

Draught of vessels. 236 

Compressibility of liquids. 236 


Hydraulics. 


Hydraulics defined. 236 

Flow in pipes. 

Practical limitations to accuracy... 236 

Head defined. 237 

Total head. Divisions of. 237 

Pressures in pipes; Piezometer; 

Hydraulic grade line. 239 

Siphon. 241 

Approximate formulae for velocity 
in, and discharge through, 

pipes. 24 o 

Rutter’s formula. 244 

To find the diameter and slope 
required for a given velocity... 245 
Table of weight of water in pipes 

1 ft long. 246 

Table of areas of cross-section, con¬ 
tents of pipes, and square roots 

of diameters. 247 

To find heads, diameters, &c, for 
given velocities and dis¬ 
charges. 248 

Table of velocities, discharges, and 

heads in cast-iron pipes. 249 

Table of 5th roots and 5th powers.. 251 
Table of square roots of 5th powers. 253 
Table of square roots of 5th powers 

in ft. 2o3 

Discharge through a compound 

pipe of different diameters. 254 

Resistance of curved bends and of 

knees.... 

Friction in pumping mams. ‘ 

Flow through openingsand adjutages 257 
Theoretical and actual velocities; 

vena contracta. *5° 

Table of, and formula for, theoret- 

ical velocities. 2o ® 

































































XXVI 


CONTENTS. 


263 


263 


PAGE 

Rules for flow from adjutages. 269 

Rules for flow from openings in thin 

partition. 260 

into air, the surface level remain¬ 
ing constant. 260 

into air; the surface level fall¬ 
ing. 262 

under water; the upper level 

remaining constant. 262 

under water; the upper level 

falling. 262 

when the contraction is incom¬ 
plete. 

when the opening is provided 
with a hor or iuclined trough, 

Ac. 

To find the time required to empty 

a pond, Ac. 264 

Discharge over Weirs. 

Francis’ formulas; Ilart and Hunk- 

ing’s modification. 265 

Tables of discharges and coeffi¬ 
cients. 266 

Author’s empirical rule, and table 

of coefficients. 266 

Bazin’s coefficients.. 267 a 

Effect of shape of weir. 2676 

Triangular notches. 2676 

Flow in open channels and streams.. 268 

Mean velocity. 268 

Velocities in different parts of the ~ 

cross section. 268 

Bottom velocity. 268 

To measure the velocity.„. 268 

To gauge a stream by means of its 

velocity. 269 

Loss by leakage. 269 

Instruments for gauging. 269 

Wheel meters.270 

Kutter’s formula for velocity. 271 

Theory of flow...." 271 

Table of coefficient n of rough- 

ness. 273 

Tables of coefficient c . 273-278 

To construct a diagram for Kut¬ 
ter’s formula. 278 

Flow in Sewers. 

Table of velocities, by Kutter’s for¬ 


mula. 


Rate at which rain-water reaches 
a sewer. Burkli-Ziegler for 


279c 


mula. 


279c 


Table of least advisable grades and 

velocities. 279d 

Weights and prices of terra-cotta 

drain-pipes. 279d 

Effect of reduction of area 4 cross- 

section of channel. 279d 

Scour.” 279 f 

Tables of heads and velocities pro¬ 
duced by obstructions. 279f 

Resistance to flow. 2 &a 

Horse-power of Water. 

Falling. Formula for. 280 

Hydraulic ram. 280 

Running. Formula for!... 281 

Dams. . . 


Primary requisites. 282 


PAGS 


Various methods of building. 282 

Abutments; Sluices ; Ground plan; 

Trembling; Cost. 285 

Height of water above crest.. 286 

'i bickuess of timber required at 
different depths. 286 


Water Supply. 




Quantity required in Cities. 287 

Reservoirs. 

Dimensions, materials, Ac. 287 

Leakage through ; Mud in. 288 

Storage reservoirs. 288 

Arrangement of reservoir ends of 

pipes, Ac; Valve towers. 289 

Compensation. 289 

Distributing reservoirs. 289 

Water-pipes. 

Diameters required. 290 

Concretions in. prevention of.. 291 

General remarks; Galvanic action, 

_ -. 293 

Standard weights of cast-iron 

pipes; Prices. 293 

Wrought-iron, gutta-percha, wood¬ 
en, and other pipes. 293, 294 

Joints. 

Converse. 293 

Standard form of. 295 

Methods of making. 295 

Flexible. 296 

Branches, sleeves, Ac. 296 

Cracks in. 

Prevention and stoppage. 296 

4o attach a pipe to one already in 

nse. 297 

Cost of pipe and laying. 297 

Air-valves. " 097 

Air-vessels. ogg 

Stand-pipes.!..."!.." 298 

Service-pipes. Methods of attach! 

... 294,299 

Stop valves. Common and fonr- 

wf- . 301 

Fire hydrants. 394 . 


Mechanics. Force in ltigid< 
Bodies. 


J 


Definitions and subdivisions. 306 

Matter; body." 399 

Motion; velocity. 397 

Unit velocity. 3 ,7 

£ orce ..!“!”!!!!"!!!! oos 

Force applied by contact. 308 

Equality of action and reaction. 309 

Acceleration. 319 

Rate of acceleration. 319 

Laws of acceleration. 319 

Mass. " 322 

Unit of mass.!. 322 

impulse. 313 

Density; inertia. 314 




































































































CONTENTS, 


XXVII 


PAGE 


Forces in opposite directions. 315 

Work. 316 

Units of work. . 316 

Power. 318 

Kinetic energy; vis viva.318a 

, Potential energy.318d 

Impact.318e 

Applied and imparted forces.318e 

Stress or strain.318A 

Composition and resolution-of forces. 319 

Forces in one plane. 319 

) Parallelogram of forces. 320 

Polygon of forces. 329 

To ascertain resultants of forces 

by means of co-ordinates. 331 

Forces in different planes, but 

tending to one point. 332 

Parallelepiped of forces. 333 

Forces in different planes, and 

tending to different points. 334 

To ascertain resultants and com¬ 
ponents of forces by means of 

tbe angles between them. 334 

Moments; Leverage. 335 

Equilibrium of Moments. 338 

Virtual Velocities. 339 

Beam. Principle of the lever ap¬ 
plied to. 339 

Arch. Principle of the lever ap¬ 
plied to. 342 

Compound levers; Gearing. 342 

Pulleys. 342 

l’iie Cord or Funicular Machine.... 344 

Parallel forces. 347 

Resultant of.. 347a 

Couples.. . 347ri 

Graphic methods .. 347c/ 

Center of pressure. 347/ 

Center of Gravity 

General rules. 348 

Special rules.351a 

Inclined Plane. 352 

Resolution ol'force upon. 353 

Stability on. 355, 356 

Moment of Stability. 357 

Stability in arches, &c. 359 

Gravity. Falling bodies. 362 

Descent on inclined planes. 363 

Pendulums. 364 

Center of Oscillation. 365 

Center of Percussion. 365 

Argular Velocity. 365 

Moment of Inertia. 365 

Table of Radii of Gyration. 366 

Centrifugal Force. 368 

Friction. 370 

Nature of. 370 

Static and kinetic. 370 

How estimated. 370 

Coefficient of.. 371 

Morin’s laws. 372 

Table of.. 373 

More recent experiments. 374 

“Adhesion ” of locomotives. 3745 

Friction under great pressures. 3745 

Rolling friction.'3745 

Friction of liquids. 374c 


PAGE 

Of lubricated surfaces. 374c 

Launching friction. 374d 

Journal friction. . 374 d 

Friction rollers. 374e 

Friction of rolling stock. 374c 

Work of overcoming friction. 374/ 

Traction on roads, canals, and rail¬ 
roads... 375 

Animal Power. 377 


MATERIALS AND THEIR 
PROPERTIES. 

Specific Gravity, Weight, Ac. 

Specific Gravity. 

Defined ; Standards of. 380 

To find specific gravities of solids 

and liquids. 380,381 

Table of specific gravities and 
weights of various substances... 381 


Weights and Measures. 


American and British Standards. 385 

Troy Weight. 386 

Foreign Coins. Table of Values of, 

in U S Currency. 386 

Gold and Silver Coins, &c. Weights, 

values, &c, of. 387 

Apothecaries’ Weight. 387 

Avoirdupois Weight. 387 

Long Measure. 387 

Lengths of various conventional 

measures. 387 

Latitude and Longitude. Lengths 

of degrees of. 387, 388 

Inches reduced to Decimals of a 

foot. Table of. 388 

Square or Land Measure. 389 

Areas of various conventional 

measures. 389 

Cubic or Solid Measure. 3b9 

Contents of various conventional 

measures. 389 

Cubic foot, inch, and yard. Equiv¬ 
alents of. 389 

Spheres, 1 foot and 1 inch diam. 

Contents of. 389 

Cylinders, 1 ft high, 1 ft and 1 in 

diam. Contents of. 390 

Liquid Measure. 390 

Basis of. The gallon and its equiv¬ 
alents. 390 

Contents of various conventional 

measures. 390 

Contents of cylinders in gallons, 

&c... 390 

British measures. To reduce U S 

measures to, and vice versa. 390 

Dry Measure. 390 

Basis of. The bushel and its equiv¬ 
alents. 390 
































































































XXV111 


CONTEXTS 


PAGE 


Contents of various conventional 

measures. 390 

British measures. To reduce U S 

measures to, and vice versa. 390 

British Imperial Measure. Liquid 

and Dry. 391 

To obtain the size of Commercial 
Measures by ineaus of the weight 

of water. 391 

Metric System. 

The metre. U S Standard. 391 

Metric Units of Length, U S and 
British Standards of Area and 

Solidity. 392 

Metric Weights. Avoirdupois 

equivalents. 393 

Old French Measures and Weights. 
Systeme Usuel. U S Equivalents.. 393 
Systeme Ancien. U S Equivalents. 393 

Russian Measures and Weights. 394 

Spanish Measures and Weights. 394 

Time. 

Civil or Clock. 395 

To regulate a watch by a star. 395 

Standard Railway Time. 396 

Dialling. 397 


Weights, Dimensions, Prop¬ 
erties, Prices, «fce, of various 
Substances ami Articles. 


Iron. 

Cast. Table, &c, of weight of. 

Weight of patterns. 398 

Cast-iron pipes. Table of weight of 399 
Wrought-iron and steel. Tables 

of weight of bars, &c. 400 

Wrought-iron aud steel. Prices 

of. 402 

Rolled Star iron. Standard sizes 

of. 402 

Sheet iron; black, galvanized, and 
corrugated, Dimensions, weights, 
prices, methods of use, painting, 

strength. 403 

Wrought-iron pipes. 

Dimensions, weights, and prices. 405 

Fittings for. 405 

Wrought-iron boiler tubes. 

Dimensions, weights, and prices. 405 
Bolts, nuts, and washers. 

Standard sizes, weights, and 

Prices. 406 

Lock-nut washers. 408 

Table of weights and strengths 


Buckled plates for bridge floors. 409 

Wire Gauges. 

Table of Birmingham, British 

(new), and American gauges. 410 

Birmingham gauge for brass, sil¬ 
ver, &c. 4 U 

Rolling mill gauge for sheet 

iron. 411 

Wire. 412 


Wire Rope. Iron and Steel. 

PAGE 
. 413 

Rope. 


Chains. 

. -*14 

. 415 

Lead, Copper, and Brass. 

Roof Copper. 

. 416 

Sheet Lead. 

. 416 

Balls of Lead, &c .. 

. 416 

Lead Pipe. 



Table of dimensions and weights. 416 

Remarks on. 416 

Tubes. Brass and Copper. 417 

Tin aud Zinc. 

Method of using tinned and terne 
plates; Kinds in use ; Properties. 418 

Prices . 418 

Tinned and Terne Plates. Table of. 419 
Zinc sheets; Zinc vessels for water. 419 
Timber. 

Table of board measure. 420 

Durability and preservation of. 425 

Prices of lumber. 4256 

Nails; Holding power of; Prices 
and sizes. 4256 


Plastering. 


Materials and Methods. 


A day's work at.. 

. 426 

Cost of.. 

. 427 

Laths. 


Slating. 


Materials and Methods. 

. 427 

Weights of Slate roofs. 

. 428 

Shingles.. 

. 429 

Painting.. 

. 429 

Glass and Glazing. 

. 431 

Draughting Materials. 


Paper, colors, lead pencils. 

. 433 


Strength of Materials. 

Strength of Materials in general. 434 

Modulus of Elasticity defined. 4346 

Table of Moduli of Elasticity, Com¬ 
pression or Stretch under various 
loads, and Elastic Limits, of va¬ 
rious substances.434e 

Fatigue of Materials. 435 


Compressive Strength. 

Compressive Strength 

of timber. 436 

of stones, bricks, masonry, con¬ 
crete, <ic. 437 

of metals. 438 

Strength of Pillars. 

Law of. Gordon’s formula. 439 

Table of least radii of gyration. 440 

Tables of breaking loads. 442-456 

for wrought iron pillars of vari¬ 
ous shapes. 442, 443 

for iron pillars, per square inch 

of cross section. . 444 

for hollow cylindrical cast-iron 

pillars. 445 

for hollow cylindrical wrought- 
iron pillars. 447 





































































CONTENTS 


XXIX 


Ad 

i 

4 ; 

4 : 

41 

4i 

41 

41 : 

41 

41 

41 


11 

H 

H 

11 

a 


a 


PAGE 


for solid cylindrical cast-iron 

pillars. 450 

for solid cylindrical wrought- 

iron pillars. 451 

for solid square cast-iron 

pillars. 452 

for solid square wrouglit-iron 

pillars.... 453 

for Rolled I beam pillars. 454 

for channel-iron pillars. 456 

Usual practice ; Cautious; Shapes 

of Capitals. 457 

Shapes of pillars; Pillars placed 
obliquely to the line of 

pressure. 457 

Steel pillars. 458 

Wooden pillars. 

Strength affected by seasoning; 

Proper factor of safety. 458 

Formula for strength ; Pine and 

Cast-Iron compared. 458 

Tables of breaking loads. 459 

Remarks. 460-462 


Tensile Strength. 


- Tensile Strength 

4 of Timber. Table of. 463 

•» of Metals. Table of. 464 

of Various Materials. Table of..... 466 
4 Diam of a round rod to bear a given 

pull. To find. 466 

4 Effect of cold on iron. 466 

■\ Riveted joints. 468 


PAGS 


Moment of Inertia. To find. 486 

Neutral axis. To find. 457 

Breaking Load l>y the above 

theory. 488 

Caution. Comparison between 

models and actual structures... 490 
Practical methods for finding 

strengths of beams. 491 

Constants for center breaking 

loads. To find. 491 

Constants for center breaking 

loads. Table of. 493 

Factors for different arrange¬ 
ments of beam and load. 494 

Breaking loads of beams of va¬ 


rious forms of cross-section. 494,495 
Allowable modifications in the 
longitudinal form of beams. 495 
Loads applied elsewhere than at 

the center of the span. 496 

Inclined beams. 496 

Triangular beams. 496 

Dimensions required in beams. Tofind.497 
Tables of safe loads and deflections 

of wooden beams. 499 

Coefficients for iron beams. 500 

Cast-Iron Beams. Rectangular and 
Cylindrical. Tables of break¬ 
ing loads...5'J2. 503 

Stone Beams. Table • f safe loads.... 504 
Elastic Limit in beams. 

Constants for. To find. 505 

Deflections of beams. 

General laws.505 b 


Shearing' Strength. 

It} 

Shearing Strength. 476 

Torsional Strength. 

Torsional Strength. 476 

it General Formulae; Constants for 

Metals and Woods. 477 

Angle of Torsion. 477 

To find Torsional Strength, having 

Shearing Strength. 477 

> Torsional Strength in shafts of 

different shapes. 477 

Shafting, Wrouglit-iron. Strength 
of. 477 

Transverse Strength. 

Theory of. 478 

Moment of Rupture. 478 

of concentrated loads. 478 

of distributed loads. 480 

General rules for... 481 

Rules for special cases ; Diagrams. 482 
Theory of Resistance in Closed 

Beams. 485 

Moment of Resistance defined... 485 

Neutral axis. 485 

Coefficient of Resistance. 485 

Moment of Resistance. Formulae 

for. 486, 488 

Moment of Inertia defined... 486, 487 


Constant for. To find. 506 

Constants for timber and metals... 507 

Amount of deflection. To find. 507 

Load to produce a given deflection. 

To find. 508 

Dimensions for a given deflection. 

To find. 508,509 

Deflection not to exceed a given frac¬ 
tion of the span . 510 

Center load. To find. 510 

Dimensions. To find. 510 

Table of loads for pine beams. 512 

Table of loads for cast-iron beams. 513 
Wooden beams for short railroad 

bridges. 514 

Continuous Beams. 515 

Hollow Beams. Results of experi¬ 
ments. 516 

Ilodgkiuson Beams. 518 

Other shapes of Cast-iron Beams. 519 

Rolled I and Channel Beams. 521 

Formulas and Tables. 521 

Rolled I Beams for short railroad 

bridges. 524 

Angle- and T-iron. 

Formulas and Tables. 525 

Beams with thin webs. 

General principles. 528 

Flange strains. Formula for. 529 

Breaking loads. Formula for. 529 

Web members. Office of. 529 

Oblique and curved flanges. 530 

















































































XXX 


CONTENTS. 


PAGE 


Vertical or ‘‘Shearing” Strains in 

Beams. 532 

Riveted Gilders. 

Flanges. Strains in. 537 

Areas of; Effective area. 537 

Widths of. 538 

Thicknesses of. 538 

Rivets. Formulae for straius and 

numbers of.. 539 

Web. 

Its office. 539 

Stiffeners. Distance apart of; 

Areas of. 539 

Shearing and buckling strains... 540 
Single and double webs com¬ 
pared. 540 

Strength. 

Formulae for ultimate loads. 540 

Factors of safety. 540 

Deflection. Formulae for. 541 

Results of experiment with a 

box girder. 541 

Weight. To find. 541 

Usual dimensions of parts. 541 

Methods of attaching stiffeners. 542 

Plate Girders for railroad bridges. 

Distauce between girders. 542 

Lateral bracing. 542 

Dimensions, &c, of girders in ac¬ 
tual use. 543 

Diagram of moving loads. 546 

Trusses. 

Introduction; Definitions. 547 

Strains in Trusses. Graphical 
methods of finding: 

in King and Queen trusses. 551 

in Warren and other trusses. 557 

Counterbracing. 564 

in roof-trusses, &c . 570 

in roof-trusses of the Fink sys¬ 
tem . 573 

Comparison between King, 
Queen, and Fink roof-trusses. 578 

Details of iron roof-trusses. 582 

in Fink bridge-trusses. 584 

in Bollman bridge-trusses. 586 

in Bowstring and Crescent 

trusses. 588 

in the Braced Arch. 592 

in Cantilevers and Swing- 

bridges. 593 

General arrangement of Sundry 
forms of truss. 

Howe bridge-truss. 594 

Pratt bridge-truss. 595 

Lattice bridge-truss. 596 

Bowstring bridge- and roof- 

trusses. 597 

Examples. 599 

Moseley bridge-truss. 600 

Burr bridge-truss. 600 

Bollman bridge-truss. 603 

Fink bridge-truss . 603 

Weights of truss bridges. 605 

Loads on truss bridges. 606 

Factors of safety. 607 

Camber. 607 




PAGl 

Falseworks. 601 [ 

Braces against overturning. 605 ; 


Distance apart of trusses. 605 [ 

Headway.. 605 |. 




Details. 

Floor girders. 615 p 

Transverse horizontal bracing... 61C 

Chord splices. 61C 

Eye Bars and Pins. 615 

Joints. 612 

Expansion rollers, &c . 614 

Suspension Bridges 
Table of data for calculation of..... 615 

General remarks. 615 

Formulae for dimensions, &c, of, 
and strains in chains and pil¬ 
lars. 616-615 

Piers and Anchorages. 62C 

Descriptions of actual bridges.. 622, <fcc 


CONSTRUCTION. 


Well Boring 1 . 


Tools for boring common wells. 626 

Tools for boring Artesian wells. 627 


Breclging. 


Cost of dredging. 631 

Dredging by bag-scoop. 632 

Weight of dredged material. 632 


Foundations. 




General remarks. 

Pierre perdue, or Rip-rap.. 

with piles. 

Cribs. 

Caissons. 

Coffer-dams. 

Wooden Piles. 

Sheet piles. 

Bearing piles. 

Grillage. 

Pile-Driving Machines. 

Ordinary steam-drum machines. 

Cost of. 

Shaw’s gunpowder pile driver.. . 
Nasmyth steam-hammer pile- 

driver. 

Bearing power of piles. 

Rules for; Examples of. 

Factors of safety. 

Splicing of piles. 

Blunt-ended piles. 

Friction of piles, and of cast-iron 

cylinders. 

Penetrability and elastic reaction 

of soils. 

Protection of feet and heads of 

piles. 

Driving below water; Follower. 

Withdrawing piles. 

Adherence of ice to piles. 


633 

634 

635 

635 

636 

637 


641 

641 

641 


641 

64]' 


642 


643 
641 

644 
044 


644 


614 


644 

645 
645 
645 





























































































CONTENTS. 


XXXI 


PAGE 

iron Piles and Cylinders. 

Brunei’s process. 645 

Screw piles. 645 

Effect of salt water on iron. 645 

Jets. 646 

Iron piles driven by percussion. 647 

Potts’ vacuum process. 647 

Plenum process. 648 

Sinking Brickwork Cylindeis. 650 

Sand-pump. 650 

Fascines. 650 

Sand-piles. 650 

Sundry Methods of forming Founda¬ 
tions . 651 

Diving-dress. Cost, &c. 651 


Stonework. 

Drilling. 

By hand; jumper and churn drill.. 651 
By machinery. 

Diamond drills. Description, Ca¬ 


pacity, Cost, Ac. 652 

Percussion drills. Description, 

Capacity, Cost, Ac... 653 

Principal points of difference in 

the more prominent makes. 656 

Hand-drilling machines. 658 

Channeling. 658 

Air Compressors and Receivers. 
Dimensions, Capacities, Costs, 

Ac. 658 


Blasting. 

By powder. 

Explosive force, weight, cost, Ac. 660 
By the modern high explosives. 
Composition,properties, methods 


of use, cost, Ac. 

of Nitro-GIycerine. 661 

of Dynamite. 662 

of foreign explosives . 664 

Methods of firing the charge. 

By common fuse. 665 

By electricity. 665 

Simultaneous fil ing. 665 

Cost of quarrying stone. 667 

Cost of dressing stone. 667 

Cost of buildings per cubic foot. 668 


Mortar, Bricks, Cements, 
Concrete, «fcc. 

Mortar. 

Proportions; Quantity required,Ac. 669 


Dime. 669 

Grout.... 670 

Strength. 670 

Effect upon wood. 670 

Adhesion to bricks. 670 

Bricks. 

Size, weight, absorption. 671 

Bricklaying and paving. 671 

Crushing strength. 671 

Enameled bricks. 671 

Tensile strength. 672 

Frozen mortar... 672 

To render brickwork impervious 

to water.'.. 672 

White efflorescence on walls. 673 


PAGB 

Hydraulic Cements. 673 

Properties. Effect of moisture on. 673 

Restoration by re-burning. 674 

Rough-casting. 674 

Pointing mortar. 674 

Tests. Rapidity of setting Ac. 674, 678 

Effect of cold. 675 

Strength of cements. 675, 678 

Cement Mortar. 

Effect of sand in. 676 

Strengths of.. 676-678 

Adhesion to bricks and stone. 677 

Voids in sand and broken stone. 677,678 

White efflorescence on walls. 678 

Cement Concrete or Beton. 

Composition, Strength, Ac. 678, 679 

Admixture of lime with'. Effect of. 680 

Ramming of Concrete. 680 

Stone Crushers... 680 

Methods of using. 680 

Mixing of; Cost of.. 681 

Coignet’s Beton..>. 681 

Concrete beams. Transverse 
strength of. 682 

Retaining' Walls. 

Practical Rules for proportioning, Ac. 
When the earth is level with the 

top of the wall. 683 

When the wall is surcharged. 685 

Theory of Retaining walls.... 686 

Thickness of walls with battered 

faces. 690 

Wharf walls. 691 

Transformation of profile. 691 

Buttresses. 692 

Protection against sliding. 692 

Counterforts; Land-ties; Curved pro¬ 
files... 692 

Revetment, counterscarp, and talus. 

Defined. 692 

Pressure as affected by width of 

backing. 692 

Table of contents per foot run of 
walls. 692 

Slone Bridges. 

Definitions. 693 

Keystone. To find depth of. 693 

Pressure sustained by arch-stones.... 694 
Table of. dimensions of existing 

arches. 695 

Use of cement in arches. 696 

Keystones of elliptic arches. 696 

Table of depths of Keystones. 697 

Abutments. To proportion. 697 

Abutment-Piers. 699 

Inclination of Courses below the 

springs. 700 

Line of Resistance, Ac. 700 

To find the length of a culvert. 702 

Tables of quantities of masonry in 

arches. 703 

Tables of quantities of masonry in 

wing-walls. 704 

Tables of quantities of masonry in 
complete bridges... 706 



























































































CONTENTS, 


xxxii 


PAGE 


Foundations. 707 

Drains. 7°7 

Drainage of roadway.... 708 

Table of contents of piers. 708 

Brick Arches. 709 

Centers for Arches. 

General principles. 711 

Arrangements for striking. 711 

Settlement; Proper time for 

striking. 713 

Pressure of Arch-stones against 

centers. 713 

Designs for Centers. 714 


RMLROADS. 

Railroad Construct ion. 

Table of acres required per mile, Ac, 


for different widths. 722 

Tables of grades.. 723-725 

Curves. 

Table of Radii, Ac, of Curves, 

in feet... 726 

in metres. 728 

Table of long chords.. 729 

Table of ordinates 5 ft apart. 730 

Earthwork. 

To prepare a table of level-cuttings. 732 

Tables of level-cuttings, Ac. 733 

Shrinkage of embankment. 741 

Cost of earthwork. 742 

Tunnels. 754 

Trestles. Wooden and iron. 755 

Roadway. 

Ballast; Ties. 759 

Rails. 7GO 

Table of middle ordinates for 

bending. 761 

Spikes. 762 

Rail-joints. 763 

Creeping of rails; Even and 

broken joints. 763 

Beveled joints,. 763 

Fish an<l angle plates. 764 

Fisher joint. 766 

Gibbon joint. 767 

Old forms of joints. 767 

Turnouts. 

Switches. 770 

Stub 8 witch . 771 

Switch levers and stands. 772 

Point switches. 774 

Stands for. 775 

Lorenz; and De Vout’s stand 

for. 775,776 

Lengths of switch rails. 776 

Three-throw point switch. 777 

Wharton switch. 778 

Frogs. 

Cast-iron. 780 

Guide-rails, &c. 781 

Rail frogs. 782 

Weir’s troir. 783 

Spring rail frogs. 784 

Laying-out of Turnouts. 785 


Railroad Equipment. 

PAGB 


Turntables. 790 

General features; Minimum length. 791 

Sellers’ cast-iron turntable. 792 

Edge Moor wrought-iron turn¬ 
table. 793 

Fritzsclie’s wrought-iron turn¬ 
table . 794 

Greenleaf’s and other turn¬ 
tables. 795,796 

Wooden turntable. 797 

Stops for turntables. 799 

Turntables with pivot at one end.. 799 

Engine-houses. Cost of. 799 

Shops. Cost of. 799 

Water Stations. 

Dimensions, capacities, &c. 800 

Burnham’s frost-proof tanks. 801 

Water supply; Capacities, &c, of 

pumps. 801 

Reservoirs. 801 

Track tanks. 802 

Evaporation by locomotives. 803 

Thicknesses of tanks. 803 

Track-scales. 803 t 

Fences; Barbed-wire fences. 803 

Stations. Cost of. 803 

Approximate estimate of cost of con¬ 
struction and equipment. 804 


Railrond Rolling-Stock and 
Operation. 

Locomotives. 

Dimensions, Weights, Ac. 805 

Performance. 

Loads hauled; Fuel consump¬ 
tion; Expense of running. 808 

Cars. 

Dimensions, Capacities, Weights, 

Cost, Ac. 811 

Resistance to motion. 812 

Wheels. Dimensions, weights, Ac- 

Cast-iron . 812 

Paper. 812 

Axles. Standard dimensions, 

weight, prices... 813 

A 

Railroad Statistics. 


For the United States. 

Plant and Operation. 814 

Items of annual expenses. 815 

Earnings and expenses of several 

lines. 816 

Statistics of several narrow-gauge 

roads. 818 

Miles of railroad in the world. 818 


tilossary of Technical 
Terms. $11 
























































































THE 


CIVIL ENGINEER’S POCKET-BOOK. 


ARITHMETIC. 


On this subject we shall merely give a few examples for refreshing the memory of those who for 
wautof constant practice cannot always recall the processes at the moment. 


Subtraction of Vulgar Fractions. 


1 1 _ n 
1 - 2 - 0 . 


3 1 _ 2 _ 1 

¥“¥“¥“ 2 - 

o4 _ i 2 — 2 5 

s 7 1 3 ~ 7 


6 2 _ 4 _ i 

8 — 8 — 8 — 2 * 


3 

¥' 


5 _ 2 7 
9 3 6 


2 0 _ 7 

3 O' — 3 A* 


5.— 75 _ 35 — 40 — 
3 “ 2 T 7T “ 2 T “ 


ill 

21 * 


Addition of Vulgar Fractions. 




1 - 2 _ 


= 1 . 


3,1—4—! 
¥ ' ¥ 3 1 

3^ 


¥ ~ *' 8 _r 8 

2 — 2-5 I 5— 75 

i a 


, 2—8 — ] 


3 i 
¥ + 


5 — 2 7 . 2 0 — ±7 — 5 

S S' * 3ft 3ft 1 


7 "p 1 3 “ 7 ~ r 3 71 


.35—110 

+ 2T “ 7T 


7 “ 3 6 

5 5 
5 2T- 


11 


3 6 


2,3.8 — 40,36.120 — 196 — ol6 - o 4 
3 + J + + 6A + 6 0“ AIT “ 

Multiplication of Vulgar Fractions. 


* w { 




1 - i 

2 “ ¥* 


1 — 3_ 


X ¥ = 


¥“T6* ¥ 


V 3 — i 

O Q 


— 12 

¥¥ 


3 

T 6 * 


5 — 15 


X o — i 

.r — - 5 - 


¥ 6 " 


5 

Ti* 


1.2 - 
3 


25 


x 4 = 


125 
2T • 


Y 8 - 48- 
A ¥“ 60 “ 


4 

W 


C * , 

1 i. 1 — 2 — , 

2 • 2 — § — 


3 „r 1 „f 5 7 — 3 v 1 V s V ? - 10 s - 21- 2 

J ™! 0 '? 01 T 2 0 “ 2 ¥ “ 8 " 

l>ivision of Vulgar Fractions. 


3il — 12 — 3 — 3 6^ 2 — 48 — 24 — 12 — 3 —3 3i5-27-, 7 

¥ • ¥“ ¥ “ T“ J ‘ 8 • A ~T6 " ¥ “ 4 “T ‘ ¥ * 1 


20 


7¥ 


34 i 1 2 — 25 i 5 — 7 5 — 1 5 — ..1 
' > n • 1 3' — 7 • -Q- — -5"K- — w — *w» 


5 if- 54 l - 40 —50 

5 • ¥ “ T * 8 “ 7 ~ °7* 


7 “ *3 ~ y * 3 “ ¥3 “ 7 “ *7* 

To find the greatest common divisor of a Vulgar Fraction. 


Ex. 1. or 


70 


701175(2 

140 


Ex. 2. Of 


84 

20 * 


20)84(4 

80 


35)70(2 

70 Ans 35. 


4)20(5 

20 Ans 4. 


To reduce a Vulgar Fraction to its lowest terms. 

First find the greatest common divisor; then divide both the numerator and denominator by It. 
Thus, in the preceding example yyjy = -5 Ans. And Jj-j = Ans. 

3 33 


33 










34 


arithmetic 


To reduce a Vulgar Fraction to a decimal form. 


Divide the numerator by the denominator. Thus, 

2 — 2 )j.0(0.5 Ans. y 3 = 4)13(3.25 Ans. |4 = 40)32.0(0.8 Ans. 

1 0 12 40 32 o 

10 

8 

~20 

20 

Reduce Synches to the decimal of a foot. There are 12 ins iu a foot; therefore, the questioi 
to reduce y% to a decimal. Therefore, 12)3.0(0.25 of a foot. Ans. 

2 4 

~60 

60 

Reduce 2 ft 3 ins to the decimal of a yard. There are 36 ins in a yard; and 27 ins in 2 ft 3 ii 
therefore, 3 y of a yard = 36)27.0(0.75 of a yd. Ans. 

25 2 


180 

180 


How many feet and ins are there in .75 of a yard ? Here 

.75 

3 ft in a yd. 


Ft 2).25 

12 ins in a ft. 


Ins 3.00 Ans 2 ft 3 ins. 

How many feet and ins are there in .0625 of a yard 7 

.0625 

3 feet in a yd. 


No feet, .1875 

12 ins in a ft. 


Ins 2.2500 


ft. ins. 
Ans. 0 2.25. 


How many cubic feet are there in .314 of a cub yard? And cub ins in .46 of a cut 
.314 4fi 

27 cub ft in a yd. 1728 cub ins a cub ft. 


2198 

628 

8.478 cub ft. Ans. 


368 

92 

322 

46 


794.88 cub ins. Ans. 


Fractions red need to exact decimals. 


1 

ST 

1 

5? 

T(T 

.015625 

.03125 

.046875 

.0625 

17 

Of 

9 

1 9 
Of 

T^T 

.265625 

.28125 

.296875 

.3125 

33 

64 

17 

5^ 

35 

64 

T7T 

.515625 

.53125 

.546875 

.5625 

49 

6¥ 

2 5 
32 

5 1 
Of 

1 3 

nr 

TTT 

Of 

i 

.078125 

.09375 

.109375 

.125 

H 

11 

3 2 

2 3 

Of 

3 

8 

.328125 

.34375 

.359375 

.375 

II 

19 

52 

3 9 

6 4' 

f 

.578125 

.59375 

.609375 

.625 

53 

64 

27 

52 

of 

1 

9 

¥¥ 

OT 

3 

TO 

.140625 

.15625 

.171875 

.1875 

25 

0 4 

1 3 
32 
27 

6 4 

7 

To 

.390625 

.40625 

.421875 

.4375 

41 

64 

21 

32 

43 

6 4 

TO' 

.640625 

.65625 

.671875 

.6875 

u 

29 

52 

5 9 
Of 
15 
TO 

13 

04 

3 ? 2 

u 

f 

.203125 

.21875 

.234375 

.25 

29 

OT 

15 

1T2 

31 

64 

1 

.453125 

.46875 

.484375 

5 

H 

2 3 
32 
42. 
64 

3 

¥ 

.703125 

.71875 

.734375 

.75 

u 

52 

Of 

0 4 

1 



ft? 


.765626 
.78125 
.796875 
.8125 i 


.828125 
.84375 
.859375 
.875 


j 


.890625 
.90625 
.921875 
.9375 « 


.953121 
.96875 
• 98437c 


1. 



























































ARITHMETIC 


35 


Decimals. 

Addition. Add together .25 and .75; also .006, 1.3172, and 43. 
.25 
.75 

1.00 Ans. 


.006 

1.3472 

43. 


44.3532 An8. 

Subtraction. Subtract .25 from .75; also .0001 from 1; also 6.30 from 9.01. 

•75 1. 9.01 

•'A> .0001 6.30 


•50 Ans. .9999 Ans. 2.71 Ans. 

Multiplication. Mult 3 X .3; also .3 X -3 ; also .3 X 03; also 4.326 X .003. 

.3 4.326 

.03 .003 


.3 

.3 


•9 Ans. .09 Ans. .009 Ans. .012978 Ans. 

Division. Divide 3. by .3 ; also .3 by .3; also .3 by .03; also 4.326 by .0003. 

.3)3.0(10. Ans. .3).3(1. Ans. .03).30(10. Ans. .0003)4.3260(14420. Ans. 


3 


3 


3 


0 

Divide 62 by 87.042. 


Divide .006 by 20. 


0 


87.042)62.0000(0.712, &c. Ans. 
60.9294 

107060~ 

87042 


13 

12 

12 

12 


200180 

20.000).0060000(0.0003 Ans. 
60000 


6 

6 


Duodecimals. 

Duodecimals refer to square feet of 144 sq ins; to twelfths of a square or duodecimal foot; each 
such twelfth being called an inch; and being equal to 12 square inches; and to twelfths, each equal 
to the 12th of a duodecimal inch, or to one square inch. The dimensions of the thing to be measd 
ire supposed to be taken in common feet, ins, and 12ths of an inch; but as ordinary measuring 
•ules are divided into 8ths of an inch, it is usually guess-work to some extent. Duodecimals are 
rery properly going out of use, in favor of decimals; we shall therefore give no rule for them. By 
Deans of our table of “ Inches reduced to Decimals of a Foot,” p. 388, all dimensions in feet, ins, 
md 8ths, &c, can be at once taken out in ft and decimals of a foot. 


Single Rule of Three; or, Simple Proportion. 

If 3 meu lay 10000 bricks in a certain time, how many could 6 men lay in the same time? They 
vill evidently lay more; therefore, the second term of the proportion must be greater than the first. 

3 : 6 :: 10000 : 20000 Ans. 

6 

3)60000 


20000 Ans. 


- -If 3 men require 10 hours to lay a certain number of bricks, how many hours would 6 men 
. equire? They will evidently require less time; therefore, the second term of the proportion must 
) 'e less than the first. 


6 : 3 :: 10 : 5 Ans. 
3 


6)30 


5 Ans 


Double Rule of Three; or. Compound Proportion. 

If three men can lay 4000 bricks in 2 days, how many men can lay 12000 in 3 days? Here we see 
j lat 4000 bricks require 3 X 2—6 days' work; therefore 12000 will require, 

4000 : 12000 :: 6 : 18 days' work. 

But there are only 3 days to do the 18 days work in ; therefore the number of men must be ^ 6 

' en. Ans. 

1 A moment’s reflection will suffice to reduce any case of double rule of three to this simple form. 


Ratio. 

1 Ratio. Simple ratio is a number denoting how often one quantity is contained in another. Thus, 

le ratio of 5 to 10 is 5 or b 5 and the rfttio of 10 to 5 is V’ or 2 ‘ WheD * of four numb « r8 . tw « 
five to each other thesame ratio that the other two have, the numbers are said to be in proportion 
) i each other. Thus, 6 has the same ratio (2) to 3, as 100 has to 50; therefore. 6. 3, 100, and 50, are 
lid to he in proportion ; or, as 6 : 3 :: 100 : 50. In other words, an equality of ratios is called pro- 
irtion. Ratio and proportion are often confounded with one another; but the error is one of no 
: upurtauce. Duplicate ratio is that of the squares of numbers. 











36 


ARITHMETIC, 


Arithmetical Progression, 

In a series of numbers, is a progressive increase or decrease in each successive number, by the ndr 
tion or subtraction of the same amount at each step; as in 1, 2. 3. 4, 5, Ac., in which 1 is added 
each step ; or 10, 8, fi. 4, Ac., in which 2 is subtracted at each step; or )4- 9$. 1. 114 Ac. aI 

such series the numbers are called its terms; and the equal increase or decrease at each step its coi 
mon difference.. 

To find the, com diff, knowing the first and last terms ; and the number of terms. Find the d 
between the first and last terms. From the number of terms subtract 1. Div tbe diff just found, 
the rent. 

To find the last term, knowing the first term ; the com diff; and the number of terms. From t 
number of terms take 1. Mult the rem by the com diff. To the prod add the first term. 

To find the 7iumi>er of terms , having the first and last ones ; and the coni diff. Take the d 
between the first and last terms. Div this diff by the com diff. To the quot add 1. 

To find the. sum of all the terms, having the first and last ones; and the number of terms. A 
together the first and last terms. Div their sum by 2. Mult the quot by the number of terms. 

Geometrical Progression, 

In a series of numbers, is a progressive increase or decrease in each successive number, by the sai 
multiplier or divisor at each step ; as 3, 9, 27, 81, Ac, where each succeeding term is increased by m 
the preceding one by 3. Or 48, 24, 12, 6, Ac, or 27. 13^, 69i, 3^, Ac, where each succeeding term 
found by dividiug the preceding one by 2. The multiplier or divisor is called the common ratio of t 
series, or progression. 

To find the last term, knowing the first one; the ratio; and the number of terms. Raise the ra 
to a power 1 less than the number of terms. Mult this power by the first term. 

Ex. First term 10; ratio 3; number of terms 8; what is the last term 7 Here the number of ter ' 
being 8, the ratio 3 must be raised to the 7th power; thus: 

3X3X3X3X3X3X3 = 2187, = 7th power. And 2187 X 10 = 21870 last term. Ans. 

A man agreed to buy 8 fine horses ; paying $10 for the first; $30 for the second; $90 for the thi 1 
Ac; how much will the last one cost him? Aus, $21870, as before. 

To find the sum of all the terms, knowing the first one; the ratio; and the number of terms. Ra 
the ratio to a power equal to the whole number of terms. From this power subtract 1. Div the r 
by 1 less than the ratio. Mult the quot by the first term. 

Ex. As before. What is the sum of all the terms? Here the ratio must be raised to the I 

power; thus, 3 X 3 X 3 X 3 X 3 X 3 X 3 X 3 = 6561 = 8th pow. And 6560 div by 1 less than the ra 

„ 6560 

3, — —j— — 3280. And 3280 X 10 (or number of terms) = 32800 = sum. Ans. 

In the foregoing case, the 8 horses would cost $32800. 

Permutation 

Shows in how many positions any number of things can be arranged in a row. To do this, nt 
together all the numbers used in counting the things. Thus, in how many positions in a row ca i 
things be placed? Here, 

1X2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 = 362880 positions. Ans. 


Combination 


Shows how many combinations of a few things can be made out of a greater number of things, 
do this, first set down that number which indicates the greater numl>er of things ; and after it, a sei i 
of numbers, diminishing by 1, until there are in all as many as the number of the few things t s 
are to form each combination. Then beginning under the last one, set down said number of 
things ; and going backward, set down another series, also diminishing by 1. until arriving under 
first of the upper numbers. Mult together all the upper numbers to form’one prod ; and all the loi 
ones to form another. Div the upper prod by the lower one. 

Ex. How many combinations of 4 figures each, can be made from the 9 figs 1, 2, 3, 4 5 6 7 8 
or from 9 any things? ‘ ' 

9 X 8 X 7 X 6 3024 , 

= 126 combinations. Ans. 


1X 2X3X4 


24 


Alligation 


Shows the value of a mixture of different ingredients, when the quantity and value of each of th 
last is known. 


Ex. What is the value of a pound of a mixture of 20 lbs of sugar worth 15 cts per lb : with 30 
worth 25 cts per lb ? 


i per 


lbs. cts. cts. 
20 X 15 = 300 
30 X 25 = 750 


1050 

Therefore, —— = 21 cts. Ans. 
o0 


50 lbs. 1050 cts. 







ARITHMETIC, 


37 


Equation of Payments. 

A owes B $1200: of which $400 are to be paid in 3 months; $500 in 4 months; and $300 in (t 
months; all bearing interest until paid ; but it has been agreed to par all at once. Now. at what tim« 
must this payment be made so that neither party shall lose any interest? 

$ months. 

400 X 3 = 1200 5000 

500 X 4 = 2000 Therefore, = 4.16, &c, months. Ans. 

300 X 6 = 1800 1200 

1200 5000 

A owes B $1000 to he paid in 12 days ; and $500 to be paid in 3 months. What would be the time 
or paying all at once ? 

$ davs. 

1000 X 12 = 12000 57000 

500 X 80 - 45000 Therefore, —— = 38 days. Ans. 


1500 57000 


Simple Interest. 

i What is the simple interest on $865.32 for one year, at 6 per ct per annum T 

Principal. Interest. Principal. Interest. 

$100 : $6 :: $865.32 : $51.9192 

6 

- $ Cts. 

100)5191.92(51.9192 Ans. =51.91^2 

What is the interest on $865.32 for 1 year, 3 months, and 10 days, at 7 per cent per annum? 

i First calculate the interest for 1 year only ; thus: 

Prin. Int. Prin. Int. 

$100 : $7 :: $865.32 : $60.5724 

i 7 

j 100)6057.24(60.5724 

Then say, If I year or 365 days give $60.5724 int, what will 465 days give? or 

Days. Int. Days. Int. 

365 : $60.5724 : : 465 : $77.16, &c. Ans. 

At 5 per ct simple interest, money doubles itself in 20 years; at 6 per ct, in 16% years; and at 7 
er ct, in 14^- years. Simple Interest is Single Rule of Three. 

Compound Interest. 

' When money is borrowed for more than a year at compound interest, find the simple interest at the 
nd of the first year, and add it to the principal, for a second principal. Fiud the simple interest on 
tiis second enlarged principal for the next year, and add it to the enlarged principal for a third prin- 
ipal; and so on for each successive year. . 

At 5 per ct compound iuterest, money doubles itself in about 14 j years ; at 6 per ct, in about 11.9 
ears; and at 7 per ct, in about 10% years. 

Discount 


, i a deduction of a part of the interest, when money at interest is paid before it is due. Or it is a 
, eduction of the whole of the interest in advance, at the time the money is lent. In the first case, if 
, borrow $100 for I year at 8 per ct. I must at the end of the year pay back $108; but if I pay at the 
id of 3 months, I must add only $2, or the interest for those 3 months, paying back $102; and the 
, iff of $6 is the discount. Therefore, to find the discount in such cases, first find the interest for 
' le full time ; then that for the short time; and take the diff. 

, In the second case, if I borrow $100 from a bank for one year, at 6 per ct. I receive but 100— 6= $94; 

ut at the end of the year I must pay back $100. By discounting in this manner, the bank actually 
| iins more than 6 per ct; for it gains $6 for the use of $94 for 1 year. In the United States, the banks 
;duct discount for 3 days more than the time stipulated in the note; these are called‘‘days of 
j ace." The borrower is not obliged to pay before the last of these 3 days. 


Commission, or Brokerage, 

a percentage (or so much per each $100) paid to commission merchants for selling our goods; or 
i brokers, or other kinds of agents, for transacting business for us. It is Single Rule of Three. 

Kx. If a broker makes purchases for me to the amount of $9362, at 2 per ct, what is his brokerage ? 

i ty, as 

Purchase. Brokerage. Purchase. Brokerage. 

$100 : $2 : : $9362 ; $187.24 


Insurance 


a percentage (called a premium) paid to a company for insuring our property against fire, Ac. 
ie company, or insurers, (called also underwriters,) deliver to the person insured, a paper bearing 
eir seal, &c, and called the Policy of Insurance ; which contains the conditions of the transaction, 
surance is calculated like Commissions, &c.; being merely Single Rule of Three. 


Fellowship. 


A puts $6000 into a business in partnership with B, who puts in $9000. At the end of a year they 
ive made $2400; how much is each one's share? Here, $6000 -)- $9000=$15000joint capital; then say, 
Joint cap. Total gain. A'scap. A’s share. 

$15000 ; $2400 : : $6000 : $960 


td 


B’s cap. B's share. 
$9000 : $1440 


$15000 


$2400 








TABLE OF LOGARITHMS. 


Lograrithnitt of Numbers, from 0 to 1000.* 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Pro] 

0 

0 

00000 

30103 

47712)60206 

69897 

77815 

84510 

90309 

95424 


10 

00000 

00432 

00860 

01283 01703 

02118 

02530 

02938 

03342 

03742 

41.' 

11 

04139 

04532 

04921 

053U7 

105690 

06069 

06445 

06818 

i 07188 

07554 

37! 

12 

07918 

08278 

08636 

08990 09342 

09691 

10037 

10380 

10721 

11059 

34! 

13 

11394 

11727 

12057 

12385 12710 

13033 

13353 

13672 

13987 

14301 

3£ 

14 

14613 

14921 

15228 

15533:15836 

16136 

16435 

16731 

17026 

17318 

30( 

15 

17609 

17897 

18184 

18469 

18752 

19033 

19312 

19590 

19865 

20139 

281 

16 

20412 

20682 

20951 

21218 

21484 

21748 

22010 

22271 

22530 

22788 

26 

17 

23045 

23299 

23552 

23804 

24054 

24303 

24551 

24797 

25042 

25285 

24! 

18 

25527 

23767 

26007 

26215 

26481 

26717 

26951 

27184 

27415 

27646 

231 

19 

27875 

2S103 

28330 

28555 

28780 

29003 

29225 

29446 

29666 

29885 

221 

20 

30103 

30319 

30535 

30749 

30963 

31175 

31386 

31597 

31806 

32014 

21 ; 

21 

32222 

32428 

32633 

32838 

33041 

33243 

3:1445 

33646 

33845 

34044 

201 

22 

34242 

3 4439 

346:35 

34S30 

35024 

35218 

35410 

35602 

35793 

35983 

19, 

23 

36173 

36361 

36548 

36735 

36921 

37106 

37291 

37474 

•3 1 65 / 

37839 

18; 

24 

38021 

38201 

33381 

38560 

38739 

38916 

39093 

39269 

39445 

39619 

177 

25 

39794 

39367 

40140 

40312 

40483 

40654 

40S24 

40993 

41162 

41330 

171 

26 

41497 

41664 

41830 

41995 

42160 

42324 

4248s 

42651 

42813 

42975 

16) 

27 

43136 

43296 

43156 

43616 

43775 

43933 

44090 

44248 

44404 

44560 

15* 

28 

44716 

44870 

45024 

45178 

45331 

45484 

45636 

45788 

45939 

46089 

151 

29 

46240 

463S9 

46538 

46686 

46834 

46982 

47129 

47275 

47421 

47567 

14* 

30 

47712 

47856 

48000 

48144 

4S287 

48430 

48572 

48713 

4S855 

48995 

141 

31 

49136 

49276 

49415 

49554 

49693 

49831 

49968 

50105 

50242 

50379 

13* 

32 

50515 

50650 

50785 

50920 

51054 

51188 

51321 

51454 

51587 

51719 

134 

33 

51351 

51982 

52113 

52244 

52374 

52504 

52633 

52763 

52891 

53020 

13( 

34 

53148 

53275 

53402 

53529 

53655 

53781 

53907 

54033 

54157 

54282 

12 ( 

35 

54407 

54530 

54654 

54777 

54900 

55022 

55145 

55266 

55388 

55509 

12 ; 

36 

55630 

55750 

55870 

55990 

56110 

56229 

56348 

56466 

56584 

56702 

11 < 

37 

56320 

56937 

57054 

57170 

57287 

57403 

57518 

57634 

57749 

57863 

lit 

38 

57978 

58092 

58206 

58319 

58433 

58546 

58658 

58771 

58883 

58995 

111 

39 

59106 

59217 

59328 

59439 

59549 

59659 

59769 

59879 

59988 

60097 

11 ( 

40 

60206 

60314 

60422 

60530 

60638 

60745 

60852 

60959 

61066 

61172 

107 

41 

61278 

61384 

61489 

61595 

61700 

61S04 

61909 

62013 

62117 

62221 

10 )’ 

42 

62325 

62428 

62531 

62634 62736 

62838 

62941 

63042 

63144 

63245 

101 

43 

63347 

63447 

63548 

63648 63749 

63848 

63948 

64048 

64147 

64246 

9! 

41 

64345 

64443 

64542 

64640 64738 

64836 

64933 

65030 

65127 

65224 

9* 

45 

65321 

65 417 

65513 

65609 65705 

65801 

65896 

65991 

660*6 

66181 

91 

46 

66276 

66370 

66164 

66558 66651 

66745 

66838 

66931 

67024 

67117 

9- 

47 

67210 

67302 

67394 

67486167577 

67669 

67760 

67851 

67942 

68033 

9: 

48 

68124 

68214 

68304 

68394 68484 

68574 

68663 

68752 

68842 

68930 

9( 

49 

69020 

69108 

69196 

69284 

69372 

69460 

69548 

69635 

69722 

69810 

8 * 

50 

69897 

69983 

70070 

70156 

70243 

70329 

70415 

70500 

70586 

70671 

8 ( 

51 

70757 

70842 

70927 

71011 

71096 

71180 

71265 

71349 

71433 

71516 

84 

52 

71600 

71683 

71767 

71850 

71933 

72015 

72098 

72181 

72263 

72345 

81 

53 

72428 

72509 

72591 

72672 

72754 

72835 

72916 

72997 

73078 

73158 

81 

54 

73239 

73319 

73399 

73480 

73559 

73639 

73719 

73798 

73878 

73957 

8 ( 

55 

74036 

74115 

74193 

74272 

74351 

74429 

74507 

74585 

74663 

74741 

7* 

56 

74813 

74896 

74973 

75050 

75127 

75204 

75281 

75358 

75434 

75511 

77 

57 

75587 

75663 

75739 

75815 

75891 

75966 

76042 

76117 

76192 

76267 

7' 

58 

76342 

76417 

76492 

76566 

76641 

76715 

76789 

76863 

76937 

77011 

74 

59 

77085 

77158 

77232 

77305 

77378 

77451 

77524 

77597 

77670 

77742 

7* 

60 

77815 

77887 

77959 

78031 

78103 

78175 

7824? 

78318 

78390 

78461 

71, 

61 

78533 

78604 

78675 

78746 

78816 

78887 

78958 

79028 

79098 

79169 

71 

62 

79239 

79309 

79379 

79448 

79518 

79588 

79657 

79726 

79796 

79865 

7( 

03 

79934 

80002 

80071 

80140 

80208 

80277 

80345 

80413 

80482 

80550 

6 ! 

64 

80618 

80685 

80753 

80821 

808S8 

80956 

81023 

81090 

81157 

81224 

6 * 

65 

81291 

81358 

81424 

81491 

81557 

81624 

81690 

81756 

81822 

81888 

67 


*Each log is supposed to have the decimal sign . before it 


























































TABLE OF LOGARITHMS, 


39 


Logarithms of Numbers, from 0 to 1000*— (Continued.) 


No. 

1 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. 

66 

81954 

82020 

82085 

82151 

82216 

82282 

82347 

8241'2 

82477 

82542 

66 

67 

82607 

82672 

82736 

82801 

82866 

82930 

82994 

83058 

83123 

83187 

65 

68 

83250 

83314 

83378 

83442 

83505 

83569 

83632 

83695 

83758 

83821 

64 

69 

88884 

83947 

84010 

84073 

84136 

84198 

84260 

84323 

84385 

84447 

63 

70 

84509 

84571 

84633 

84695 

84757 

84818 

84880 

84941 

85003 

85064 

62 

71 

85125 

85187 

8524S 

85309 

85369 

85430 

85491 

85551 

85612 

85672 

61 

72 

857 33 

85793 

85853 

85913 

85973 

86033 

86093 

86153 

86213 

86272 

60 

73 

86332 

86391 

86451 

86510 

86569 

86628 

86687 

86746 

86805 

86864 

59 

74 

86923 

86981 

87040 

87098 

87157 

87215 

87273 

87332 

87390 

87448 

58 

75 

87506 

87564 

87621 

87679 

87737 

87794 

87852 

87909 

87966 

88024 

57 

76 

88081 

88138 

88195 

88252 

88309 

8S366 

88422 

88479 

88536 

88592 

56 

77 

88649 

88705 

88761 

88818 

88874 

88930 

88986 

89042 

89098 

89153 

56 

78 

89209 

89265 

89320 

89376 

89431 

89487 

89542 

89597 

89652 

89707 

55 

79 

89762 

89S17 

89872 

89927 

89982 

90036 

90091 

90145 

90200 

90254 

54 

80 

90309 

90363 

90417 

90471 

90525 

90579 

90633 

90687 

90741 

90794 

54 

81 

90848 

90902 

90955 

91009 

91062 

91115 

91169 

91222 

91275 

91328 

53 

82 

91381 

91434 

91487 

91540 

91592 

91645 

91698 

91750 

91S03 

91855 

53 

S3 

91907 

91960 

92012 

92064 

92116 

92168 

92220 

92272 

92324 

92376 

52 

84 

92427 

92479 

92531 

92582 

92634 

92685 

92737 

92788 

92839 

92890 

51 

85 

92941 

92993 

93044 

93095 

93146 

93196 

93247 

93298 

93348 

93399 

61 

86 

93449 

93500 

93550 

93601 

93651 

93701 

93751 

93802 

93852 

93902 

50 

S7 

93951 

94001 

94051 

94101 

94151 

94200 

94250 

94300 

94349 

94398 

49 

88 

94448 

94497 

94546 

94596 

94645 

94094 

94743 

94792 

94841 

94890 

49 

89 

94939 

94987 

95036 

95085 

95133 

95182 

95230 

95279 

95327 

95376 

48 

90 

95424 

95472 

95520 

95568 

95616 

95664 

95712 

95760 

95808 

95856 

48 

91 

95904 

95951 

95999 

96047 

96094 

9C142 

96189 

96236 

96284 

96331 

48 

92 

96.378 

98426 

96473 

96520 

96567 

96614 

96661 

96708 

96754 

96801 

47 

93 

£16848 

96895 

96941 

96988 

97034 

97081 

97127 

97174 

97220 

97266 

47 

94 

97312 

97359 

97405 

97451 

97497 

97543 

97589 

97635 

97680 

97726 

46 

95 

97772 

97818 

97863 

97909 

97954 

98000 

98045 

98091 

98136 

98181 

46 

96 

98227 

98272 

98317 

98362 

98407 

98452 

98497 

98542 

98587 

98632 

45 

97 

98677 

98721 

98766 

98811 

98855 

98900 

98945 

989S9 

99033 

99078 

45 

98 

99122 

99166 

99211 

99255 

99299 

99343 

99387 

99431 

99475 

99519 

44 

99 

99563 

99607 

99651 

99694 

99738 

99782 

99825 

99869 

99913 

99956 

. 

44 


♦ Each log is supposed to have the decimal sign . before it. 


The log of 2870 is 3.45788 
“ “ “ 287 is 2.45788 

“ “ “ 28.7 is 1.45788 

“ “ « 2.87 is 0.45788 


The log of .287 is — 1.45788 

“ “ “ .028 is — 2.44716 

“ “ “ .002 is — 3.30103 

“ “ “ .0002 is — 4.30103 


What is the log of 2873 ? 

Here, log of 2870 = 3.45788 
And prop 153 X 3 = 459 

3.458339 


To find roots divide the log (with its index) of the given number, by that 

umber which expresses the kind of root. The quotient will be the log of the required root. 

Example. What is the cube root of 2870? 

3.45788 

ere, the log of 2870, with its index, is 3.45788. And —-= 1.15263. Hence the cube root is 14.2. 

3 

The Hyperbolic, or Napierian logarithm is the common log of 
ie table multiplied by 2.3025851. 



















































40 


SQUARE AND CUBE ROOTS 


Square Roots and Cube Roots of Numbers from .1 to 2S. 

No errors 


No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

No. 

Sq. Rt. 

C. Rt. 

NO. 

Sq. Rt. 

C. R 

.1 

.01 

.001 

.316 

.464 

.7 

2.387 

1.786 

.4 

3.661 

2.37; 

.15 

.0225 

.0034 

.387 

.531 

.8 

2.408 

1.797 

.6 

3.688 

2.38 

.2 

.04 

.008 

.447 

.585 

.9 

2.429 

1.807 

.8 

3.715 

2.39! 

.‘25 

.0625 

.0156 

.500 

.630 

6. 

2.449 

1.817 

14. 

3.742 

2.411 

.3 

.09 

.027 

.548 

.669 

.1 

2.470 

1.827 

.2 

3.768 

2.42 , 

.36 

.1225 

.0429 

.592 

.705 

.2 

2.490 

1.837 

.4 

3.795 

2.43! 

.4 

.16 

.064 

.633 

.737 

.3 

2.510 

1.847 

.6 

3.821 

2.44 

.45 

.2025 

.0911 

.671 

.766 

.4 

2.530 

1.857 

.8 

3.847 

2.45! 

.5 

.25 

.125 

.707 

.794 

.5 

2.550 

1.866 

15. 

3.873 

2.461 

.55 

.3025 

.1664 

.742 

.819 

.6 

2.569 

1.876 

.2 

3.899 

2.47‘ 

.6 

.36 

.216 

.775 

.843 

.7 

2.588 

1.885 

.4 

3.924 

2.48) 

.65 

.4225 

.2746 

.806 

.866 

.8 

2.608 

1.895 

.6 

3.950 

2.49! 

.7 

.49 

.343 

837 

.888 

.9 

2.627 

1.904 

.8 

3.975 

2.501 

.75 

.5625 

.4219 

.866 

.909 

7. 

2.646 

1.913 

16. 

4. 

2.521 

.8 

.64 

.512 

.894 

.928 

.1 

2.665 

1.922 

.2 

4.025 

2.531 

.85 

.7225 

.6141 

.922 

.947 

.2 

2.683 

1.931 

.4 

4.050 

2.54 

.9 

.81 

.729 

.949 

.965 

.3 

2.702 

1.940 

.6 

4.074 

2.551 

.95 

.9025 

.8574 

.975 

,9a3 

.4 

2.720 

1.949 

.8 

4.099 

2.561 

1. 

1.000 

1.000 

1.000 

1.000 

.5 

2.739 

1.957 

17. 

4.123 

2.5711 

.05 

1.103 

1.158 

1.025 

1.016 

.6 

2.757 

1.966 

.2 

4.147 

2.581 

1.1 

1.210 

1.331 

1.049 

1.032 

.7 

2.775 

1 975 

.4 

4.171 

2.59: 

.15 

1.323 

1.521 

1.072 

1.048 

.8 

2.793 

1.983 

.6 

4.195 

2.601 

1.2 

1.440 

1.728 

1.095 

1.063 

.9 

2.811 

1.992 

.8 

4.219 

2.61l| 

.25 

1.563 

1.963 

1.118 

1.077 

8. 

2.828 

2.000 

18. 

4.243 

2.621 k 

1.3 

1.690 

2.197 

1.140 

1.091 

.1 

2.846 

2.008 

.2 

4.266 

2.631 

.35 

1.823 

2.460 

1.162 

1.105 

.2 

2.864 

2.017 

.4 

4.290 

2 641 

1.4 

1.960 

2.744 

1.183 

1.119 

.3 

2.881 

2.025 

.6 

4.313 

2.651 

.45 

2.103 

3.049 

1.204 

1.132 

.4 

2.898 

2.033 

.8 

4.336 

2.65! 

1.5 

2.250 

3.375 

1.225 

1.145 

.5 

2.915 

2.041 

19. 

4.359 

2.66! 

.55 

2.403 

3.724 

1.245 

1.157 

.6 

2.933 

2.0(9 

.2 

4.382 

2.671 

1.6 

2.560 

4.096 

1.265 

1.170 

.7 

2.950 

2.057 

.4 

4.405 

2.681 

.65 

2.723 

4.492 

1.285 

1.182 

.8 

2.966 

2.065 

.6 

4.427 

2.691 

1.7 

2.890 

4.913 

1.304 

1.193 

.9 

2.983 

2.072 

.8 

4.450 

2.701: 

.75 

3.063 

5.359 

1.323 

1.205 

9. 

3. 

2.080 

20. 

4.472 

2.714 

1.8 

3.240 

5.832 

1.342 

1.216 

.1 

3.017 

2.0»8 

.2 

4.494 

2.72! 

.85 

3.423 

6.332 

1.360 

1.228 

.2 

3.033 

2.095 

.4 

4.517 

2.731 

1.9 

3.610 

6.859 

1.378 

1.239 

•3 

3.050 

2.103 

.6 

4.539 - 

2.741 

.95 

3.803 

7.415 

1.396 

1.249 

.4 

3.066 

2.110 

.8 

4.561 

2.754 

2. 

4.000 

8.000 

1.414 

1.260 

.5 

3.082 

2.118 

21. 

4.583 

2.75! 

.1 

4.410 

9.261 

1.449 

1.281 

.6 

3.098 

2.125 

.2 

4.604 

2.708 

.2 

4.840 

10.65 

1.483 

1.301 

.7 

3.114 

2 133 

.4 

4.626 

2.77f 

.3 

5.290 

12.17 

1.517 

1.320 

.8 

3.130 

2.140 

.6 

4.648 

2.785: 

.4 

5.760 

13.82 

1.549 

1.339 

.9 

3.146 

2.147 

.8 

4.669 

2.79! 

.5 

6.250 

15.63 

1.581 

1.357 

10. 

3.162 

2.154 

22. 

4.690 

2. SOI : 

.6 

6.760 

17.58 

1.612 

1.375 

.1 

3.178 

2.162 

.2 

4.712 

2.810 

.7 

7.290 

19.68 

1.643 

1.392 

.2 

3.194 

2.169 

.4 

4.733 

2.819 ; 

.8 

7.840 

21.95 

1.673 

1.409 

.3 

3.209 

2.176 

.6 

4.754 

2.827 ; 

.9 

8.410 

24.39 

1.703 

1.426 

.4 

3.225 

2.183 

.8 

4.775 

2.836 

S. 

9. 

27. 

1.732 

1.442 

.5 

3.240 

2.190 

23. 

4.796 

2.841 

.1 

9.61 

29.79 

1.761 

1.458 

.6 

3.256 

2.197 

.2 

4.817 

2.85! 

.2 

10.24 

32.77 

1.789 

1.474 

.7 

3.271 

2.204 

.4 

4.837 

2.86( 

.3 

10.89 

35.94 

1.817 

1.489 

.8 

3.286 

2.210 

.6 

4.&58 

2.868 

.4 

11.56 

39.30 

1.844 

1.504 

.9 

3.302 

2.217 

.8 

4.879 

2.876 

.5 

12.25 

42.88 

1.871 

1.518 

11. 

3.317 

2.224 

24. 

4.899 

2.884 

.6 

12.96 

46.66 

1.897 

1.533 

.1 

3.332 

2.231 

.2 

4.919 

2.892. 

.7 

13.69 

50.65 

1.924 

1.547 

.2 

3.347 

2.237 

.4 

4.940 

2.90C 

.8 

14.44 

54.87 

1.949 

1.560 

.3 

3.362 

2.244 

.6 

4.960 

2.9081 

.9 

15.21 

59.32 

1.975 

1.574 

.4 

3 376 

2.251 

.8 

4.980 

2.916 

4. 

16. 

64. 

2. 

1.587 

.5 

3.391 

2.257 

25. 

5. 

2.924 

.1 

16.81 

68.92 

2.025 

1.601 

.6 

3.406 

2.264 

.2 

5.020 

2.932 

.2 

17.64 

74.09 

2.049 

1 613 

.7 

3.421 

2.270 

.4 

5.040 

2.940 

.3 

18.49 

79.51 

2.074 

1.626 

.8 

3.435 

2.277 

.6 

5.060 

2.947 

.4 

19.36 

85.18 

2.098 

1.639 

.9 

3.450 

2.283 

.8 

5.079 

2.955 

.5 

20.25 

91.13 

2.121 

1.651 

12. 

3.464 

2 289 

26. 

5 099 

2.962 

.6 

21.16 

97.34 

2.145 

1.663 

.1 

3.479 

2.296 

.2 

5.119 

2.970 

.7 

22.09 

103.8 

2.168 

1.675 

.2 

3.493 

2.302 

.4 

5.138 

2.978 

.8 

23.04 

110.6 

2.191 

1.687 

.3 

3.507 

2.308 

.6 

5.158 

2.985 

.9 

24.01 

117.6 

2.214 

1.698 

.4 

3.521 

2.315 

.8 

5.177 

2.993 

B. 

25. 

125. 

2.236 

1.710 

.5 

3.536 

2.321 

27. 

5.196 

3.000 

.1 

26.01 

132.7 

2.258 

1.721 

.6 

3.550 

2.327 

.2 

5.215 

3.007 

.2 

27.04 

140.6 

2.280 

1.732 

.7 

3.564 

2.333 

.4 

5.235 

3.015 

.3 

28.09 

148.9 

2.302 

1.744 

.8 

3.578 

2.339 

.6 

5.254 

3.022 

.4 

29.16 

157.5 

2.324 

1.754 

.9 

3.592 

2.345 

.8 

5.273 

3.029 

.5 

30-25 

166.4 

2.345 

1.765 

13. 

3.606 

2.351 

28. 

5.292 

3.037 

.6 

31.36 

175.6 

2.366 

1.776 

.2 

3.633 

2.363 

.2 

5.310 

3.044 





























41 


SQUARES, CUBES, AND ROOTS, 


TABLE of Squares. Cubes, Square Roots, a ml Cube Roots, 
of Numbers from 1 to lOOO. 

Remark on the following Table. Wherever the effect of a fifth decimal in the roots would be to 
add 1 to the fourth aud final decimal iu the table, the addition has been made. No errors. 


No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

1 

1 

1 

1 .0000 

1.0000 

61 

3721 

226981 

7.8102 

3.9365 

2 

4 

8 

1.4142 

1.2599 

62 

3844 

238328 

7.8740 

3.9579 

3 

9 

27 

1.7321 

1.4422 

63 

3969 

250047 

7.9373 

3.9791 

4 

16 

64 

2.0000 

1.5874 

64 

4096 

262144 

8.0000 

4. 

5 

25 

125 

2.2361 

1.7100 

65 

4225 

274625 

8.0623 

4.0207 

6 

36 

216 

2.4495 

1.8171 

66 

4356 

287496 

8.1240 

4.0412 

7 

49 

343 

2.6458 

1.9129 

67 

4489 

300763 

8.1854 

4.0615 

8 

64 

512 

2.8284 

2.0000 

68 

4624 

314432 

8.2162 

4.0817 

y 

81 

729 

3.0000 

2.0801 

69 

4761 

328509 

8.3066 

4.1016 

10 

100 

1000 

3.1623 

2.1544 

70 

4900 

343000 

8.3666 

4.1213 

11 

121 

1331 

3.3166 

2.2240 

71 

5041 

357911 

8.4261 

4.1408 

12 

141 

1728 

3.4641 

2.2894 

72 

5184 

373248 

8.4853 

4.1602 

13 

169 

2197 

3.6056 

2.3513 

73 

5329 

389017 

8.5440 

4.1793 

14 

196 

2744 

3.7417 

2.4101 

74 

5476 

405224 

8.6023 

4.1983 

15 

225 

3375 

3.8730 

2.4662 

75 

5025 

421875 

8.6603, 

4.2172 

16 

256 

4096 

4.0000 

2.5198 

76 

5776 

438976 

8.7178 

4.2358 

17 

289 

4913 

4.1231 

2.5713 

77 

5929 

456533 

8.7750 

4.2543 

18 

324 

5832 

4.2426 

2.6207 

78 

6084 

474552 

8.8318 

4.2727 

19 

361 * 

6859 

4.3589 

2.6684 

79 

6241 

493039 

8.8882 

4.2908 

20 

400 

8000 

4.4721 

2.7144 

80 

6400 

512000 

8.9443 

4.3089 

21 

441 

9261 

4.5826 

2.7589 

81 

6561 

531441 

9. 

4.3267 

22 

484 

10648 

4.6904 

2.8020 

82 

6724 

551368 

9.0554 

4.3445 

23 

529 

12167 

4.7958 

2.8439 

83 

6889 

571787 

9.1104 

4.3621 

24 

576 

13824 

4.8990 

2.8845 

84 

7056 

592704 

9.1652 

4.3795 

25 

625 

15625 

5.0000 

2.9240 

85 

7225 

614125 

9.2195 

4.3968 

26 

676 

17576 

5.0990 

2.9625 

86 

7396 

636056 

9.2736 

4.4140 

27 

729 

19683 

5.1962 

3.0000 

87 

7569 

658503 

9.3274 

4.4310 

28 

784 

21952 

5.2915 

3.0306 

88 

7744 

681472 

9.3808 

4.4480 

29 

841 

21383 

5.3852 

3.0723 

89 

7921 

704969 

9.4340 

4.4647 

30 

900 

27000 

5.4772 

3.1072 

90 

8100 

729000 

9.4868 

4.4814 

31 

961 

29791 

5.5678 

3.1414 

91 

8281 

753571 

9.5394 

4.4979 

32 

1024 

32768 

5.6569 

3.1748 

92 

8464 

778688 

9.5917 

4.5144 

33 

1089 

35937 

5.7446 

3.2075 

93 

8649 

804357 

9.6437 

4.5307 

34 

1156 

39304 

5.8310 

3.2396 

94 

8836 

830584 

9.6954 

4.5468 

35 

1225 

42875 

5.9161 

3.2711 

95 

9025 

857375 

9.7468 

4.5629 

36 

1296 

46656 

6.0000 

3.3019 

96 

9216 

884736 

9.7980 

4.5789 

37 

1369 

50653 

6.0828 

3.3322 

97 

9409 

912673 

9.8489 

4.5947 

38 

1414 

54872 

6.1644 

3.3620 

98 

9604 

941192 

9.8995 

4.6104 

39 

1521 

59319 

6.2450 

3.3912 

99 

9801 

970299 

9.9499 

4.6261 

40 

1600 

64000 

6.3246 

3.4200 

100 

10000 

1000000 

10. 

4.6416 

41 

1681 

68921 

6.4031 

3.4482 

101 

10201 

1030301 

10.0499 

4.6570 

42 

1764 

74088 

6.4807 

3.4760 

102 

10404 

1061208 

10.0995 

4.6723 

*3 

1849 

79507 

6.5574 

3.5034 

103 

10609 

1092727 

10.1489 

4.6875 

44 

1936 

85184 

6.6332 

3.5303 

104 

10816 

1124864 

10.1980 

4.7027 

45 

2025 

91125 

6.7082 

3.5569 

105 

11025 

1157625 

10.2470 

4.7177 

46 

2116 

97336 

6.7823 

3.5830 

106 

11236 

1191016 

10.2956 

4.7326 

47 

2209 

103823 

6.8557 

3.6088 

107 

11449 

1225043 

10.3441 

4.7475 

48 

2304 

110592 

6.9282 

3.6342 

108 

11664 

1259712 

10.3923 

4.7622 

49 

2401 

117619 

7.0000 

3.6593 

109 

11881 

1295029 

10.4403 

4.7769 

50 

2500 

125000 

7.0711 

3.6840 

110 

12100 

1331000 

10.4881 

4.7914 

51 

2601 

132651 

7.1414 

3.7084 

111 

12321 

1367631 

10.5357 

4.8059 

52 

2704 

140608 

7.2111 

3.7325 

112 

12544 

1404928 

10.5830 

4.8203 

53 

2809 

148877 

7.2801 

3.7563 

113 

12769 

1442897 

10.6301 

4.8346 

51 

2916 

157464 

7.3485 

3.7798 

114 

12996 

1481544 

10.6771 

4.8488 

55 

3025 

166375 

7.4162 

3.8030 

115 

13225 

1520875 

10.7238 

4.8629 

56 

3136 

175616 

7.4833 

3.8259 

116 

13456 

1560896 

10.7703 

4.8770 

57 

3249 

185193 

7.5498 

3.8485 

117 

13689 

1601613 

10.8167 

4.8910 

58 

3364 

195112 

7.6158 

3.8709 

118 

13924 

16430.32 

10.8628 

4.9049 

59 

3481 

205379 

7.6811 

3.8930 

119 

14161 

1685159 

10.9087 

4.9187 

CO 

3600 

216000 

7.7460 

3.9149 

120 | 

14400 

1728000 

10.9545 

4.9324 





































SQUARES, CUBES, AND ROOTS, 


4 V 




TABLE of Squares. Cubes, Square Roots, and Cube Roots 
of Numbers front 1 to 1000— (Continued.) 


. 


No. 

Square. 

Cube. 

Sq. Itt. 

C. Rt. 

No. 

Square. 

Cube. 

Sq. Rt. 

' 

C. Rt 

121 

14641 

1771561 

..... 

11. 

4.9461 

186 

34596 

6434856 

13.6382 

5.708 

122 

14884 

1815848 

11.0454 

4.9597 

187 

34969 

6539203 

13.6748 

5.718 

123 

15129 

1860867 

11.0905 

4.9732 

188 

35344 

6644672 

13.7l13 

5.728 

124 

15376 

1906624 

11.1355 

4.9866 

189 

35721 

675126-9 

13.7477 

5.738 

125 

15625 

1953125 

11.1803 

5. 

190 

36100 

6859000 

13.7840 

5.748 

120 

15876 

2000376 

11.2250 

5.0133 

191 

36481 

6967871 

13.8203 

5.759 

127 

16129 

2048383 

11.2694 

5.0265 

192 

36864 

7077888 

13.8564 

.(». 

128 

16384 

2097152 

11.3137 

5.0397 

193 

37249 

7189057 

13.8924 

5. i <9 

129 

16641 

2146689 

11.3578 

5.0528 

194 

37636 

7301384 

13.9284 

5.789 

130 

16900 

2197000 

11.4018 

5.0658 

195 

38025 

7414875 

13.9642 

5.79t 

131 

17161 

2248091 

11.4455 

5.0788 

196 

38416 

7529536 

14. 

5.80? 

13*2 

17424 

2299968 

11.4891 

5.0916 

197 

38809 

7615373 

14.0357 

5.8b 

133 

17689 

2352637 

11.5326 

5.1045 

198 

39204 

7762392 

14.0712 

5.82,' 

134 

17956 

2406104 

11.5758 

5.1172 

199 

39601 

7880599 

14.1067 

5.83b 

135 

18225 

2460375 

11.6190 

5.1299 

200 

40000 

8000000 

14.1421 

5.84? 

136 

18496 

2515456 

11.6619 

5.1426 

201 

40401 

8120601 

14.1774 

5.85- 

137 

18769 

2571353 

11.7047 

5.1551 

202 

40804 

8242408 

14.2127 

5.86, 

138 

19044 

2628072 

11.7473 

5.1676 

203 

41209 

8:465427 

14.2478 

5.87, 

139 

19321 

2685819 

11.7898 

5.1801 

204 

41616 

8489664 

14.2829 

5.88t 

140 

19600 

2744000 

11.8322 

5.1925 

205 

42025 

8615125 

14.3178 

5.89< 

141 

19881 

2803221 

11.8743 

5.2048 

206 

42436 

8741816 

14.3527 

5.90. 

142 

20164 

2863288 

11.9164 

5.2171 

207 

42849 

8869743 

14.3875 

5.91; 

143 

20119 

2921207 

11.9583 

5.2293 

208 

43264 

8998912 

14.4222 

5.92; 

144 

20736 

2985984 

12. 

5.2415 

209 

43681 

9129329 

14.4568 

5.93* 

145 

21025 

3048625 

12.0416 

5.2536 

210 

44100 

9261000 

14.4914 

5.94, 

146 

21316 

3112136 

12.0830 

5.2656 

211 

44521 

9393931 

14.5258 

5.95: 

147 

21609 

3176523 

12.1244 

5.2776 

212 

44944 

9528128 

14.5602 

5.90: 

148 

21904 

3211792 

12.1655 

5.2896 

213 

45369 

9663597 

14.5945 

5.97' 

149 

22201 

3307949 

12.2036 

5.3015 

214 

45796 

9800344 

14.6287 

5.9K 

150 

22500 

3375000 

12.2474 

5.3133 

215 

46225 

9938375 

14.6629 

5.99; 

151* 

22801 

3442951 

12.2882 

5.3251 

216 

46656 

10077696 

14.6969 

6. 

152 

23101 

3511808 

12.3288 

5.3368 

217 

47089 

10218313 

14.7309 

6.00' 

153 

23409 

3581577 

12.3693 

5.3485 

218 

47524 

10360232 

14.7648 

6.01 

154 

23716 

3652264 

12.4097 

5.3601 

219 

47961 

10503459 

14.7986 

6.02 

155 

24025 

3723875 

12.4499 

5.3717 

220 

48400 

10648000 

14.8324 

6.03i 

156 

24336 

3796416 

12.4900 

5.3832 

221 

48841 

10793861 

14.8661 

6.04. 

157 

21619 

3869893 

12.5300 

5.3947 

222 

49284 

10941048 

14.8997 

6.05; 

158 

24964 

3944312 

12.5698 

5.4061 

223 

49729 

11089567 

14.9332 

6.06 

159 

25281 

4019879 

12.6095 

5.4175 

224 

50176 

11239424 

14.9666 

6.07 

160 

25600 

4096000 

12.6491 

5.4288 

225 

50625 

11390625 

15. 

6.08 

161 

25921 

4173281 

12.6886 

5.4401 

226 

51076 

11543176 

15.0333 

6.09 

162 

26214 

4251528 

12.7279 

5.4514 

227 

51529 

11697083 

15.0665 

6.10 

163 

26569 

4330747 

12.7671 

5.4626 

228 

51984 

11852352 

15.0997 

6.10 

164 

26896 

4110944 

12.8062 

5.4737 

229 

52441 

12008989 

15.1327 

6.11 

165 

27225 

4492125 

12.8452 

5.4848 

230 

52900 

12167000 

15.1658 

6.12' 

166 

27556 

4574296 

12.8841 

5.4959 

231 

53361 

12326391 

15.1987 

6.13 

167 

27889 

4657 463 

12.9228 

5.5069 

232 

53824 

12487168 

15.2315 

6.14 

168 

28224 

4741632 

12.9615 

5.5178 

233 

54289 

12649337 

15.2643 

6.15 

169 

28561 

4826809 

13. 

5.5288 

234 

54756 

12812904 

15.2971 

6.16 

170 

28900 

4913000 

13.0384 

5.5397 

235 

55225 

12977875 

15.3297 

6.17 

171 

29241 

5000211 

13.0767 

5.5505 

236 

55696 

13144256 

15.3623 

6.17 

172 

29584 

5088448 

13.1149 

5.5613 

237 

56169 

13312053 

15.3948 

6,18 

173 

29929 

5177717 

13.1529 

5.5721 

238 

56644 

13481272 

15.4272 

6.19 

174 

30276 

5268024 

13.1909 

5.5828 

239 

57121 

13651919 

15.4596 

6.20 

175 

30625 

5359375 

13.2288 

5.5934 

240 

57600 

13824000 

15.4919 

6.21 

176 

30976 

5451776 

13.2665 

5.6011 

241 

58081 

13997521 

15.5242 

6.22 

177 

31329 

5545233 

13.3041 

5.6147 

242 

58564 

14172488 

15.5563 

6.23 

178 

31684 

5639752 

13.3417 

5.6252 

243 

59049 

14318907 

15.5885 

6.24 

179 

32041 

5735339 

13.3791 

5.6357 

244 

59536 

14526784 

15.6205 

6.24 

180 

32400 

5832000 

13.4164 

5.6462 

245 

60025 

14706125 

15.6525 

6.25 

181 

32761 

5929741 

13.4536 

5.6567 

246 

60516 

14886936 

15.6844 

6.26 

182 

33124 

6028568 

13.4907 

5.6671 

247 

61009 

15069223 

15.7162 

6.27 

183 

33489 

6128487 

13.5277 

5.6774 

248 

61504 

15252992 

15.7480 

6.28 

184 

33856 

6229504 

13.5647 

5.6877 

249 

62001 

15438249 

75.7797 

3.29 

185 

34225 

6331625 

13.6015 

5.6980 

250 

62500 

15625000 

15.8114 

| 6.29 























































SQUARES, CUBES, AND ROOTS 


43 


TABLE of Squares, Cubes, Square Roots, and Cube Roots, 
ot Numbers from 1 to 1000 — (Continued.) 


No. 

Square. 

j 

Cube. 

Sq. Rt. 

C. Rt 

No. 

Square 

Cube. 

Sq. Rt. 

C. Rt. 

251 

63001 

15813251 

15.8430 

6.3030 

316 

99856 

31554496 

17.7764 

6.8113 

252 

63504 

16003008 

15.8745 

6.3164 

317 

100489 

31855013 

17.8045 

6.8185 

25! 

64009 

16194277 

15.9060 

6.3217 

318 

101124 

32157432 

17.8326 

6.8256 

‘25+ 

64516 

16387064 

15.9374 

6.3330 

319 

101761 

32461759 

17.8606 

6.8328 

255 

65025 

16581375 

15.9687 

6.3413 

320 

102400 

32768000 

17.8885 

6.8399 

256 

65536 

16777216 

16. 

6.3496 

321 

103041 

33076161 

17.9165 

6.8470 

257 

66049 

16474593 

16.0312 

6.3579 

322 

103684 

33386248 

17.9 444 

6.8541 

25* 

tk)o(>4 

17173512 

16.0624 

6.3601 

323 

104329 

33698267 

17.9722 

6.8612 

259 

67081 

17373979 

16.0935 

6.3743 

324 

104976 

34012224 

18. 

6.8683 

260 

67600 

17576000 

16.1245 

6.3825 

325 

105625 

34328125 

18.0278 

6.8753 

261 

68121 

17779581 

16.1555 

6.3907 

326 

106276 

34645976 

18.0555 

6.8824 

262 

68644 

17984728 

16.1864 

6.398* 

327 

106929 

34965783 

18.0831 

6.8894 

262 

69169 

18191447 

16.2173 

6.4070 

328 

107584 


18.1108 

6.8964 

261 

69696 

18399744 

16.2481 

6.4151 

329 

108241 

35611289 

18.1384 

6.9034 

265 

70225 

18609625 

16.2788 

6.4232 

330 

108900 

35937000 

18.1659 

6.9104 

266 

70756 

18821096 

16.3095 

6.4312 

331 

109561 

36264691 

18.1934 

6.9174 

267 

71289 

19034163 

16.3401 

6.4393 

332 

110224 

36594368 

18.2209 

6.9244 

268 

71824 

19248832 

16.3707 

6.4473 

333 

110889 

36926037 

18.2483 

6.9313 

269 

72361 

19465109 

16.4012 

6.4553 

334 

111556 

37259704 

18.2757 

6.9382 

270 

72900 

19683000 

16.4317 

6.4633 

335 

112225 

37595375 

18.3030 

6.9451 

271 

73441 

19902511 

16.4621 

6.4713 

336 

112896 

37933056 

18.3303 

6.9521 

272 

73984 

20123618 

16.4924 

6.4792 

337 

113569 

38272753 

18.8576 

6.9589 

272 

74529 

20346417 

16.5227 

6.4872 

338 

114244 

38614472 

18.3848 

6.9658 

271 

75076 

20570824 

16.5529 

6.4951 

339 

114921 

38958219 

18.4120 

6.9727 

275 

75625 

20796875 

16.5831 

6.5030 

340 

115600 

39304000 

18.4391 

6.9795 

276 

76176 

21024576 

16.6132 

6.5108 

341 

116281 

39651821 

18.4662 

6.9864 

277 

76729 

21253933 

16.6433 

6.5187 

342 

116364 

40001688 

18.4932 

6.9932 

278 

77284 

21484952 

16.6733 

6.5265 

343 

117649 

40353607 

18.5203 

7. 

27‘J 

77841 

21717639 

16.7033 

6.5343 

344 

118336 

40707584 

18.5472 

7.0008 

280 

78400 

21952000 

16.7o.J2 

6.5421 

345 

119025 

41063625 

18.5742 

7.0136 

281 

78961 

22188041 

16.7631 

6.5499 

346 

119716 

41421736 

18.6011 

7.0203 

282 

79524 

22425768 

16.7929 

6.5577 

347 

120409 

41781923 

18.6279 

7.0271 

283 

80089 

22665187 

16.8228 

6.5654 

348 

121104 

42144192 

18.6548 

7.0338 

281 

80656 

22906304 

16.8523 

6.5731 

349 

121801 

42508549 

18.6815 

7.0406 

285 

81225 

23149125 

16.8819 

6.5808 

350 

122500 

42875000 

18.7083 

7.0473 

286 

81796 

23393656 

16.9115 

6.5885 

351 

123201 

43243551 

18.7350 

7.0540 

287 

82369 

23639903 

16.9411 

6.5962 

352 

123904 

43614208 

18.7617 

7.0607 

288 

82944 

23887872 

16.9706 

6.6039 

353 

124609 

43986977 

18.7883 

7.0674 

289 

83521 

24137569 

17. 

6.6115 

354 

125316 

44361864 

18.8149 

7.0740 

2iX> 

84100 

24389000 

17.0294 

6.6191 

355 

126025 

44738875 

18.8414 

7.0807 

291 

81681 

24642171 

17.0587 

6.6267 

356 

126736 

45118016 

18.8680 

7.0873 

292 

85264 

24897088 

17.0880 

6.6343 

357 

127449 

45499293 

18.8944 

7.0940 

293 

85849 

25153757 

17.1172 

6.6419 

358 

128164 

45882712 

18.9209 

7.1006 

291 

86436 

25412184 

17.1464 

6.6494 

359 

128881 

46268279 

18.9473 

7.1072 

295 

87025 

25672375 

17.1756 

6.6569 

360 

129600 

46656000 

18.9737 

7.1138 

296 

87616 

25934336 

17.2047 

6.6644 

361 

'130321 

47045881 

19. 

7.1204 

297 

88209 

26198073 

17.2337 

6.6719 

362 

131044 

47437928 

19.0263 

7.1269 

298 

88804 

26463592 

17.2627 

6.6794 

363 

131769 

47832147 

19.0526 

7.1335 

299 

89401 

26730899 

17.2916 

6.6869 

364 

132 496 

48228544 

19.0788 

7.1400 

300 

90000 

27000000 

17.3205 

£6943 

365 

133225 

48627125 

19.1050 

7.1466 

301 

90601 

27270901 

17.3494 

6.7018 

366 

133956 

49027896 

19.1311 

7.1531 

302 

91204 

27543608 

17.3781 

6.7092 

367 

134689 

49430863 

19.1572 

7.1596 

303 

91809 

27818127 

17.4069 

6.7166 

368 

135424 

49836032 

19.1833 

7.1661 

301 

92416 

28094464 

17.4356 

6.7240 

369 

136161 

50243409 

19.2094 

7.1726 

305 

93025 

28372625 

17.4642 

6.7313 

370 

136900 

50653000 

19.2354 

7.1791 

306 

93636 

28652616 

17.4929 

6.7387 

371 

137641 

51064811 

19.2614 

7.1855 

307 

94249 

28934443 

17.5214 

6.7460 

372 

138384 

51478848 

19.2873 

7.1920 

308 

94864 

29218112 

17.5499 

6.7533 

373 

139129 

51895117 

19.3132 

7.1984 

309 

95481 

29503629 

17.5784 

6.7606 

374 

139876 

52313624 

19.3391 

7.2048 

310 

96100 

29791000 

17.6068 

6.7679 

375 

140625 

52734375 

19.3649 

7.2112 

311 

96721 

30080231 

17.6352 

6.7752 

376 

141376 

53157376 

19.3907 

7.2177 

312 

97344 

30371328 

17.6635 

6.7824 

377 

142129 

53582633 

19.4165 

7.2240 

313 

97969 

30664297 

17.6918 

6.7897 

378 

142884 

54010152 

19.4422 

7.2304 

314 

98596 

30959144 

17.7200 

6.7969 

379 

143641 

54439939 

19.4679 

7.2368 

315 

99225 

31255875 

17.7482 

6.8044 

380 

144400 

54872000 

19.4936 

7.2432 




















































44 


SQUARES, CUBES, AND BOOTS 


TABLE of S< 3 ««*ires, Cubes, Square Roots, and Cube Roots, 
of Numbers from 1 to 1000 —(Continued.) 


No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

381 

145161 

55306341 

19.5192 

7.2495 

446 

198916 

88710536 

21.1187 

7.6403 

382 

145924 

65742968 

19.5448 

7.2558 

417 

199809 

89314023 

21.1424 

7.6160 

383 

146689 

56181887 

19.5704 

7.2622 

418 

200704 

89915392 

21.1660 

7.6517 

384 

147156 

66623104 

19.5959 

7.2685 

419 

201601 

90518849 

21.1896 

7.6574 

385 

148225 

57066625 

19.6214 

7.2748 

450 

202500 

91125000 

21.2132 

7.6634 

386 

148996 

57512456 

19.6469 

7.2811 

451 

203401 

91733851 

21.2368 

7.6688 

387 

119769 

57960603 

19.6723 

7.2874 

452 

204304 

92345408, 

21.2603 

7.(>744 

388 

150544 

58411072 

19.6977 

7.2936 

453 

205209 

92959677 

21.2838 

7.6801 

38!) 

151321 

58863869 

19.7231 

7.2999 

451 

206116 

93570664 

21.3073 

7.6857 

390 

152100 

59319000 

19.7484 

7.3061 

455 

207025 

94196375 

21.3307 

7.6914 

391 

152881 

59776471 

19.7737 

7.3124 

456 

207936 

94818816 

21.3542 

7.6970 

392 

153664 

60236288 

19.7990 

7.3186 

457 

208849 

95443993 

21.3776 

7.7026 

393 

154449 

60698157 

19.8242 

7.3248 

458 

209764 

90071912 

21.4009 

7.7082 

394 

155236 

61162984 

19.8494 

7.3310 

459 

210681 

96702579 

21.4243 

7.7138 

395 

156025 

61629875 

19.8746 

7.3372 

460 

211600 

97336000 

21.4476 

7.7194 

396 

156816 

62099136 

19.8997 

7.3434 

461 

212521 

97972181 

21.4709 

7.7250 

397 

157609 

62570773 

19.9249 

7.3496 

462 

21.3444 

98611128 

21.4942 

7.7306 

398 

158404 

63044792 

19.9499 

7.3558 

463 

214369 

99252847 

21.5174 

7.7368 

899 

159201 

63521199 

19.9750 

7.3619 

464 

215296 

99897344 

21.5407 

7.7418 

400 

160000 

64000000 

20. 

7.3681 

465 

216225 

100544625 

21.5639 

7.7473 

401 

160801 

64481201 

20.0250 

7.3742 

466 

217156 

101191696 

21.5870 

7.7529 

402 

161604 

61964808 

20.0499 

7.3803 

467 

21S089 

101847563 

21.6102 

7.7584 

403 

162109 

65450827 

20.0749 

7.3864 

468 

219024 

102503232 

21.6333 

7.7639 

404 

163216 

65939264 

20.0998 

7.3925 

469 

219961 

103161709 

21.6564 

7.7695 

405 

161025 

66430125 

20.1246 

7.3986 

470 

2209100 

103823000 

21.6795 

7.7750 

406 

164836 

66923416 

20.1494 

7.4047 

471 

221841 

104487111 

21.7025 

7.7805 

407 

165649 

67419143 

20.1742 

7.4108 

472 

222784 

105154018 

21.7256 

7.7860 

403 

166164 

67917312 

20.1990 

7.4169 

473 

223729 

105823817 

21.7486 

7.7915 

409 

167281 

63417929 

20.2237 

7.4229 

474 

224676 

106496124 

21.7715 

7.7970 

410 

168100 

68921000 

20.2485 

7.4290 

475 

225625 

107171875 

21.7945 

7.8025 

411 

168921 

69426531 

20.2731 

7.4350 

47fi 

226576 

107850176 

21.8174 

7.8079 

412 

169744 

69934528 

20.2978 

7.4410 

477 

227529 

108531333 

21.8403 

7.8134 

413 

170569 

70444997 

20.3224 

7.4470 

478 

228484 

109215352 

21.8632 

7.8188 

411 

171396 

70957944 

20.3470 

7.4530 

479 

229441 

109902239 

21.8861 

7.8243 

415 

172225 

71473375 

20.3715 

7.4590 

480 

230400 

110592000 

21.9089 

7.8297 

416 

173056 

71991296 

20.3961 

7.4650 

481 

231361 

111281641 

21.9317 

7.8352 

417 

173889 

72511713 

20.4206 

7.4710 

482 

232324 

111980168 

21.9545 

7.K10G 

418 

171724 

73034632 

20.4450 

7.4770 

483 

233289 

112678587 

21.9773 

7.8460 

419 

175561 

73560059 

20.4095 

7.4829 

484 

234256 

113379904 

22. 

7.8514 

420 

176400 

74088000 

20.4939 

7.4S89 

485 

235225 

114084125 

22.0227 

7.8568 

421 

177241 

74618461 

20.51R3 

7.4948 

486 

236196 

114791256 

22.0454 

7.8622 

422 

178084 

75151-448 

20.5426 

7.5007 

487 

237169 

115501303 

22.0681 

7.8676 

423 

178929 

75686967 

'2().5(j70 

7.5067 

488 

238144 

116214272 

22.0907 

7.8730 

424 

179776 

76225024 

20.5913 

7.5126 

489 

239121 

116930169 

22.1133 

7.8784 

425 

180625 

76765625 

20.6155 

7.5185 

490 

240100 

117649000 

22.1359 

7.8837 

426 

181476 

77308776 

20.6398 

7.5244 

491 

241081 

118370771 

22.1585 

7.8891 

427 

182329 

77854483 

20.6640 

7.5302 

492 

242064 

119095488 

22.1811 

7.8944 

428 

183184 

78402752 

20.6882 

7.5361 

493 

243049 

119823157 

22.2036 

7.8998 

429 

184041 

78953589 

20.7123 

7.5420 

494 

241036 

120553784 

22.2261 

7.9051 

430 

184900 

79507000 

20.7364 

7.5478 

495 

245025 

121287375 

22.2486 

7.9105 

431 

185761 

80062991 

20.7605 

7.5537 

496 

246016 

122023936 

22.2711 

7.9158 

432 

18o6'24 

80621568 

20.7846 

7.5595 

497 

247009 

122763473 

22.2935 

7.9211 

433 

187489 

81182737 

20.8087 

7.5654 

498 

248004 

123505992 

22.3159 

7.9264 

434 

188356 

81746504 

20.8327 

7.5712 

499 

249001 

124251199 

22.3383 

7.9317 

435 

189225 

82312875 

20.8567 

7.5770 

500 

250000 

125000000 

22.3607 

7.9370 

436 

190096 

82881856 

20.8806 

7.5828 

501 

251001 

125751501 

22.3830 

7.9423 

437 

190969 

83453453 

20.9045 

7.5886 

502 

252004 

126506008 

22.4054 

7.9476 

438 

191844 

84027672 

20.9284 

7.5944 

503 

253009 

127263527 

22.4277 

7.9528 

439 

192721 

84604519 

20.9523 

7.6001 

504 

251016 

128024064 

22.4499 

7.9581 

440 

193600 

85184000 

20.9762 

7.6059 

505 

255025 

126787625 

22.4722 

7.9634 

441 

194481 

85766121 

21. 

7.6117 

506 

256036 

129554216 

22.4944 

7.9086 

412 

195364 

86350888 

21.0238 

7.6174 

507 

257019 

130323848 

22.5167 

7 9739 

4(3 

196249 

86938307 

21.0476 

7.6232 

508 

258064 

131096512 

22.5389 

7.9791 

414 

197136 

87528384 

21.0713 

7.6289 

509 

259081 

131872229 

22.5610 

7.9843 

445 

198025 

88121125 

21.0950 

7.6346 

510 

260100 

132651000 

22.5832 

7.9896 
































SQUARES, CUBES, AND ROOTS 


45 


TABLE of Squares, Cubes, Square Roots, and Cube Roots, 
of Numbers from 1 to 1000 — (Continued.) 


No. 

Square 

Cube. 

Sq. Ht. 

C. Et. 

No. 

Square. 

Cube. 

Sq. Et. 

C. Et. 

511 

261121 

133432831 

22.6053 

7.9948 

576 

331776 

191102976 

24. 

8.3203 

512 

262144 

131217724 

22.6274 

8. 

577 

332929 

192100033 

24.0208 

8.3251 

513 

263169 

135005697 

22.6495 

8.0052 

578 

334084 

193100552 

24.0416 

8.3300 

511 

261196 

135796744 

22.6716 

8.0104 

579 

335241 

194104536 

24.0624 

8.3348 

515 

265225 

136590875 

22.6936 

8.0156 

580 

336400 

195112000 

24.0832 

8.3396 

510 

266256 

137388096 

22.7156 

8.0208 

581 

337561 

196122941 

24.1039 

8.3443 

517 

267289 

13818841: 

22.7376 

8.0260 

582 

338724 

197137368 

24.1247 

8.3491 

518 

268324 

138991832 

22.7596 

8.0311 

583 

339889 

198155287 

2 4.1454 

8.3539 

519 

269361 

139798359 

22.7816 

8.0363 

584 

341056 

199176704 

24.1661 

8.3587 

520 

270400 

140608000 

22.8035 

8.0415 

585 

342225 

200201625 

24.1868 

8.3634 

521 

271441 

141420761 

22.8254 

8.0466 

586 

343396 

201230056 

24.2074 

8.3682 

522 

272484 

142236648 

22.8473 

8.0517 

587 

344569 

202262003 

24.2281 

8.3730 

523 

273529 

143055667 

22.8692 

8.0569 

588 

345744 

203297472 

24.2487 

8.3777 

521 

274576 

143877824 

22.8910 

8.0620 

589 

346921 

204336469 

24.2693 

8.3825 

525 

275625 

144703125 

22.9129 

8.0671 

590 

348100 

205379000 

24.2899 

8.3872 

52(5 

276676 

145531576 

22.9347 

8.0723 

591 

349281 

206125071 

24.3105 

8.3919 

527 

277729 

146363183 

22.9565 

8.0774 

592 

350464 

207474688 

24.3311 

8.3967 

628 

278784 

147197952 

22.9783 

8.0825 

593 

351649 

208527857 

24.3516 

8.4014 

529 

279841 

148035889 

23. 

8.0876 

594 

352836 

209584584 

24.3721 

8.4061 

530 

280900 

148877000 

23.0217 

8.0927 

595 

354025 

210644875 

24.3926 

8.4108 

531 

281961 

149721291 

23.0434 

8.0978 

596 

355216 

211708736 

24.4131 

8.4155 

532 

283024 

150568768 

23.0651 

8.1028 

597 

356409 

212776173 

24.4336 

8.4202 

533 

284089 

151419437 

23.0868 

8.1079 

598 

357604 

213847192 

24.4540 

8.4249 

531 

285156 

152273304 

23.1084 

8.1130 

599 

358801 

214921799 

24.4745 

8.4296 

535 

286225 

153130375 

23.1301 

• 

8.1180 

600 

360000 

216000000 

24.4949 

8.4343 

536 

287296 

153990656 

23.1517 

8.1231 

601 

361201 

217081801 

24.5153 

8.4390 

537 

288369 

154854153 

23.1733 

8.1281 

602 

362404 

218167208 

24.5357 

8.4437 

538 

289444 

155720872 

23.1948 

8.1332 

603 

363609 

219256227 

24.5561 

8.4484 

539 

290521 

156590819 

23.2164 

8.1382 

604 

364816 

220348864 

24.5764 

8.4530 

540 

291600 

157464000 

23.2379 

8.1433 

605 

366025 

221445125 

24.5967 

8.4577 

541 

292681 

158340421 

23.2594 

8.1483 

606 

367236 

222545016 

24.6171 

8.4623 

542 

293764 

159220088 

23.2809 

8.1533 

607 

368449 

223648543 

24.6374 

8.4670 

513 

294849 

160103007 

23.3024 

8.1583 

608 

369664 

224755712 

24.6577 

8.4716 

541 

295936 

160989184 

23.3238 

8.1633 

609 

370881 

225866529 

24.6779 

8.4763 

545 

297025 

161878625 

23.3452 

8.1683 

610 

372100 

226981000 

24.6982 

8.4809 

546 

298116 

162771336 

23.3666 

8.1733 

611 

373321 

228099131 

24.7184 

8.4856 

547 

299209 

163667323 

23.3880 

8.1783 

612 

374544 

229220928 

24.7386 

8.4902 

548 

300304 

164566592 

23.4094 

8.1833 

613 

375769 

230346397 

24.7588 

8.4948 

549 

301401 

165469149 

23.4307 

8.1882 

614 

376996 

231475544 

24.7790 

8.4994 

550 

302500 

166375000 

23.4521 

8.1932 

615 

378225 

232608375 

24.7992 

8.5040 

551 

303601 

167284151 

23.4734 

8.1982 

616 

379456 

233744896 

24.8193 

8.5086 

552 

304704 

168196608 

23.4947 

8.2031 

617 

380689 

234885113 

24.8395 

8.5132 

553 

305809 

169112377 

23.5160 

8.2081 

618 

381924 

236029032 

24.8596 

8.5178 

554 

306916 

170031464 

23.5372 

8.2130 

619 

383161 

237176659 

24.8797 

8.5224 

555 

308025 

170953875 

23.5584 

8.2180 

620 

384400 

238328000 

24.8998 

8.5270 

556 

309136 

171879616 

23.5797 

8.2229 

621 

385641 

239483061 

24.9199 

8.5316 

557 

310249 

172808693 

23.6008 

8.2278 

622 

386884 

240611848 

24.9399 

8.5362 

558 

311364 

173741112 

23.6220 

8.2327 

623 

388129 

241804367 

24.9600 

8.5408 

559 

312481 

174676879 

23.6432 

8.2377 

624 

389376 

242970624 

24.9800 

8.5453 

560 

313600 

175616000 

23.6643 

8.2426 

625 

390625 

244140625 

25. 

8.5499 

561 

314721 

176558481 

23.6854 

8.2475 

626 

391876 

245314376 

25.0200 

8.5544 

562 

315844 

17750 4328 

23.7065 

8.2524 

627 

393129 

246491883 

25.0400 

8.5590 

563 

316969 

178453547 

23.7276 

8.2573 

628 

394384 

247673152 

25.0599 

8.5635 

504 

318096 

179406144 

23.7487 

8 2621 

629 

395641 

248858189 

25.0799 

8.5681 

565 

319225 

180362125 

23.7697 

8.2670 

630 

396900 

250047000 

25.0998 

8.5726 

566 

320356 

181321496 

23.7908 

8.2719 

631 

398161 

251239591 

25.1197 

8.5772 

567 

321489 

182284263 

23.8118 

8.2768 

632 

399424 

252435968 

25.1396 

8.5817 

568 

322624 

183250432 

23.8328 

8.2816 

633 

400689 

253636137 

25.1595 

8.5862 

569 

323761 

184220009 

23.8537 

8.2865 

634 

401956 

254840104 

25.1794 

8.5907 

570 

324900 

185193000 

23.8747 

8.2913 

635 

403225 

256047875 

25.1992 

8.5952 

571 

326041 

186169411 

23.8956 

8.2962 

636 

404496 

257259456 

25.2190 

8.5997 

572 

327184 

187149248 

23.9165 

8.3010 

637 

405769 

258474853 

25.2389 

8.6043 

573 

328329 

188132517 

23.9374 

8.3059 

638 

407044 

259694072 

25.2587 

8.6088 

571 

329476 

189119224 

23.9583 

8.3107 

639 

408321 

260917119 

25.2784 

8.6132 

575 

330625 

1901093751 

23.9792 

8.3155 

640 

409600 

262144000 

25.2982 

8.6177 












































46 


SQUARES, CUBES, AND ROOTS, 


TABLE of Squares, ('nbcs. Square Roots, ami Cube Roots, 

of Numbers from I to 1000 — (Continued.) 


No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

No. 

Square. 

Cube. 

. 

Sq. Rt. 

C. Rt. 

641 

410881 

263374721 

25 3180 

8.6222 

706 

498436 

351895816 

26.5707 

8.9043 

642 

412164 

261609288 

25.3377 

8.6267 

707 

499849 

353393243 

26.5895 

8.9085 

643 

413449 

265847707 

25.3574 

8.6312 

708 

501264 

354894912 

26.6083 

8.9127 

614 

414736 

267089984 

25.3772 

8.6357 

709 

502681 

356400829 

26.6271 

8.9169 

645 

416025 

268336125 

25.3969 

8.6401 

710 

504100 

357911000 

26.6458 

8.9211 

646 

417316 

269586136 

25.4165 

8.6446 

711 

505521 

359425431 

26.6646 

8.9253 

617 

418609 

270840023 

25.4362 

8.6490 

712 

506944 

360944128 

26.6833 

8.9295 

618 

419904 

272097792 

25.4558 

8.6535 

713 

508369 

362467097 

26.7021 

8.9337 

619 

421201 

273359449 

25.4755 

8.6579 

714 

509796 

363994344 

26.7208 

8,9378 

650 

#22500 

274625000 

25.4951 

8.6624 

715 

511225 

365525875 

26.7395 

8.9420 

651 

423801 

275894451 

25.5147 

8.6668 

716 

512656 

367061696 

26.7582 

8.9462 

652 

425104 

277167808 

25.5343 

8.6713 

717 

514089 

368601813 

26.7769 

8.9503 

653 

426109 

278445077 

25.5539 

8.6757 

718 

515524 

370146232 

26.7955 

8.9545 

651 

427716 

279726264 

25.5734 

8.6801 

719 

516961 

371694959 

26.8142 

8.9587 

655 

429025 

281011375 

25.5930 

8.6845 

720 

518400 

373248000 

26.8328 

8.9628 

656 

430336 

282300416 

25.6125 

8.6890 

721 

519841 

374805361 

26.8514 

8.9670 

657 

431649 

283593393 

25.6320 

8.6934 

722 

521284 

376367048 

26.8701 

8.9711 

658 

432964 

284890312 

25.6515 

8.6978 

723 

522729 

377933067 

26.8887 

8.9752 

659 

434281 

286191179 

25.6710 

8.7022 

724 

524176 

379503424 

26.9072 

8.9794 

660 

435600 

287496000 

25.6905 

8.7066 

725 

525625 

381078125 

26.9258 

8.9835 

661 

436921 

288804781 

25.7099 

8.7110 

726 

527076 

382657176 

26.9444 

8.9876 

662 

438244 

290117528 

25.7294 

8.7154 

727 

528529 

384240583 

26.9629 

8.9918 

663 

439569 

291434247 

25.7488 

8.7198 

728 

529984 

385828352 

26.9815 

8.9959 

664 

440896 

292754944 

25.7682 

8.7241 

729 

531441 

387420489 

27. 

9. 

665 

442225 

294079625 

25.7876 

8.7285 

730 

532900 

389017000 

27.0185 

9.0041 

666 

443556 

295408296 

25.8070 

8.7329 

731 

534361 

390617891 

27.0370 

9.0082 

667 

441889 

296740963 

25.8263 

8.7373 

732 

535824 

392223168 

27.0555 

9.0123 

668 

446224 

298077632 

25.8457 

8.7416 

733 

537289 

393832837 

27.0740 

9.0164 

669 

447J61 

299118309 

25.8650 

8.7460 

734 

538756 

395446904 

27.0924 

9.0205 

670 

448900 

300763000 

25.8844 

8.7503 

735 

540225 

397065375 

27.1109 

9.0246 

671 

450241 

302111711 

25.9037 

8.7547 

736 

541696 

398688256 

27.1293 

9.0287 

672 

451584 

303464448 

25.9230 

8.7590 

737 

543169 

400315553 

27.1477 

9.0328 

673 

452929 

304821217 

25.9422 

8.7634 

738 

544644 

401947272 

27.1662 

9.0369 

674 

454276 

306182024 

25.9615 

8.7677 

739 

546121 

403583419 

27.1816 

9.0410 

675 

455625 

307546875 

25.9808 

8.7721 

740 

547600 

405224000 

27.2029 

9.0450 

676 

456976 

308915776 

26. 

8.7764 

741 

549081 

406869021 

27.2213 

9.0491 

677 

458329 

310288733 

26.0192 

8.7807 

742 

550564 

408518488 

27.2397 

9.0532 

678 

459684 

311(365752 

26.0384 

8.7850 

743 

552049 

410172407 

27.2580 

9.0572 

679 

461041 

313046839 

26.0576 

8.7893 

744 

553536 

411830784 

27.2764 

9.0613 

680 

462400 

314432000 

26.0768 

8.7937 

745 

555025 

413493625 

27.2947 

9.0654 

681 

463761 

315821241 

26.0960 

8.7980 

746 

556516 

415160936 

27.3130 

9.0694 

682 

465124 

317214568 

26.1151 

8.8023 

747 

558009 

416832723 

27.3313 

9.0735 

683 

466489 

318611987 

26.1343 

8.8066 

748 

559504 

418508992 

27.3496 

9.0775 

684 

467856 

320013504 

26.1534 

8.8109 

749 

561001 

420189749 

27.3679 

9.0816 

685 

469225 

321419125 

26.1725 

8.8152 

750 

562500 

421875000 

27.3861 

9.0856 

686 

470596 

322828856 

26.1916 

8.8194 

751 

564001 

423564751 

27.4044 

9.0896 

687 

471969 

324242703 

26.2107 

8.8237 

752 

565504 

425259008 

27.4226 

9.0937 

688 

473344 

325660672 

26.2298 

8.8280 

753 

567009 

426957777 

27.4408 

9.0977 

689 

474721 

327082769 

26.2488 

8.8323 

754 

568516 

428661064 

27.4591 

9.1017 

690 

476100 

328509000 

26.2679 

8.8386 

755 

570025 

430368875 

27.4773 

9.1057 

691 

477481 

329939371 

26.2869 

8.8408 

756 

571536 

432081216 

27.4955 

9.1098 

692 

478864 

331573888 

26.3059 

8.8451 

757 

573049 

433798093 

27.5136 

9.1138 

693 

480249 

332812557 

26.3249 

8.8493 

758 

574564 

435519512 

27.5318 

9.1178 

694 

481636 

334255384 

26.3439 

8.8536 

759 

576081 

437245479 

27.5500 

9.1218 

#95 

483025 

£35702375 

26.3629 

8.8578 

760 

577600 

438976000 

27.5681 

9.1258 

696 

484416 

337153536 

26.3818 

8.8621 

761 

579121 

440711081 

27.5862 

9.1298 

697 

485809 

338608873 

26.4008 

8.8663 

762 

580644 

442450728 

27.6043 

9.1338 

698 

487204 

340068392 

26.4197 

8.8706 

763 

582169 

444194947 

27.6225 

9.1378 

699 

488601 

341532099 

26.4386 

8.8748 

764 

583696 

445943744 

27.6405 

9.1418 

700 

490000 

343000000 

26.4575 

8.8790 

765 

585225 

447697125 

27.6586 

9.1458 

701 

491401 

344472101 

26.4761 

8.8833 

766 

586756 

449455096 

27.6767 

9.1498 

702 

492804 

345948408 

26.4953 

8.8875 

767 

588289 

451217663 

27.6948 

9.1537 

703 

491209 

347428927 

26.5141 

8 8917 

768 

589824 

452984832 

27.7128 

9.1577 

704 

495616 

348913664 

26.5330 

8.8959 

769 

591361 

454756609 

27.7308 

9.1617 

705 

497025 

350402625 

26.5518 

8.9001 

770 

592900 

456533009 

27.7489 

9.1657 




























SQUARES, CUBES, AND ROOTS, 


47 


TABLE of Squares, Cubes, Square Roots, and Cube Roots, 
of .Numbers from 1 to 1000 — (Continued.; 


4o. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

771 

594441 

458314011 

27.7669 

9.1696 

772 

595984 

460099648 

27.7849 

9.1736 

773 

597529 

461889917 

27.8029 

9.1775 

774 

599076 

463684824 

27.8209 

9.1815 

775 

600625 

465484375 

27.8388 

. 9.1855 

776 

602176 

467288576 

27.8568 

9.1894 

777 

603729 

469097433 

27.8747 

9.1933 

778 

605284 

470910952 

27.8927 

9.1973 

779 

606841 

472729139 

27.9106 

9.2012 

780 

608100 

474552000 

27.9285 

9.2052 

781 

609961 

476379541 

27.9464 

9.2091 

782 

611524 

478211768 

27.9643 

9.2130 

T 83 

613089 

480018687 

27.9821 

9.2170 

784 

614656 

481890304 

28. 

9.2209 

785 

616225 

483736625 

28.0179 

9.2248 

786 

617796 

485587656 

28.0357 

9.2287 

787 

619369 

487443403 

28.0535 

9.2326 

788 

620944 

489303872 

28.0713 

9.2365 

'89 

622521 

491169069 

28.0891 

9.2404 

790 

624100 

493039000 

28.1069 

9.2443 

791 

625681 

494913671 

28.1247 

9.2482 

'92 

627264 

496793088 

28.1425 

9.2521 

■93 

6288 49 

498677257 

28.160.3 

9.2560 

'94 

630436 

500566184 

28.1780 

9.2599 

'95 

632025 

502459875 

28.1957 

9.2638 

96 

633616 

504.358336 

28.2135 

9.2677 

97 

635209 

506261573 

28.2312 

9.2716 

98 

636804 

508169592 

28.2489 

9.2754 

99 

638401 

510082399 

28.2666 

9.2793 

100 

640000 

512000000 

28.2843 

9.2832 

01 

641601 

513922401 

28.3019 

9.2870 

02 

64320 4 

515849608 

28.3196 

9.2909 

03 

641809 

517781627 

28.3373 

9.2948 

04 

646416 

519718164 

28.3549 

9.2986 

05 

648025 

521660125 

28.3725 

9.3025 


No. 

Square. 

Cube. 

Sq. Rt. 

C. R r , 

836 

698896 

584277056 

28.9137 

9.4204 

837 

700569 

586376253 

28.9310 

9.4241 

838 

702244 

588180472 

28.9182 

9.4279 

839 

703921 

590589719 

28.9655 

9.4316 

840 

705600 

592701000 

28.9828 

9.4354 

841 

707281 

591823321 

29. 

9.4391 

842 

708964 

596947688 

29.0172 

9.4429 

813 

710649 

599077107 

29.0345 

9.4466 

811 

712336 

601211584 

29.051 1 

9.4503 

845 

714025 

603351125 

29.0689 

9.4541 

846 

715716 

605495736 

29.0861 

9.4578 

847 

717409 

607645423 

29.1033 

9.4615 

848 

719104 

609800192 

29.1204 

9.4652 

849 

720801 

611960049 

29.1376 

9.4690 

850 

722500 

614125000 

29.1548 

9.4727 

851 

724201 

616295051 

29.1719 

9.4764 

852 

725904 

618470208 

29 1890 

9.4801 

853 

727609 

620650477 

29.2062 

9 4838 

854 

729316 

622835864 

29.2233 

9.4875 

855 

731025 

625026375 

29.2404 

9.4912 

856 

732736 

627222016 

29.2575 

9.4949 

857 

731449 

629422793 

29.2746 

9.4986 

858 

736164 

631628712 

29.2916 

9.5023 

859 

737881 

633839779 

29.3087 

9.5060 

860 

739600 

63G056000 

29.3258 

9.5097 

861 

741321 

638277381 

29.3428 

9.5134 

862 

743044 

640503928 

29.3598 

9.5171 

863 

744769 

642735647 

29.3769 

9.5207 

864 

746496 

644972544 

29.3939 

9.5244 

865 

748225 

647214625 

29.4109 

9.5281 

866 

749956 

649461896 

29.4279 

9.5317 

867 

751689 

651714363 

29.4449 

9.5354 

868 

753424 

653972032 

29.4618 

9.5391 

869 

755161 

656234909 

29.4788 

9.5427 

870 

756900 

658503000 

29.4958 

9.5404 


06 

•07 

08 

09 

10 


649636 

651249 

652864 

651481 

656100 


523606616 
525557943 
527514112 
529475129 
531441000 


28.3901 

28.4077 

28.4253 

28.4429 

28.4605 


9.3063 

9.3102 

9.3140 

9.3179 

9.3217 


871 

872 

873 

874 

875 


758641 
760384 
762129 
763876 
765625 


660776311 

663054848 

665338617 

667627624 

669921875 


29.5127 

29.5296 

29.5466 

29.5635 

29.5804 


9.5501 

9.5537 

9.5574 

9.5610 

9.5647 


11 

12 

13 

14 

15 


657721 

659344 

660969 

662596 

664225 


533411731 

535387328 

537367797 

539353144 

54134.3375 


28.4781 

28.4956 

28.5132 

28.5307 

28.5482 


9.3255 

9.3294 

9.3332 

9.3370 

9.3408 


876 

877 

878 

879 

880 


767376 

769129 

770884 

772641 

774400 


672221376 

674526133 

676836152 

679151439 

681472000 


29.5973 
29.6142 
29.6.311 
29.6479 
29.6648 


9.5683 

9.5719 

9.5756 

9.5792 

9.5828 


17 

18 

19 

20 


665856 

667189 

669124 

670761 

672400 


543338196 

545338513 

547343432 

549353259 

551368000 


28.5657 

28.5832 

28.6007 

28.6182 

28.6356 


9.3147 

9.3485 

9.3523 

9.3561 

9.3599 


881 

882 

883 

884 

885 


776161 

777924 

779689 

781456 

783225 


683797841 

686128968 

688465387 

690807104 

693154125 


29.6816 

29.6985 

29.7153 

29.7321 

29.7489 


9.5865 

9.5901 

9.5937 

9.5973 

9.6010 


21 

22 

23 

21 

25 


674011 

675684 

677329 

678976 

680625 


553387661 

555412248 

557441767 

559176224 

561515625 


28.6531 

28.6705 

28.6880 

28.7054 

28.7228 


9.3637 

9.3675 

9.3713 

9.3751 

9.3789 


886 

887 

888 

889 

890 


784996 
786769 
788511 
790321 
792100 


695506456 

697864103 

700227072 

702595369 

704969000 


29.7658 

29.7825 

29.7993 

29.8161 

29.8329 


9.6016 

9.6082 

9.6118 

9.6154 

9.6190 


26 

27 

28 

29 

30 


682276 

683929 

685584 

687211 

688900 


563559976 

565609283 

567663552 

569722789 

571787000 


28.7402 

28.7576 

28.7750 

28.7924 

28.8097 


9.3827 

9.3865 

9.3902 

9.3910 

9.3978 


891 

892 

893 
891 
895 


793881 

795664 

797449 

799236 

801025 


707347971 

709732288 

712121957 

714516984 

716917375 


29.8496 

29.8664 

29.8831 

29.8998 

29.9166 


9.6226 

9.6262 

9.6298 

9.6334 

9.6370 


31 

32 

33 

34 

35 


690561 

692224 

693889 

695556 

697225 


573856191 

575930368 

578009537 

580093704 

582182875 


28.8271 

28.8444 

28.8617 

28.8791 

28.8964 


9.4016 

9.4053 

9.4091 

9.4129 

9.4166 


896 

897 

898 

899 

900 


802816 

804609 

806404 

808201 

810000 


719323136 

721734273 

724150792 

726572699 

729000000 


29.9333 

29.9500 

29.9666 

29.9833 

30. 


9.6106 
9.6442 
9.6477 
9 6513 

9.6549 





































48 


SQUARES, CUBES, AND ROOTS, 


TABLE of Squares, Cubes, Square Roots, anti Cube Root 

of Numbers from 1 to 1000 — (Continued.) 


No. 

Square 

Cube. 

Sq. Rt. 

C. Rt. 

No. 

Square. 

Cube. 

Sq. Rt. 

C. I 

901 

811801 

731432701 

30.0167 

9.6585 

951 

901101 

860985351 

30.8383 

9.8:: 

902 

813604 

733870808 

30.0333 

9.6620 

952 

906304 

802.',0140s 

30.8545 

9.83 

903 

815109 

736314327 

30.0500 

9.6656 

953 

908209 

805523177 

S0.8707 

9.84 

901 

817216 

73876326- 

30.0666 

9.6692 

954 

910116 

868250664 

30.8869 

9.81 

905 

819025 

74121762a 

30.0832 

9.6727 

955 

912025 

870983875 

30.9031 

9.84 

906 

820836 

743677416 

30.0998 

9.6763 

956 

913936 

873722816 

30.9192 

9.85 

907 

822619 

746142643 

30.1164 

9.6799 

957 

915849 

87G467493 

30.9354 

9.85 

90S 

821164 

748613312 

30.1330 

9.6834 

958 

917764 

879217912 

30.9516 

9.85 

903 

826281 

751089429 

30.1496 

9.6870 

959 

919C81 

881974079 

30.9677 

9.8C 

910 

828100 

753571000 

30.1662 

9.6905 

960 

921600 

884736000 

30.9839 

9.8C 

911 

829921 

756058031 

30.1828 

9.6941 

961 

923521 

887503681 

31. 

9.8f 

912 

831744 

758550528 

30.1993 

9.6976 

962 

925444 

890277128 

31.0161 

9.87 

913 

833569 

76 1 0 48 497 

30.2159 

9.7012 

963 

927369 

893056347 

31.0322 

9.87 

914 

835396 

763551944 

30.2324 

9.7017 

964 

929296 

895841344 

31.0483 

9.87. 

915 

837225 

766060875 

30.2490 

9.7082 

965 

931225 

898632125 

31.0644 

9.88 

916 

839056 

768575296 

30.2655 

9.7118 

966 

933156 

901428696 

31.0805 

9.88 

917 

810889 

771095213 

30.2820 

9.7153 

967 

935089 

904231063 

31.0966 

9.88 

918 

812724 

773620632 

30.2985 

9.7188 

968 

937024 

907039232 

31.1127 

9.8£ 

919 

844561 

776151559 

30.3150 

9.7224 

969 

938961 

909853209 

31.1288 

9.89 

920 

846100 

778688000 

30.3315 

9.7259 

970 

940900 

912673000 

31.1448 

9.891 

921 

818241 

781229961 

30.3480 

9.7294 

971 

942841 

91519S611 

31.1609 

9.9C 

922 

850084 

783777448 

30.3645 

9.7329 

972 

944784 

918330048 

31.1709 

9.90 

923 

851929 

786330467 

30.3809 

9.7364 

973 

946729 

921167317 

31.1929 

9.90 

924 

853776 

788889024 

30.3974 

9.7400 

971 

918676 

924010124 

31.2090 

9.91 

925 

855625 

791453125 

30.4138 

9.7435 

975 

950625 

926859375 

31.2250 

9.91: 

926 

857476 

794022776 

30.4302 

9.7470 

976 

952576 

929714176 

31.2410 

9.91 

927 

859329 

796597983 

30.4167 

9.7505 

977 

954529 

932574833 

31.2570 

9.92 

928 

861184 

799178752 

30.4631 

9.7540 

978 

956484 

935441352 

31.2730 

9.92 

929 

863011 

801765089 

30.4795 

9.7575 

979 

958141 

938313739 

31.2890 

9.92 

930 

861900 

804357000 

30.4959 

9.7610 

980 

960400 

941192000 

31.3050 

9.93 

931 

866761 

806954191 

30.5123 

9.7645 

981 

962361 

944076141 

31.3209 

9.93 

932 

868624 

809557568 

30.5287 

9.7680 

982 

964324 

946966168 

31.3369 

9.93 

933 

870489 

812166237 

30.5150 

9.7715 

983 

966289 

949862087 

31.3528 

9.94 

934 

872356 

814780501 

30.5614 

9.7750 

984 

968256 

952763904 

31.3688 

9.94 

935 

874225 

817400375 

30.5778 

9.7785 

985 

970225 

955671625 

31.3847 

9.94 

936 

876096 

820025856 

30.5941 

9.7819 

986 

972196 

958585256 

31.4006 

9.95 

9.47 

K779H9 

822656953 

80.6105 

9.7854 

987 

974169 

961504803 

81.4166 

9.95 

938 

87981-4 

825293672 

30.6268 

9.7889 

988 

976144 

964430272 

31.4325 

9.95 

939 

881721 

827936019 

80.6431 

9.7924 

989 

978121 

967361669 

31.4484 

9.90 

910 

883600 

830584000 

30.6594 

9.7959 

990 

980100 

970299000 

31.4643 

9.96 

911 

885 481 

833237621 

30.6757 

9.7993 

991 

982081 

973242271 

31.4802 

9.90 \ 

91*2 

887.10 ( 

835896888 

30.6920 

9.8028 

992 

984064 

976191488 

31.4960 

9.91 \ 

944 

889249 

838561807 

30.7083 

9.8063 

993 

986049 

979146657 

31.5119 

9JllM 

9(4 

891136 

811232384 

30.7216 

9.8097 

994 

988036 

982107*84 

31.5278 

9.9K ] 

945 

893025 

843908625 

30.7409 

9.8132 

995 

990025 

985074875 

31.5436 

9.98,1 

916 

894916 

846590536 

80.7571 

9.8167 

996 

992016 

988047936 

31.5595 

q 98 J 

947 

896809 

849278123 

30.7731 

9.8201 

997 

994009 

991026973 

31 5753 

<1 99 W 

918 

898704 

851971392 

30.7896 

9.8236 

998 

996004 

994011992 

31.5911 

<) 99 

919 

900601 

854670349 

30.8058 

9.8270 

999 

998001 

997002999 

31. (>070 

9 99 

950 

902500 

857375000 

30.8221 

9.8305 

1000 

1000000 

1000000000 

31.6228 

10 .' 


To find the square or cube of any whole number endii 
wall* ciphers. First, omit all the final ciphers. Take from the table 

square or cube (as the case may be) of the rest of the number. To this square add twice a?av 
cn-hers as there were (inu ciphers in the original number. To the cube add three times as mani 
IV H«<i original number. Thus, for 90o002 ; 9052 = 819025. Add twice 2 ciphers obtain^ R1 wwvw 
tor .I0a003, 90a3 -- 711217625. Add 3 times 2 ciphers, obtaining 741217625000000. ^ 

For tables of fifth roots, fifth powers, and square roots of Ilf 
powers, see pp 251 to 253. 


















































SQUARE AND CUBE ROOTS, 


49 


Si kill a re Roots and Cube Roots of Numbers from 1000 to 10000. 


Num. 


1005 
1010 
4 1015 

a 1020 
; 1025 
?! 1030 

;! 1035 
1010 
1015 
? 1050 

I 1055 
?11060 
!!l065 
1,1 1070 
,J 1075 

; loso 
!! 1085 
H 1090 
: 1095 

II 100 
J 105 
' ; no 
1 115 
: 120 

125 
l! | 130 
135 

no 

145 

150 

•j!55 

4160 

, ><*5 

'170 
75 
,80 
85 
90 
95 
00 
» 05 

I 10 
15 

120 
25 
30 
!5 
10 
15 
50 
>5 
.0 
j >5 
■o 
'5 
!0 
5 
0 
5 
0 
5 
0 
5 
0 
5 
9 
5 
1 
5 
) 

> 

) 

5 
) 
i 
I 
i 
t 


Sq. Rt. 

Cu. Rt. 

Num. 

31.70 

10.02 

1405 

31.78 

10.03 

1410 

31.86 

10.05 

1 1415 

31.94 

10.07 

1420 

32.02 

10.08 

1425 

32.09 

10.10 

1430 

32.17 

10.12 

1435 

32.25 

10.13 

1440 

32.33 

10.15 1 

1445 

32.40 

10.16 

1450 

32.48 

10.18 | 

1455 

32.56 

10.20 

1460 

32.63 

10.21 | 

1465 


I 


32.71 
32.79 
32.86 
32.94 
33.02 
33.09 
33.17 
33.24 
33.32 
33.39 
33.47 
33.54 
33.62 
33.69 
33.76 
33.84 
33.91 
33.99 
34.06 
34.13 
34.21 
31.28 
34.35 
34.42 
34.50 
34.57 
31.64 
34.71 
34.79 
34.86 
34.9.3 
35.00 
35.07 
35.14 
35.21 
35.28 
35.36 
35.43 
35.50 
35.57 
35.64 
35.71 
35.78 
35.85 
35.92 
35.99 
36.06 
36.12 
36.19 
86.26 
86.33 
36.40 
36.47 
36.54 
36.61 
36.67 
36.74 
36.81 
36.88 
36.95 
37.01 
37.08 
37.15 
37.22 
37.28 
37.35 
37.12 


10.23 

10.24 
10.26 
10.28 
10.29 

10.31 

10.32 

10.34 

10.35 

10.37 

10.38 
10.40 

10.42 

10.43 

10.45 

10.46 

10.48 

10.49 

10.51 

10.52 

10.54 

10.55 

10.57 

10.58 
10.60 
10.61 

10.63 

10.64 
10.66 
10.67 

10.69 

10.70 

10.71 

10.73 

10.74 
10.76 


1470 

1475 

1480 

1485 

1490 

1495 

1500 

1505 

1510 

1515 

1520 

1525 

1530 

1535 

1540 

1545 

1550 

1555 

1560 

1565 

1570 

1575 

1580 

1585 

1590 

1595 

1600 

1605 

1610 

1615 

1620 

1625 

1630 

1635 

J640 

1645 


37.48 
37.55 
37.62 
37.68 
37.75 
37.82 
37.88 
37.95 
38.01 
38.08 
38.14 
38.21 
38.28 
38.34 
38.41 
38.47 
38.54 
38.60 
38.67 
38.73 
38.79 
38.86 
38.92 
38.99 
39.05 
39.12 
39.18 
39.24 
39.31 
39.37 
39.43 
39.50 
39.56 
39.62 
39.69 
39.75 
39.81 
39.87 
39.94 
40.00 
40.06 
40.12 
40.19 
40.25 
40.31 
40.37 
40.44 
40.50 
40.56 


11.20 
11.21 

11.23 

11.24 

11.25 

11.27 

11.28 
11.29 

11.31 

11.32 

11.33 

11.34 

11.36 

11.37 

11.38 

11.40 

11.41 

11.42 

11.43 

11.45 

11.46 

11.47 

11.49 

11.50 

11.51 

11.52 

11.54 

11.55 

11.56 

11.57 
11.79 
11.60 
11.61 
11.62 
11.63 

11.65 

11.66 

11.67 

11.68 

11.70 

11.71 

11.72 

11.73 

11.74 

11.76 

11.77 

11.78 

11.79 

11.80 


1805 

1810 
1815 
1820 
1825 
1830 
1835 
1840 
1845 
1850 
1855 
1860 
1865 
1870 
1875 
1880 
1885 
1890 
1895 
1900 
1905 
1910 
1915 
1920 
1925 
1930 
1935 
1940 
1945 
1950 
1955 
1960 
1965 
1970 
1975 
1980 
1985 
1990 
1995 
2000 
2005 
2010 
2015 
2020 
2025 
2030 
2035 
2040 
2045 


Sq. Rt 

Cu. Rt 

I Num. 

Sq. Rt. 

42.49 

12.18 

2205 

46.96 

42.54 

12.19 

2210 

47.01 

42.60 

12.20 

2215 

47.06 

42. 

12.21 

2220 

47.12 

42.72 

12.22 

2225 

47.17 

42.78 

12.23 

2230 

47.22 

42.84 

12.24 

2235 

47.28 

42.90 

12.25 

2240 

47.33 

42.95 

12.26 

2245 

47.38 

43.01 

12.28 

2250 

47.43 

43.07 

12.29 

2255 

47.49 

43.13 

12.30 

2260 

47.54 

43.19 

12.31 

2265 

47.59 

43.24 

12.32 

2270 

47.64 

43.30 

12.33 

2275 

47.70 

43.36 

12.34 

2280 

47.75 

43.42 

12.35 

2285 

47.80 

43.47 

12.36 

2290 

47.85 

43.53 

12.37 

2295 

47.91 

43.59 

12.39 

2300 

47.96 

43.65 

12.40 

2305 

48.01 

43.70 

12.41 

2310 

48.06 

43.76 

12.42 | 

2315 

48.11 

43.82 

12.43 

2320 

48.17 

43.87 

12.44 

2325 

48.22 

43.93 

12.45 

2330 

48.27 

43.99 

12.46 

2335 

48.32 

44.05 

12.47 

2340 

48-37 

44.10 

12,48 

2345 

48.43 

44.16 

12.49 

2350 

48.48 

44.22 

12.50 

2355 

48.53 

44.27 

12.51 

2360 

48.58 

44.33 

12.53 

2365 

48.63 

44.38 

12.54 

2370 

48.68 

44.44 

12.55 

2375 

48.73 

44.50 

12.56 

2380 

48.79 

44.55 

12.57 

2385 

48.84 

44.61 

12.58 I 

2390 

48.89 

44.67 

12.59 

2395 

48.94 


Cu. Rt. 


44.72 

44.78 

44.83 

44.89 

44.94 

45.00 

45.06 

45.11 

45.17 

45.22 


12.60 
12.61 
12.62 

12.63 

12.64 

12.65 

12.66 

12.67 

12.68 


10.77 

1650 

40.62 

11.82 

2050 

45.28 

12.71 

10.79 

1655 

40.68 

11.83 

2055 

45.33 

12.7 

10.80 

1660 

40.74 

11.84 

2060 

45.39 

12 .7‘ 

10.82 

1665 

40.80 

11.85 

2065 

45.44 

12.7; 

10.83 

1670 

40.87 

11.86 

2070 

45.50 

12.74 

10.84 

1675 

40.93 

11.88 

2075 

45.55 

12.75 

10.86 

1680 

40.99 

11.89 

I 2080 

45 61 

12.77 

10.87 

1685 

41.05 

11.90 

2085 

45.66 

12.78 

10.89 

1690 

41.11 

11.91 

2090 

45.72 

12.79 

10.90 

1695 

41.17 

11.92 

2095 

45.77 

12.80 

10.91 

1700 

41.23 

11.93 

2100 

45.83 

12.81 

10.93 

! 1705 

41.29 

11.95 

2105 

45.88 

12.82 

10.94 

1710 

41.35 

11.96 

| 2110 

45.93 

12.83 

10.96 

1715 

41.41 

11.97 

2115 

45.99 

12.84 

10.97 

1720 

41.47 

11.98 

2120 

46.04 

12.85 

10.98 

1725 

41.53 

11.99 

2125 

46.10 

12.86 

11.00 

1730 

41.59 

12.00 

2130 

46.15 

12.87 

11.01 

1735 

41.65 

12.02 

2135 

46.21 

12.88 

11.02 

1740 

41.71 

12.03 

2140 

46.26 

12.89 

11.04 

1745 

41.77 

12.04 

2145 

46.31 

12.90 

11.05 

1750 

41.83 

12.05 

2150 

46.37 

12.91 

11.07 

1755 

41.89 

12.06 

2155 

46.42 

12.92 

11.08 

1760 

41.95 

12.07 

2160 

46.48 

12.93 

11.09 

1765 

42.01 

12.09 

2165 

46.53 

12.94 

11.11 I 

1770 

42.07 

12.10 

2170 

46.58 

12.95 

11.12 

1775 

42.13 

12.11 1 

2175 

46.64 

12.96 

11.13 

1780 

42.19 

12.12 

2180 

46.69 

12.97 

11.15 I 

1785 

42.25 

12.13 

2185 

46.74 

12.98 

11.16 

1790 

42.31 

12.14 

2190 

46.80 

12.99 

11.17 

1795 

42.37 

12.15 

2195 

46.85 

13.00 

11.19 | 

1800 

42.43 

12.16 

2200 

46.90 1 

13.01 


2400 
2405 
2410 
2415 
2420 
2425 
2430 
2435 
2440 
2445 
2450 
2460 
2470 
2480 
2490 
2500 
2510 
2520 
2530 
2540 
2550 
2560 
2570 
2580 
2590 
2600 
2610 
2620 
2630 
2640 
2650 
2660 
2670 
2680 
2690 
2700 
2710 
2720 
2730 
2740 
2750 


48.99 
49.04 
49.09 
49.14 
49.19 
49.24 

49.30 
49.35 

49.40 
49.45 
49.50 

49.60 

49.70 
49.80 
49.90 
50.00 
50.10 
50.20 

50.30 

50.40 
50 50 

50.60 

50.70 
50.79 
50.89 
50.99 
51.09 
51.19 
51.28 
51.38 
51.48 
51.58 
51.67 
51.77 
51.87 
51.96 
52.06 
52.15 
52.25 
52.35 
62.44 


13.02 

13.03 

13.04 

13.05 

13.05 

13.06 

13.07 

13.08 

13.09 

13.10 

13.11 

13.12 

13.13 

13.14 

13.15 

13.16 

13.17 

13.18 

13.19 

13.20 

13.21 

13.22 

13.23 

13.24 

13.25 

13.26 

13.27 

13.28 

13.29 

13.30 

13.30 

13.31 

13.32 

13.33 

13.34 

13.35 

13.36 

13.37 


13.39 

13.40 

13.41 

13.42 

13.43 

13.43 

13.44 

13.45 

13.46 

13.47 

13.48 
13.50 
13.52 

13.54 

13.55 
13.57 
13.59 
13.61 

13.63 

13.64 
13.66 
13 68 
13.70 

13.72 

13.73 
13.75 
13.77 

13.79 

13.80 
13.82 
13.84 

13.86 

13.87 
13.89 

13.91 

13.92 
13.94- 
13.96 

13.98 

13.99 
14.01 
















































































50 


SQUARE AND CUBE ROOTS, 


Square Roots anti Cube Root* of Numbers from lOOO tolOt] 

—(Continued.) 


Niim. 

Sq. Rt 

Cti. Rt 

2760 

52.54 

14.03 

2770 

52.63 

14.01 

2780 

52.73 

14.06 

2790 

52.82 

14.08 

2800 

52.92 

14.09 

2810 

53.01 

14.11 

2820 

53.10 

14.13 

2830 

53.20 

14.14 

2810 

53.29 

14.16 

2850 

53.39 

14.18 

2860 

53 48 

14.19 

2870 

53.57 

14.21 

2880 

53.67 

14.23 

2890 

53.76 

14.24 

2900 

53.85 

14.26 

2910 

53.94 

14.28 

2920 

54.04 

14.29 

2950 

51.13 

14.31 

2910 

54.22 

1 4.33 

2950 

54.31 

14.34 

2960 

54.41 

14.36 

2970 

54.50 

14.37 

2980 

54.59 

14.39 

2990 

54.68 

14.41 

3000 

51.77 

14.42 

3010 

54.86 

14.44 

3020 

51.95 

14.45 

3030 

55.05 

14.47 

3010 

55.14 

14.49 

3050 

55.23 

14.50 

3060 

55.32 

14.52 

3070 

55.41 

14.53 

3080 

55.50 

14.55 

3090 

55.59 

1 4.57 

3100 

55.68 

14.58 

3110 

55.77 

14.60 

3120 

55.86 

14.61 

3130 

55.95 

14.63 

3110 

56.04 

14.64 

3150 

56.12 

14.66 

3160 

66.21 

14.67 

3170 

56.30 

14.69 

3180 

56.39 

14.71 

3190 

56.48 

14.72 

3200 

56.57 

14.74 

3210 

56.66 

14.75 

3220 

56.75 

14.77 

3230 

56.83 

14.78 

3230 

56.92 

14.80 

3250 

57.01 

14.81 

3260 

57.10 

14.83 

3270 

57.18 

14.84 

3280 

57.27 

14.86 

3290 

57.36 

14.87 

3300 

57.45 

14.89 

3310 

57.53 

14.90 

3320 

57.62 

14.92 

3330 

57.71 

14.93 

3310 

67.79 

14.95 

3350 

57.88 

14.96 

3360 

57.97 

14.98 

3370 

58.05 

14.99 

3380 

58.11 

15,01 

3390 

58.22 

15.02 

3400 

58.31 

15.04 

3410 

58.10 

15.05 

3420 

58.48 

15.07 

3430 

58.57 

15.08 

3140 

58.65 

15.10 

3450 

58.74 

15.11 

3160 

58.82 

15.12 

3170 

58.91 

15.14 

3180 

58.99 

15.15 

3490 

59.08 

15.17 

3500 

59.16 

15.18 

3310 

59.25 

15.20 

3520 

59.33 

15.21 

3530 

59.41 

15.23 

3540 

59.50 

15.24 


Nnm. 

Sq. Rt 

Cn. Rt 

3550 

59.58 

15.25 

3560 

59.67 

15.27 

3570 

59.75 

15.28 

3580 

59.83 

15.30 

3590 

59.92 

15.31 

3600 

60.00 

15.33 

3610 

60.08 

15.34 

3620 

60.17 

15.35 

3630 

60.25 

15.37 

3640 

60.33 

15.38 

3650 

60.42 

15.40 

3660 

60.50 

15.41 

3670 

60.58 

15.42 

3680 

60.66 

15.44 

3690 

60.75 

15.45 

3700 

60.83 

15.47 

3710 

60.91 

15.48 

3720 

60.99 

15.49 

3730 

61.07 

15.51 

3740 

61.16 

15.52 

3750 

61.24 

15.54 

3760 

61.32 

15.55 

3770 

61.40 

15.56 

3780 

61.48 

15.58 

3790 

61.56 

15.59 

3800 

61 64 

15.60 

3810 

61.73 

15.62 

3820 

61.81 

15.63 

3830 

61.89 

15.65 

3840 

61.97 

15.66 

3850 

62.05 

15.67 

3860 

62.13 

15.69 

3870 

62.21 

15.70 

3880 

62.29 

15.71 

3890 

62.37 

15.73 

3900 

62.45 

15.74 

3910 

62.53 

15.75 

3920 

62.61 

15.77 

3930 

62.69 

15.78 

3940 

62.77 

15.79 

3950 

62.85 

15.81 

3960 

62.93 

15.82 

3970 

63.01 

15.83 

3980 

63.09 

15.85 

3990 

63.17 

15.86 

4000 

63.25 

15.87 

4010 

63.32 

15.89 

4020 

63.40 

15.90 

4030 

63.48 

15.91 

4040 

63.56 

15.93 

4050 

63.64 

15.94 

4060 

63.72 

15.95 

4070 

63 80 

15.97 

4080 

63.87 

15.98 

4090 

63.95 

15.99 

4100 

64.03 

16.01 

4110 

64.11 

16.02 

4120 

64.19 

16.03 

4130 

64.27 

16.04 

4140 

64.34 

16.06 

4150 

64.42 

16.07 

4160 

64.50 

16.08 

4170 

64.58 

16.10 

4180 

64.65 

16.11 

4190 

64.73 

16.12 

4200 

64.81 

16.13 

4210 

64.88 

16.15 

4220 

64.96 

16.16 

4230 

65.04 

16.17 

4240 

65.12 

16.19 

4250 

65.19 

16.20 

4260 

65.27 

16.21 

4270 

65.35 

16.22 

4280 

65.42 

16.24 

4290 

65.50 

16.25 

4300 

65 57 

16.26 

4310 

65.65 

16.27 

4320 

65.73 

16.29 

4330 

65.80 

16.30 


Num. 

Sq. Rt. 

Cu. Rt 

4340 

65.88 

16.31 

4350 

65.95 

16.32 

4360 

66.03 

16.34 

4370 

66.11 

16.35 

4380 

66.18 

16.36 

4390 

66.26 

16.37 

4400 

66.33 

16.39 

4410 

66.41 

16.40 

4420 

66.48 

16.41 

4130 

66.56 

16.42 

4440 

66.63 

16.41 

4450 

66.71 

16.45 

4460 

66.78 

16.46 

4470 

66.86 

16.47 

4480 

66.93 

16.49 

4490 

67.01 

16 50 

4500 

67.08 

16 51 

4510 

67.16 

16.52 

4520 

67.23 

16.53 

4530 

67.31 

16.55 

4540 

67.38 

16.56 

4550 

67.45 

16.57 

4560 

67.53 

16.58 

4570 

67.60 

16.59 

4580 

67.68 

16.61 

4590 

67.75 

16.62 

4600 

67.82 

16.63 

4610 

67.90 

16.64 

4620 

67.97 

16.66 

4630 

68.04 

16.67 

4640 

68.12 

16.68 

4650 

68.19 

16.69 

4660 

68.26 

16.70 

4670 

68.34 

16.71 

4680 

68.41 

16.73 

4690 

68.48 

16.74 

4700 

68 56 

16.75 

4710 

68.63 

16.76 

4720 

68.70 

16.77 

4730 

68.77 

16.79 

4740 

68.85 

16.80 

4750 

68.92 

16.81 

4760 

68 99 

16.82 

4770 

69.07 

16.83 

4780 

69.14 

16.85 

4790 

69 21 

10.86 

4800 

69.28 

16.87. 

4810 

69 35 

16.88 

4820 

69.43 

16 89 

4830 

69.50 

16.90 

4840 

69.57 

16.92 

4850 

69.64 

16.93 

4860 

69.71 

16.94 

4870 

69.79 

16.95 

4880 

69.86 

16.96 

4890 

69.93 

16.97 

4900 

70.00 

16.98 

4910 

70.07 

17.00 

4920 

70.14 

17.01 

4930 

70.21 

17.02 

4940 

70.29 

17.03 

4950 

70.36 

17.04 

4960 

70.43 

17.05 

4970 

70.50 

17.07 

4980 

70.57 

17.08 

4990 

70.64 

17.09 

5000 

70.71 

17.10 

5010 

70.78 

17.11 

5020 

70.85 

17.12 

5030 

70.92 

17.13 

5040 

70.99 

17.15 

5050 

71.06 

17.16 

5060 

71.13 

17.17 

5070 

71.20 

17.18 

5080 

71.27 

17.19 

5090 

71.34 

17.20 

5100 

71.41 

17.21 

5110 

71.48 

1 7.22 

5120 

71.55 | 

17.24 


Num. 

Sq. Rt. 

On 

5130 

71.62 

17 

5140 

71.69 

17 

5150 

71.76 

17 

6160 

71.83 

17 

5170 

71.90 

17 

5180 

71.97 

17 

5190 

72.04 

17 

5200 

72.11 

17 

5210 

72.18 

17 

5220 

72.25 

17 I 1 

5230 

72.32 

17 

5240 

72.39 

17 

5250 

72.46 

17 

5260 

72.53 

17i 

5270 

72.59 

17i 

5280 

72.66 

17 

5290 

72.73 

17! 

5300 

72.80 

17- 

5310 

72.87 

17. 

5320 

72.94 

171 

5330 

73.01 

17' 

5340 

73.08 

171 

5350 

73.14 

171 

5360 

73.21 

17 1 

5370 

73.28 

171 

5360 

73.35 

17! 

5390 

73.42 

171 

5400 

73.48 

17 

5410 

73.55 

17. 

5420 

73.62 

17' 

5430 

73.69 

171 

5440 

73.76 

17t 

5450 

73.82 

17 * 

5460 

73.89 

17 

5470 

73.96 

171 

5480 

74.03 

17 

5490 

74.09 

17 

5500 

74.16 

17- 

5510 

74.23 

171 

5520 

74.30 

17 

5530 

74.36 

17! 

5540 

74.43 

17' 

5550 

74.50 

17 

5560 

74.57 

17 

5570 

74.63 

17. 

5580 

74.70 

17 

5590 

74.77 

17 

5600 

74.83 

17. 

5610 

74.90 

17 

5620 

74.97 

17 ii 

5630 

75.03 

1*; 1 

5640 

75.10 

lA 

5650 

75.17 

r. 

5660 

75.23 

17 ; 

5670 

75.30 

17 1 

5680 

75.37 

17 

5690 

75.43 

17 

5700 

75.50 

17 

5710 

75.56 

17 

5720 

75.63 

17 

5730 

75.70 

17 

5740 

75.76 

17 

5750 

75.83 

17 

5760 

75.89 

17 

5770 

75.96 

17 

5780 

76.03 

17 

5790 

76.09 

17 

5800 

76.16 

17 

5810 

76.22 

17 

5820 

76.29 

17 

5830 

76.35 

18 

5840 

76.42 

18- 

5850 

76.49 

18 

5860 

76.55 

18 

5870 

76.62 

18 

5880 

76.68 

18 

5890 

76.75 

18 

5900 

76.81 

18 

5910 

76.88 

lb 















































SQUARE AND CUBE ROOTS. 51 


Square Roots and Cube Roots of Numbers from 1000 to 10000 

__ —(Continue!).) 


Num. 

Sq. RtJcu. R 

1 

I Num. 

Sq. Rt. Cu. R 

• 1 Num. 

Sq. Rt 

. Cu. Rt 

I Num. 

Sq. Rt 

. Cu. Rt 

5920 

76 94 

18.09 

| 6710 

81.91 

18.86 

7500 

86.60 

19.57 

8290 

91.05 

20.24 

593C 

77.01 

18.10 

| 6720 

81.98 

18.87 

7510 

86.66 

19.58 

8300 

91.10 

20.25 

5910 

77.07 

18.11 

| 6730 

82.04 

18.88 

7520 

86.72 

19.59 

8310 

91.16 

20.26 

D9oO 

/ 7.14 

18.12 

j 6740 

82.10 

18.89 

7530 

86.78 

19.60 

8320 

91.21 

20.26 

5900 

77.20 

18.13 

6750 

82.16 

18.90 

7540 

86.83 

19.61 

1 8330 

91.27 

20.27 

59 70 

77.27 

18.14 

| 6760 

82.22 

18.91 

7550 

86.89 

19.62 

1 8340 

91.32 

20.28 

5980 

1 7.33 

18.15 

| 6770 

82.28 

18.92 

7560 

86.95 

19.63 

1 8350 

91.38 

20.29 

5990 

77.40 

18.16 

| 6780 

82.34 

18.93 

7570 

87.01 

19.64 

1 8360 

91.43 

20.30 

6000 

77.46 

18.17 

6790 

82.40 

18.94 

7580 

87.06 

19.64 

1 8370 

91.49 

20.30 

6010 

77.52 

18.18 

6800 

82.46 

18.95 

7590 

87.12 

19.65 

8380 

91.54 

20.31 

6020 

77.59 

18.19 

| 6810 

82.52 

18.95 

7600 

87.18 

19.66 

1 8390 

91.60 

20.32 

6000 

77.65 

18.20 

6820 

82.58 

18.96 

7610 

87.24 

19.67 

8400 

91.65 

20.33 

GO 10 

77.72 

18.21 

| 6830 

82.64 

18.97 

7620 

87.29 

19.68 

8410 

91.71 

20.34 

60. <0 

77.78 

18.22 

| 6840 

82.70 

18.98 

7630 

87.35 

19.69 

8420 

91.76 

20.34 

6060 

77.35 

18.23 

6850 

82.76 

18.99 

7640 

87.41 

19.70 

8430 

91.82 

20.35 

6070 

77.91 

18.24 

[ 6860 

82.83 

19.00 

7650 

87.46 

19.70 

8440 

91.87 

20.36 

60*0 

77.97 

18.25 

| 6870 

82.89 

19.01 

7660 

87.52 

19.71 

8450 

91.92 

20.37 

60.)0 

78.04 

18.20 

6880 

82.95 

19.02 

7670 

87.58 

19.72 

1 8460 

91.98 

20.38 

6100 

78.10 

18.27 

6890 

83.01 

19.03 

7680 

87.64 

19.73 

1 8470 

92.03 

20.38 

6110 

<8.17 

18.28 

6900 

83.07 

19.04 

7690 

87.69 

19.74 

8480 

92.09 

20.39 

6120 

75.23 

18.29 

6910 

83.13 

19.05 

7700 

87.75 

19.75 

8490 

92.14 

20.40 

Gl.fO 

78.29 

18.30 

6920 

83.19 

19.06 

7710 

87.81 

19.76 

8500 

92.20 

20.41 

6140 

78.36 

18.31 

6930 

83.25 

19.07 

7720 

87.8t> 

19.76 

8510 

92.25 

20.42 

6150 

78.42 

18.32 

6940 

83.31 

19.07 

7730 

87.92 

19.77 

8520 

92.30 

20.42 

6160 

78.49 

18.33 

6950 

83.37 

19.08 

7740 

87.98 

19.78 

8530 

92.36 

20.43 

6170 

78.55 

18.34 

6960 

83.43 

19.09 

7750 

88.03 

19.79 

8540 

92.41 

20.44 

6180 

78.61 

18.35 

6970 

83.49 

19.10 

7760 

88.09 

19.80 

8550 

92.47 

20.45 

6190 

78.68 

18.36 

6980 

83.55 

19.11 

7770 

88.15 

19.81 

8560 

92.52 

20.46 

6200 

78./4 

18.37 

| 6990 

83.61 

19.12 

7780 

88.20 

19.81 

8570 

92.57 

20.46 

6210 

78.80 

18.38 

| 7000 

83.67 

19.13 

7790 

88.26 

19.82 

8580 

92.63 

20.47 

6220 

78.87 

18.39 

7010 

83.73 

19.14 

7800 

88.32 

19.83 

1 8590 

92.68 

20.48 

62.10 

78.93 

18.40 

| 7020 

83.79 

19.15 

7810 

88.37 

19.84 

8600 

92.74 

20.49 

6210 

78.99 

18.41 

7030 

83.85 

19.16 

7820 

88.43 

19.85 

8610 

92.79 

20.50 

>250 

79.06 

18.42 

7040 

83.90 

19.17 

7830 

88.49 

19.86 

8620 

92.84 

20.50 

5260 

79.12 

18.43 

7050 

83.»o 

19.17 

7840 

88.54 

19.87 

8630 

92.90 

20.51 

1270 

79.18 

18.44 

7060 

84.02 

19.18 

7850 

88.60 

19.87 

8640 

92.95 

20.52 

5280 

79.25 

18.45 

7070 

84.08 

19.19 

7860 

88.66 

19.88 

8050 

93.01 

20.53 

5290 

79 .11 

18.46 

7080 

84.14 

19.20 

7870 

88.71 

19.89 

8660 

93.06 

20.54 

5.100 

79.37 

18.47 

7090 

84.20 

19.21 

7880 

88.77 

19.90 


93.11 

20.54 

5.110 

79.44 

18.48 

7100 

84.26 

19.22 

7890 

88.83 

19.91 

8680 

93.17 

20.55 

5920 

79.50 

18.49 

7110 

84.32 

19.23 

7900 

88.88 

19.92 

8690 

93.22 

20.56 

>330 

79.56 

18.50 | 

7120 

84.38 

19.24 

7910 

88.94 

19.92 

8700 

93.27 

20.57 

)340 

79.62 

18.51 

7130 

84.44 

19.25 1 

7920 

88.99 

19.93 

8710 

93.33 

20.57 

>350 

79.69 

18.52 

7140 

84.50 

19.26 j 

7930 

89.05 

19.94 

8720 

93.38 

20.58 

360 

79.75 

18.53 

7150 

84.56 

19.26 

7940 

89.11 

19.95 

8730 

93.43 

20.59 

370 

79.81 

18.54 

7160 

84.62 

19.27 

7950 

89.16- 

19.96 

8740 

93.49 

20.60 

380 

79.87 

18.55 

7170 

84.68 

19.28 

7960 

89.22 

19.97 

8750 

93.54 

20.61 

390 

79.94 

18.56 

7180 

84.73 

19.29 

7970 

89.27 

19.97 

8760 

93.59 

20.61 

100 

80.00 

18.57 I 

7190 

84.79 

19.30 

7980 

89.33 

19.98 

8770 

93.65 

20.62 

110 

80.06 

18.58 

7200 

84.85 

19.31 

7990 

89.39 

19.99 

8780 

93.70 

20.63 

120 

80.12 

18.59 

7210 

84.91 

19.32 

8000 

89.44 

20.00 

8790 

93.75 

20.64 

430 

80.19 

18.60 

7220 

84.97 

19.33 

8010 

89.50 

20.01 

8800 

93.81 

20.65 

no 

80.25 

18.60 

7230 

85.03 

19.34 

8020 

89.55 

20.02 

8810 

93.86 

20.65 

150 

80.31 

18.61 

7240 

85.09 

19.35 

8030 

89.61 

20.02 

8820 

93.91 

20.66 

460 

80.37 

18.62 

7250 

85.15 

19.35 

8040 

89.67 

20.03 

8830 

93.97 

20.67 

170 

80.44 

18.63 

7260 

85.21 

19.36 

8050 

89.72 

20.04 

8840 

94.02 

20.68 

480 

80.50 

18.64 

7270 

85.26 

19.37 

8060 

89.78 

20.05 

8850 

94.07 

20.68 

190 

80.56 

18.65 

7280 

85.32 

19.38 

8070 

89.83 

20.06 

8860 

94.13 

20.69 

500 

80.62 

18.66 

7290 

85.38 

19.39 

8080 

89.89 

20.07 

8870 

94.18 

20.70 

510 

80.68 

18.67 1 

7300 

85.44 

19.40 

8090 

89.94 

20.07 

8880 

94.23 

20.71 

520 

80.75 

18.68 

7310 

85.50 

19.41 

8100 

90.00 

20.08 

8890 

94.29 

20.72 

>30 

80.81 

18.69 

7320 

85.56 

19.42 

8110 

90.06 

20.09 

8900 

94.34 

20.72 

>40 

80.87 

18.70 

7330 

85.62 

19.43 

8120 

90.11 

20.10 

8910 

94.39 

20.73 

550 

80.93 

18.71 

7340 

85.67 

19.43 

8130 

90.17 

20.11 

8920 

94.45 

20.74 

>60 

80.99 

18.72 

7350 

85.73 

19.44 

8140 

90.22 

20.12 

8930 

94.50 

20.75 

>70 

81.06 

18.73 

7360 

85.79 

19.45 

8150 

90.28 

20.12 

8940 

94.55 

20.75 

>80 

81.12 

18.74 

7370 

85.85 

19.46 

8160 

90.33 

20.13 

8950 

94.60 

20.76 

>90 

81.18 

18.75 

7380 

85.91 

19.47 

8170 

90.39 

20.14 

8960 

94.66 

20.77 

500 

81.24 

18.76 

7390 

85.97 

19.48 

8180 

90.44 

20.15 

8970 

94.71 

20.78 

>10 

81.30 

18.77 

7400 

86.02 

19.49 

8190 

90.50 

20.16 

8980 

94.76 

20.79 

>20 

81.36 

18.78 

7410 

86.08 

19.50 

8200 

90.55 

20.17 

8990 

94.82 

20.79 

>30 

81.42 

18.79 

7420 

86.14 

19.50 

8210 

90.61 

20.17 

9000 

94.87 

20.80 

>40 

81.49 

18.80 

7430 

86.20 

19.51 

8220 

90.66 

20.18 

9010 

94.92 

20.81 

50 

81.55 

18.81 

7440 

86.26 

19.52 

8230 

90.72 

20.19 

9020 

94.97 

20.82 

60 

81.61 

18.81 

7450 

86.31 

19.53 | 

8240 

90.77 

20.20 

9030 

95.03 

20.82 

70 

81.67 

18.82 

7460 

86.37 

19.54 

8250 

90.83 

20.21 

9040 

95.08 

20.83 

80 

81.73 

18.83 | 

7470 

86.43 

19.55 

8260 

90.88 

20.21 

9050 

95.13 

20.84 

90 

81.79 

18.84 | 

7480 

86.49 

19.56 

8270 

90.94 

20.22 

9060 

95.18 

20.85 

00 

81.85 

18.85 | 

7490 

86.54 

19.57 | 

8280 

90.99 

20.23 | 

9070 

95.24 

20.85 





























































52 


SQUARE AND CUBE ROOTS 


Square Roots ami Cube Roots of Numbers from 1000 to 100 ) 

— (Continued.) 


Num. 

Sq. Rt. 

Cu. Rt 

Num. 

| Sq. Rt. 

Cu. Rt. 

Num. 

^ Sq. Rt. 

Cu. Rt. 

Num. 

Sq. Rt. 

I 

Cu. 

9080 

95.29 

20.86 

9320 

96.54 

I 21.04 

9550 

97.72 

21.22 

9780 

98.89 

21 

9090 

95.34 

20.87 

9330 

96.59 

21.05 

9560 

97.78 

21.22 

9790 

98.94 

21 

9100 

95.39 

20.88 

9340 

96.64 

21.06 

9570 

97.83 

21.23 

9800 

98.99 

21 

9110 

95.46 

20.89 

9350 

96.70 

21.07 

9580 

97.88 

21.24 

9810 

99.05 

21 

9120 

95.50 

20.89 

9360 

96.75 

21.07 

9590 

97.93 

21.25 

9820 

99.10 

21 

9130 

95.5» 

20.90 

9370 

96.80 

21.08 

9600 

97.98 

21.25 

9830 

99.15 

21 

9140 

96.00 

20.91 

9380 

96.85 

21.09 

9610 

98.03 

21.26 

9840 

99.20 

21 

9150 

95.66 

20.92 

9390 

96.90 

21.10 

9620 

98.08 

21.27 

9850 

99.25 

21 

9160 

95.71 

20.92 

9400 

96.95 

21.10 

9630 

98.13 

21.28 

9860 

99.30 

21 

9170 

95.76 

20.93 

9410 

97.01 

21.11 

9640 

98.18 

21.28 

9870 

99.35 

21 

9180 

95.81 

20.94 

9420 

97.06 

21.12 

9650 

98.23 

21.29 

9880 

99.40 

21 

#1*0 

95.86 

20.95 

9430 

97.11 

21.13 

9660 

98.29 

21.30 

9890 

99.45 

21 

9200 

95.92 

20.95 

9440 

97.16 

21.13 

9670 

98.34 

21.30 

9900 

99.50 

21 

9210 

95.97 

20.96 

9450 

97.21 

21.14 

9680 

98.39 

21.31 

9910 

99.55 

21. 

9220 

96.02 

20.97 

9460 

97.26 

21.15 

9690 

98.44 

21.32 

9920 

99.60 

21 i 

9230 

96.07 

20.98 

9470 

97.31 

21.16 

9700 

98.49 

21.33 

9930 

99.65 

21i 

9240 

96.12 

20.98 

9480 

9(.37 

21.16 

9710 

98.54 

21.33 

9940 

99.70 

21 i 

9250 

96.18 

20.99 

9490 

97.42 

21.17 

9720 

98.59 

21.34 

9950 

99.75 

21 

9260 

96.23 

21.00 

9500 

97.47 

21.18 

9730 

98.64 

21.35 

9960 

99.80 

21 1 

9270 

96.28 

21.01 

9510 

97.52 

21.19 

9740 

98.69 

21.36 

9970 

99.85 

211 

9280 

96.33 

21.01 

9520 

97.57 

21.19 

9750 

98.74 

21.36 

9980 

99.90 

211 

9290 

96.38 

21.02 

9530 

97.62 

21.20 

9760 

98.79 

21.37 

9990 

99.95 

21- 

9300 

9310 

96.44 

96.49 

21.03 

21.04 

9540 

97.67 

21.21 

9770 

98.84 

21.38 

10000 

100.00 

• 21} 

r 


To find Square or Cube Roots of larg^e numbers not eoi 
tained in the column of numbers of the table. 


Such roots may sometimes be taken at once from the table, by merelv regarding the column 
powers as being columns of numbers; and those of numbers as’being those of roots. Thus, if 
sq rt of 25281 is reqd, first find that number in the column of squares; and opposite to it, in 
column of numbers, is its sq rt 159. For the cube rt of 857375, find that number in the colum 
cubes ; and opposite to it, in the col of numbers, is its cube rt 95. When the exact number is not 
tained in the column of squares, or cubes, as the case may be, we may use instead the number neti 
to it, if no great accuracy is reqd. But when a considerable degree of accuracy is necessary, 
following very correct methods may be used. 


For the square root. 

This rule applies both to whole numbers, and to those which are partly (not wholly) decimal. F 
in the foregoing manner, take out the tabular number, which is nearest to the given one • and als 

tolklilor* on ■»* \ f ,, It tlw., • - l... 1 V. 1 o . . i -a . . , . _ " 


I 


tabular sq rt. Mult this tabular number by 3; to the prod add the given number. Call the sui 
Then mult the given number by 3; to the prod add the tabular number. Call the sum B. Then 


s m 


A : B : 

Ex. Let the given number be 946.53. 
tabular sq rt 30.7734. Hence, 

947 = tab num 
3 


; Tabular root : Reqd root. 

Here we find the nearest tabular number to be 947; an 


U 


2841 

946.53 = given num. 


3787.53 = A. 


and 


946.53 = given num. 
3 


2839.59 

947 = tab num. 


(3786.59 = B. 


T , , B ‘ Tab root. Reqd root. 

Then 3787.53 ; 3 ( 86.59 :: 30.7734 : 30.7657 -}-. 

The root as found by actual mathematical process is also 30.7657 -}-. 

For the cube root. 

This rule applies both to whole numbers, and to those which are partly decimal. First take on 

number A,** ^ th ? 8 ' VCn one : an,i a,so its tabular cube rt. Mult this tab 

w‘S^ e . r „j y J’, h and t0 , th ?, rir .' )d a,i . d * hc 8 lven number. Call the sum A. Then mult the given nui 
by 2, and to the prod add the tabular number. Call the sum B. Then * 

A : B : : Tabular root : Reqd root. , 

tubes'; {; e le^9! V and n its m ta e b r ular SSrtW™ Hence? ** (,n the C0,ura { 

6859= tab num. 't ’ r 7368 = given num. 

2 2 


B. 

3786.59 


Tab root. 
30.7734 


\ 


13718 

7368 = given num. 


► and 


14736 

6859 = tab num. 


21595 = B. 


21086 = A. 

H. Tab Root. 

Then, as 21086 : 21595 : : 19 : 

The root as found by correct mathematical process is 19.4588 


Reqd Rt. 

19.4585 


The engineer rarely requires 













































SQUARE AND CUBE ROOTS. 


53 


i her which is wholly 


abm d iou?ou°e r aCCUraCy ' ^ Ms purpoaes ’ t^refore, this process is greatly preferable to the ordinary 

To find the square root of a mi 

decimal. 

Z7L^i^%T l L°ZZ Ct r° the , ii th l r<i , numeral pg^e inclusive. If the number does not contain at 
; e l . dYer .h- ,U. y ;/'; the / ir * t . »«««*<. and including it, add one or more ciphers to make 

hen l n number is not separable into twos, add another cipher to make it so. 

u h tL b taMe hod the m, h ounleral figure, and including it, assume the number to be a whole one. 
o the table hud the number nearest to this assumed one ; take out its tabular sq rt • move the deci¬ 
le figures?^ ^ bu ar root 10 tbe left ’ hal * as mau >’ P la ces as the Anally modified decimal number 

® x \ " bat is the «q rt of the decimal .002? Here, in order to have at least five decimal figures 
muting from the first numeral (2). and including it, add ciphers thus, .00.20,00 0. But as itls not 
ow separable into twos, add another cipher, thus, .00,20,00,00. Then beginning at the first numeral 

nd H?e U so b V he W h°!f n . umber 20000 °- The uearest to this it the table is 190809 

, u ' e sl J rt of t, h‘ s » s « 7 V. 8 !°! r * the decuual number as finally modified, namely. .00,20 00 00 has 
gilt figures ; one-half of which is 4; therefore, move the decimal point of the root 447 four places to 
.e left; making it .0447. This is the reqd sq rt of .002, correct to the third numeral 7 included 

fo find tlie cube coot of <i number wliich is wholly decimal. 

Very simple, and correct to the third numeral inclusive. 

I tm e ^» l !, 1 i )erd0e8 n o tC0 ? tain . at ! eaSt ti , v , e fig ures ’ counting from the first numeral, and including 
’ ... tith' 8 C1 P berS “ ake fiv e- If. after that, the number is not separable into threes, add 

it mintw 3 8,p i!i ers “ ake 11 *°* Then beginning at the first numeral, and including it, assume 
fi^'tt 8 i t0 b8 i, !1 . Wh »i e ° De : - t le table find the number nearest to this assumed one, and take 

iali^odi^ ° f tbis « * tbe left - 33 p ' a - - the 

F ' x -.Y bat , i3 the cub * ft of the decimal .002 ? Here, in order to have at least five figures, counting 
fi” 1 anueral (2), and including it, add ciphers thus, .002 000,0. But as it is not now separ- 
•'!? '“‘“oil i add c! . phe , rs to make it so; thus, .002.000,000. Then beginning with the 

st numeral (2), assume the decimal to be the whole number 2000000. The nearest cube to this in 
i.>L ab v* 1D *J? e of cubeSl IS 2000376; and its tabular cube rt as found in the col of numbers 

12b. Now, the decimal number as finally modified, namely, .002 000 000, has nine figures; one-third 
13 3; therefore, move the decimal point of the root 1‘26, three places to the left, making it 
2b. This is the reqd cube rt of the decimal .00*2, correct to the third uumeral 6 included. 

To find roots by logarithms, see p 39. 

For tables of fifth roots, fifth powers, and square roots of fifth 
•owers, see pp 251 to 253. 




54 


GEOMETKY. 




GEOMETEY. 


Linos. Fig-tiros. Solids, defined. Strictly speaking a geometrical lin 

Is simply length, or distance. The lines we draw on paper have not only length, but breadth l 
thickness ; still they are the most convenient symbol we can employ for denoting a geometrical li j 

Straight lines are also called right lines. A vortical line is one that poiu 
toward the center of the earth; and a horizontal one is at right angles t i 
vert one. A piano figure is merely any flat surface or area entirely enclosl 

by lines either straight or curved ; which arc called its outliue, boundary, circumf. or periphery ■ 
ofteu confound the outline with the fig itself as when we speak of draw ing circles, squares, Ac; ; 
we actually draw only their outlines. Geometrically speaking, a fig has length and breadth only • > 

thickness. A solid is any body; it has length, breadth, and thickness. 

Geometrically similar figs or solids, are not necessarily of the sai\ 
size; but only of precisely the same shape. Thus, any two squares are,son . 

tifical)y speaking, similar to each other ; so also any two circles, cubes, Ac, no matter how differ 
they may he in size. When they are not only of the same shape, but or the same size, they are s 1 

to be similar, and equal. 


The quantities of linos are to each other simply as their leng'tlis; l, 
the quantities, or areas, or surfaces of similar figures, are as, or in proporti 
to, the squares of any one of the corresponding lines or sides which enclose t- 
figures, or which may be drawn upon them; and the quantities, or solidities 
similar solids, are as the cubes of any of the corresponding lines which foi 

their edges, or the figures by which they are enclosed. 

Hem.— Simple as the following operations appear, it is only hv care, and good instruments t! 

“! M * e to K‘'e accurate results. Several of them can lie much better performed by means o , 
metallic triaugle having one perfectly accurate right angle. In the field, the tape-liue chain o ^ 
measuring-rod wiU take the place of the dividers aud ruler used indoors. ’ 



To divide a given line, a b, into two equal part 


From its ends a and b as centers, and with anv rad greater than one-half of a 
describe the arcs c and d. and join e/. If the llne a l is very long, first lav . 
equal cists « o and b g. each way from thp ends, so as to approach convenient 
near to each other ; and then proceed as if o g were the line to be divided. • - 
measure a b by a scale, aud thus ascertaiu its center. 



To divide a given line, m n, into an 
given n ii in her of equal parts. 


From to and n draw any two parallel lines to o and n 
to an indefinite dist; and on them, from to and n step off t 
reqd number of equal parts of any convenient length : fin: 
l.v, join the corresponding points thus stepped off. Or on 
one line, as mo. may he drawn and stepped off. as to 
then join s n; and draw the other short lines parallel to 


To divide a given line, m n, into two pnrts which shall hav 
a given proportion fo each other. 

^ti^ro^i^Vf iITT"*? ft 1 10 3 ‘ Pi ™ ^ 

„«•! U U,™ .Up* Job/. „ ; „ d p„.5.i ! t *uT„‘iTlTu*?’ ‘ ‘ 


angles. 


Angles. When two straight, or right lines meet each other at anv inelin 
tion, the inclination is called an angle; and is measure! hv ti,„ ^ inc ‘ m 
tained in the arc of a circle described from the point of meeting as a center Si'nof* nil co 

large or small, are supposed to be divided into 3H0 degrees t fohowa that .nv nn^ k c "l c ' es - whetl 

to the Inclination e»».then the Two Vum “are '3d' *t? & ■>»«'» 

perpendicular to each other; and the angles on a and 
o n b, are called right angles ; and are each measd by or 
are equal to. SH>° or one-fourth part of the circumf of a circle. Any I’gle 
as c e d, smaller than a right angle, is called acute or sharp* 

him!t° n vrf ef ' la VP r than a ri - ht a,, g le - is called obt use, or 

•?.««« **•-“ » 



side of SibS'liSarccaito COniigno^^uXl^t' *5.“ « 
V u w are adjacent; also tus and tuw; sut anil * u v • w u t and 1 *i Ub » V u ? an 

angles is always equal to two right ancles • or m ikiio' T i. , Bdwuv. The sum of two adjacei 

grees coutaiued in one of them, aud subtract it from' 180°, wo^obtLiu’the'othcr,‘ U nU “ lber ° f d ' 










GEOMETRY. 


55 



When two straight lines cross each other, forming four 
angles, either pair of those angles which point in exactly 
opposite directions are called opposite, or vertical 
angles ; thus, the pair s u t, and v n w are opposite an¬ 
gles; also the pair s nv and t n w. The opposite angles 
of any pair are always equal to each other. 




When a straight line a b crosses two parallel lines c d, 
ef , the alternate angles which form a kind of 7> are 
equal to each other. Thus, the angles don , and o n /, are 
equal: as are also c o it, and one. Also the sum of the 
two internal angles on the same side of a b, is equal to two 
right angles, or 180°; thus, con -f- o nf — 180°; also 
don-\-one = 180°. 

An interior angle. 

In any fig, is any angle formed inside of that fig, by the meet¬ 
ing of two of its sides, as the angles c a b, a b c, 6 c a, of this 
triangle. All the interior angles of any straight-lined figure of 
any number of sides whatever, are together equal to twice aa 
many right angles minus four, as the figure has sides. Thus, a 
triangle has 3 sides; twice that number is 6; and 6 right angles, 
or 6 X 90° = 540°; from which take4 right angles, or 360°; and 
there remain 180°, which is the number of degrees in every 
plane, or straight-lined triangle. This principle furnishes an 
easy means of testing our measurements of the augles of any 
fig; for if the sum of all our measurements does not agree with 
e sum given by the ruie, it is a proof that we have committed some error. 

An exterior angle 

Of any straight-lined figure, is any angle, as a b d. formed outside of that fig, 
by the meeting of any side, as a b, with the prolongation of an adjacent side, 
as c 5; so likewise the augles eat, and b c w.* All the exterior angles of any 
straight-lined fig, no matter how many sides it may have, amount to 360°; 

but if any of the angles are re-entering, i e, pointing 

inwards, as g ij, the supplements, as g \ x, corresponding to such, must be 
taken as negative or minus. Thus abd-\-bcw-\-cas~ 360°; and y hj -f- 
zji — g ix i g w~ 360°. Angles, as a, b, c, g, A, tmdj, which point out¬ 
ward, are called salient. 


rom any given point, p, on a line s t, 
to draw a perp, p a. 

From p, with any convenient opening of the dividers, step off the 
juals po,p g. From o and g as centers, with any opening greater 
urn half o g, describe the two short arcs b and c; and join a p. 
r still better, describe four arcs, and join a y. 

Or from p with any convenient scale describe two 

tort arcs g and c either one of them with a radius 3, and the other 
ith a rad 4. Then from g with rad 5 describe the arc b. Join pa. 




f the point p is at one end of the line, 
or very near it, 

ixtend the line, if possible, and proceed as above. But if this 
an not be done, then from auy convenient point, w, open the divid- 
rs to p, and describe the semicircle, sp o; through o w draw o w 
; join p s. 

Or use the last foregoing process with 

ads 3, 4, and 5. 



From a given point, o, to let fall a 
perp os, to a given line, nvn. 

From o, measure to the line m n, any two equal dists, o c, 
e; and from c and e as centers, with any opening greater 
ian half of c e, describe the two arcs a and b ; join o t. Or 
•om any point, as d on the line, open the dividers to o, and 
escribe the arc o g ; make i x equal tot'o; and join a,x. 


* An exterior angle a b d, or y hj, is the supplement (p 5fi) 
f the interior angle a b c, or g hj, at the same point b, or A. 

















56 


GEOMETRY, 


If the line, a b, is on the ground. 

And a perp is reqd to be drawn from c, first measure off any two 
equal dists, c to, c n. At to and n, hold the ends of a piece of string, ' 
tape line, or chain, m * n ; then tighten out the string, Ac, as shown 
by to s n ; * being its center. Then will « c be the reqd perp. Or if 
‘.he perp x z is to be drawn from the end of the line w x, first measure x y 
upon the line, and equal to three feet; then holding the end of a tape- 
line at x, and its nine feet mark at y, hold the four feet mark at z keep- 
lug xx and zy equally stretched. Then zx will be the reqd perp, because 
S, 4, and 5, make the sides of a right-angled triangle. Instead of 3, 4, and 
o’ *"y ' nu ’ t, I ,les of those numbers may be used, such as 6, 8, and 10 ; or 
12 , 15, Ac: also instead of feet, we inay use yards, chains, Ac. 


Through a given point, a, to draw a „ 
line, a c, parallel to another line, 

«/. ! 

H ith the perpdist, a e, from any point, n, in e f, describe 
an arc, (; draw a c just touching the arc. —i 




8 




n 


At any point, «, in a line a b, 
lo make an angle en /'.equal 
to a given angle, m n o. 

From n and a, with any convenient rad, describe 
She arcs s /, d e; measure s t, anrt make t d equal 
to it; through a d draw a c. 




To bisect, or divide any angle, tr or y, inti 
two equal parts. 

From x set off any two equal dists, xr, x ». From r and $ with any r 
describe two arcs intersecting, as at o; and join o x. If the two sides 
the angle do not meet, as c / and g h, either first extend them until tli 
do meet; or else draw lines x w, and xy, parallel to them, and at eqi 
dists from them, so as to meet; then proceed as before. 



All angles, as n a m, n o m. at the circnmf of a semicircle, and stan 
Ing on its diam n m, are right angles; or, as it is usually expresse i 

nil angles in j» semicircle nre right, ang'le.* 

An^angle n s x at the centre of a circle, is twice as great as au angle 
to x at the circumf, when both stand upon the same arc n x. 



All angles, as y dp, y e p, y g p, at the circnmf of a circle, and standir 
upon the same arc. as y p, are equal to each other; or, as usually expresse- 

all angles in the same segment of a circlear 
equal. 


The complement Of an angle is what it lacks of 90°. Thus the i 

plemeiit of is 90° — 80° ~~ 10°; and that of 210° is 90° 2UK 3 ~r l 

The supplement of an angle is what it lacks of 180°. Thus the sui 
ment ot 80° is 180° _ 80° = 100°; and that of 210° is 180° -210° = 
But ordiuanly we may ueglect the signs -J- and before complements 
supplements, and call the complement of an uDgle its diff from 90° • 
the supplement its dijf from 180°. 6 v 













ANGLES 


57 


Angles in a Parallelogram. 

A parallelogram is any four-sided straight-lined fig¬ 
ure whose opposite sides are equal, as abed ; or a 
square, &c. Any line drawn across a parallelogram 
between 2 opposite angles, is called a diagonal, as a c, 
or b d. A diag divides a parallelogram into two equal 
parts; as does also any line m n drawn through the 
center of either diag; and moreover, the line m n 
itself is div into two equal parts by the diag. Two 
diags bisect each other; they also divide the parallel¬ 
ogram into four triangles of equal areas. The sum 
of the two angles at the ends of any one side is = 180°; thus, dab -\-abc — ab c + 
bed — 180°; and the sum of the four angles, d a b, a b c, b c d, c d a = 360°. 

The sum of the squares of the four sides, is equal to the sum of the squai'es of tho 
two diags. 

To reduce Minutes and Seconds to Degrees and decimals 

of a Degree, etc. 

In any given angle—- 

Number of degrees — Number of minutes CO. 

= Number of seconds ~~ 3000. 

Number of minutes = Number of degrees X GO. 

= Number of seconds GO. 

Number Of seconds Number of degrees X 3600. 

= Number of minutes X 60. 


a 



Table of Minutes and Seconds in decimals of a Degree. 


Min. 

Deg. 

Min. 

Deg. 

Min. 

Deg. 

Sec. 

Deg. 

Sec. 

Deg. 

Sec. 

Deg. 

i 

.016666 

21 

.350000 

41 

.683333 

1 

.000278$ 

21 

.005833 

41 

.011389 

2 

.033333 

22 

. 366666 

42 

.700000 

2 

.000556 

22 

.006111 

42 

.011667 

3 

.050000 

23 

.383333 

43 

.716666 

3 

.000833 

23 

.006389 

43 

.011944 

4 

.066666 

24 

.400000 

44 

.733333 

4 

.001111 

24 

.006667 

44 

.012222 

5 

.083333 

25 

.416666 

45 

.750000 

5 

.001389 

25 

.006914 

45 

.012500 

6 

.100000 

26 

.433333 

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58 


ANGLES, 




To Measure Angles by a 2 Ft Rule, etc. 


The lour lingers of the hand, held at right angles to the arm, and a 

arms-leiigih from the eye. cover about 7 degrees. Aud 7 C corresponds to about 12.2 ft iu 100 ft; or ti • 
36.6 ft iu 100 yds; or to 615 ft iu a mile; or iu the same proportion as the distance. 


The following: Table 

may sometimes lie found useful for the rough measurement of angles, either on a drawing, or be ' 
tween distant objects in the field. If the inner edges of a common two-foot rule be opened to the ex 
tent shown iu the column of inches, its edges will be inclined at the angles shown in the columns o 
angles. Since an opening of of an inch up to 19 inches or about 105°, corresponds to from abon 
% to 1°, no great accuracy is to be expeeted; and beyond 105° still less; the liability to error in 
creasing very rapidly as the opening becomes greater. Thus, the last X inch corresponds to about 12 c 
As to the talde itself, angles for openings intermediate of those therein given, may be calculated t J 
the nearest minute or two, by simple proportion, up to 23 inches of opening, or about 147°. 


Table of Angles corresponding' to openings of a 2-foot rule 

(Original.) 

D, degrees; M, minutes. Correct. 


Ins. 

D. 

M. 

Ins. 

D. 

M. 

Ins. 

D. 

M. 

Ins. 

D. 

M. 

Ins. 

D. 

M. 

Ins. 

I>. 

M 

X 

1 

12 

*X 

20 

24 

&X 

40 

13 

V2X 

61 

23 

16^ 

85 

14 

20‘i 

115 

5 


1 

48 


21 



40 

51 


62 

5 

86 

3 


116 

12 

Vi 

2 

24 

X 

21 

37 

X 

41 

29 

X 

62 

47 

X 

86 

52 


117 

20 

X 

3 

00 


22 

13 


42 

7 

63 

28 

87 

41 

118 

30 

3 

36 

X 

22 

50 

X 

42 

46 

X 

61 

11 

X 

88 

31 

X 

119 

46 


4 

11 


23 

27 


43 

21 

61 

53 

89 

21 

120 

521 

1 

4 

47 

5 

24 

3 

9 

44 

3 

13 

65 

35 

17 

90 

12 

21 

122 

6 


5 

23 


24 

39 


44 

42 


66 

18 


91 

3 


123 

20 

X 

5 

58 

X 

25 

16 

X 

45 

21 

X 

67 

1 

X 

91 

54 

X 

124 

36 

X 

6 

34 


25 

53 


45 

59 

67 

44 

92 

46 

125 

54 

7 

10 

X 

26 

30 

X 

46 

38 

X 

68 

28 

X 

93 

38 

X 

127 

14 

X 

7 

46 


27 

7 


47 

17 

69 

12 

94 

31 

128 

35 

8 

22 

X 

27 

44 

X 

47 

56 

X 

69 

55 

X 

95 

24 

X 

129 

52 


8 

5H 


28 

21 


48 

35 

70 

38 

96 

17 

131 

25 

2 

9 

34 

6 

28 

58 

10 

49 

15 

14 

71 

22 

18 

97 

11 

22 

132 

53 


10 

10 


29 

35 


49 

54 


72 

6 


98 

0 


134 

ti 

x 

10 

46 

X 

30 

n 

X 

50 

34 

X 

72 

51 

X 

99 

00 

X 

135 

5> 


11 

22 


30 

49 


51 

13 

73 

36 

99 

55 

137 

35 

X 

11 

58 

X 

31 

26 

X 

51 

53 

X 

74 

21 

X 

100 

51 

X 

139 

H 

X 

12 

34 


32 

3 


52 

33 

75 

6 

101 

4« 

141 

1 

13 

10 

X 

32 

40 

X 

53 

13 

X 

75 

51 

X 

102 

45 

X 

142 

51 


13 

46 


33 

17 

53 

53 

75 

36 

103 

43 

144 

•if 

3 

14 

22 

7 

33 

54 

11 

54 

34 

15 

77 

22 

19 

104 

41 

23 

146 

4> 

y* 

14 

58 


34 

33 


55 

14 


78 

8 


105 

40 


148 

5." 

15 

34 

X 

35 

10 

X 

55 

55 

X 

78 

54 

X 

106 

39 

X 

151 

17 

X 

16 

10 


35 

47 


56 

35 


79 

40 

107 

40 

153 

4f i 

16 

46 

X 

3(5 

25 

X 

57 

16 

X 

80 

27 

X 

108 

41 

X 

156 

34 

X 

17 

22 


37 

3 


57 

57 


81 

14 

109 

43 

159 

4 : 

17 

59 

X 

37 

41 

X 

58 

38 

X 

82 

2 

x 

110 

46 

X 

163 

27 


18 

35 

8 

38 

19 

59 

19 

82 

49 


Ill 

49 

16M 

n 

4 

19 

12 

38 

57 

12 

60 

oo 

;6 

83 

37 

20 

112 

53 

24 

180 

04 


19 

48 


39 

35 


60 

41 


84 

26 


113 

58 



Or this table may be used thus. From any point measure 12 ( 

towards each object, and place marks. Measure the dist in ft between thes 
marks. Sojipo.se the first cols in the table to be ft instead of ins; then opposit 
the dist in ft will be the angle. One-eightli of a ft is 1.5 ins. 

The following is a good way to measure an angle. Measur 

100 or any other number of ft towards each object, and place marks. Measure th 
dist between the marks. Then 


As dist measured . 1 . . Half the dist . nat sine of 
toward one object •-*-•• between marks • Half the angle. 

Find this nat sine in the table of nat sines, take out the corresponding an<d( 
and multiply it by 2. See near foot of p 114. y b ° 























SINES, TANGENTS, ETC 


59 


Sines, Tangents, Ac. 



Sine, a s, of any angle, a c ft, or which is the same thing, the sine of any circular arc, a 6, 
which subtends or measures the angle, is a straight line drawn from one eud, as a, ot the arc, at right 
angles to, and terminating at, the rad c ft, drawn to the other end b of the arc. It is, therefore, equal 
to half the chord a n, of the arc ahn, which is equal to twice the arc ab ; or, the sine of an angle is 
always equal to half the chord of twice that angle; and vice versa, the chord of an angle is always 
equal to twice the sine of half the angle. 

The sine t c of an angle teb, or of an arc 
{ a b, of 90°, is equal to the rad of the arc 
3 r of the circle ; and this sine of 90° is 
greater than that of any other angle. 

Cosine c s of an angle a cb, 

is that part of the rad which lies between 
the sine and the center of the circle. It 
is always equal to the sine y a of the 
complement tea of a c b; or of what a 
c b wants of being90°. The prefix co be¬ 
fore siues, &c, means complement; thus, 
cosine means sine of the complement. 

Versed si n e s b of any angle 

a cb, is that part of the diam which lies 
between the siue, and the outer end b. 

It is very common, but erroneous, when 
speaking of bridges, &c, to call the rise 
or height 3 b of a circular arch a b n, its 
versed sine: while it is actually the versed 
sine of only half the arch. This absurdity 
should cease; for the word rise or height 
is not only more expressive,but is correct. 

Tangen t b wor a d, of any angle 
a c b, is a line drawn from, and at right 
augles to, the end 6 or a of either rad c b, 
or c a, which forms one of the legs of the 
angle; and terminating as at w. or d, in « 
the prolongation of the rad which forms 
the other leg. This last rad thus pro¬ 
longed, that is, cic, or c d,as the case may 

be, is the secant of the angle 

a c b. The angle t cb being supposed 

to be equal to 90°, the angle tea becomes the complement of the angle a cb, or what a c b wants 
of being 90°; and the sine y a of this complement; its versed sine t y ; its tangent t o ; and its secant 
c o, are respectively the co sine, co-versed sine; co tangent; and co-secant, of the angle a c b. Or, 
vice versa, the sine, &c, of a c b, are the cosine, &c, of tea; because the angle a c b is the comple¬ 
ment of the angle lea. When the rad c ft, c a, or c t, is assumed to be equal to unity, or 1, the cor¬ 
responding sines, tangents, &c, are called natural ones; and their several lengths for ditf angles, 
for said rad of unity, have been calculated: constituting the well-known tables of nat siues, &c. In 
any circle whose rad is either larger or smaller than 1, the sines, &c, of the augles will be in the 
same proportion larger or smaller than those in the tables, and are consequently found by mult the 
sine, &c, of the table, by said larger or smaller rad. 


The following: table of natural sines. «Src. does not contain nat 
versed sines, co-versed sines, secants, nor cosecants, but these may be found thus; 
for any angle not exceeding 90 degrees. 

Versed Sine. From 1 take the nat cosine. 

Co-versed Sine, l-’rom 1 tal:e the nat sine. 

Secant. Divide 1 hy the nat cosine. 

Cosecant. Divide 1 by the nat sine. 


For angles exceeding 90° ; to find the sine, cosine, tangent, cotang, secant, or cosec, (but not 
the versed sine or co-versed sine), take the angle from 180°: if between 180° and 270° take 1S0° from 
the angle : if bet 270° and 360°, take the angle from 360°. Then in each case take from the table the 
sine, cosine, tang, or cotang of the remainder. Find its secant or cosec as directed above, for the 
versed sine; if between 90° aud 270°, add cosine to 1; if bet 270° and 360°, take cosine from 1. ( lie 
engineer seldom needs sines, &c, exceeding 180°. 

To find the nat sine, cosine, tang:, secant, versed sine, Ac, 
of’ Hit containing 1 soconiJs. First find that duo to the given deg 

and miu ; then the next greater one. Take their diff. Then as 60 sec are to this diff, so are the sec 

only of the given angle to a dec quantity to be acltictl to the one first taken out 
if it is a sine, taug. secant, Ac ; or to be subtracted from it if it is a cosine, 

cotang, cosecant, Ac. , . . . „ 

The tangents in the table are strict trig’ononietrical ones, that is, 

tamrents to coven anqles ; and which must extend to meet the secants of the augles 
tfwUich Ihly Slo Ordinary, or t-. ometrical tangents, «. t .oso on 
Pl« uiay extern! ns far as we plearo. In the Held practice of railroad 
eurvcs. two trigonometrical tangents terminate where they meet each other. 
Each of these tangs is the tang of half the curve It is usually, hut improperly 
called “tin* tang of the curve." “Apex dist of the curve,” as suggested by Mr 
Shunk, would he hotter. 









NATURAL SINES AND TANGENTS TO A RADIUS 1. 


60 





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TABLE OF CHORDS. 


105 



Tlie table Of chords, below, furnishes the means of laying down angles on 
| P il P (,r more accurately than by an ordinary protractor. To do this, after having drawn 
and measured the first side (say ac) of the figure that is 
to be plotted; from its end c as a center, describe an arc 
ny of a circle of sufficient extent to subtend the angle at 
that point. The rad cn with which the arc is described 
should be as great as convenience will permit; and it is to 
be assumed as unity or 1; and must be decimally divided, 
and subdivided, to be used as a scale for laying down the 
chords taken from the table, in which their lengths are 
given in parts of said rad 1. Having described the arc, find 
in the table the length of the chord n t corresponding to 
the angle act. Let us suppose this angle to be 45°; then 
we find that the tabular chord is .7654 of our rad 1. There¬ 
fore from n we lay off the chord nt, equal to .7654 of our radius-scale; and the line 
cs drawn through the point t will form the reqd angle act of 45°. And so at each 
angle. The degree of accuracy attained will evidently depend on the length of the 
rad, and the neatness of the drafting. The method becomes preferable to the com¬ 
mon protractor in proportion as the lengths of the sides of the angles exceed the rad 
of the protractor. With a protractor of 4 to 6 ins rad, and with sides of angles not 
much exceeding the same limits, the protractor will usually be preferable. The di- 
■ viders in boxes of instruments are rarely fit for accurate arcs of more than about 6 
j ins diam. In practice it is not necessary to actually describe the whole arc, but 
merely the portion near t, as well as can be judged by eye. We thus avoid much use 
!j the India-rubber, and dulling of the pencil-point. For larger radii we may dis- 
; pense with the dividers, and use a straight strip of paper with the length of the rad 
marked on one edge; and by laying it from c towards, and at the same time placing 
another strip (with one edge divided to a radius-scale) from n toward t, we can 
by trial find their exact point of intersection at the required point t. In such mat¬ 
ters, practice and some ingenuity are very essential to satisfactory results. We can¬ 
not devote more space to the subject. 


CHORDS TO A RADIUS 1. 


M. 

o° 

1 10 

| 2° 

| 3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

M, 

0 

.001(0 

.0175 

.0349 

.0524 

.0698 

.0872 

.1047 

.1221 

.1395 

.1569 

.1743 

O' 

2 

.0008 

.0180 

.0355 

.0529 

.0704 

.0878 

.1053 

.1227 

.1401 

.1575 

.1749 

2 

4 

.0012 

.0183 

.0361 

.0535 

.0710 

.0884 

.1058 

.1233 

.1407 

.1581 

.1755 

4 

6 

.0017 

.0192 

.0366 

.0541 

.0715 

.0890 

.1064 

.1238 

.1413 

.1587 

.1761 

6 

8 

.0023 

.0198 

.0372 

.0547 

.0721 

.0896 

.1070 

.1244 

.1418 

.1592 

.1766 

8 

10 

.002.9 

.0204 

.0378 

.0553 

.0727 

.0901 

.1076 

.1250 

.1424 

.1598 

.1772 

10 

12 

.0035 

.0209 

.0384 

.0558 

.0733 

.0907 

.1082 

.1256 

.1430 

.1604 

.1778 

12 

14 

.0041 

.0215 

.0390 

.0564 

.0739 

.0913 

.1087 

.1262 

.1436 

.1610 

.1784 

14 

16 

.0017 

.0221 

.0396 

.0.570 

.0745 

.0919 

.1093 

.1267 

.1442 

.1616 

.1789 

16 

18 

.0052 

.0227 

.0101 

.0576 

.0750 

.0925 

.1099 

.1273 

.1447 

.1621 

.1795 

18 

20 

.0058 

.0233 

.0407 

.0582 

.0756 

.0931 

.1105 

.1279 

.1453 

.1627 

.1801 

20 

22 

.0064 

.0239 

.0113 

.0588 

.0762 

.0936 

.1111 

.1285 

.1459 

.1633 

.1807 

22 

21 

.0070 

.0244 

.0419 

.0593 

.0768 

.0942 

.1116 

.1291 

.1465 

.1639 

.1813 

24 

26 

.0076 

.0250 

.0425 

.0599 

.0774 

.0948 

.1122 

.1296 

.1471 

.1645 

.1818 

26 

28 

.0081 

.0256 

.04.30 

.0805 

.0779 

.0954 

.1128 

.1302 

.1476 

.1650 

.1824 

28 

30 

.0087 

.0262 

.0436 

.0611 

.0785 

.0960 

.1134 

.1308 

.1482 

.1656 

.1830 

30 

32 

.0093 

.0268 

.0442 

.0617 

.0791 

.0965 

.1140 

.1314 

.1488 

.1662 

.1836 

32 

34 

.0099 

.0273 

.0448 

.0622 

.0797 

.0971 

.1145 

.1.320 

.1494 

.1668 

.1842 

34 

36 

.0105 

.0279 

.0454 

.0623 

.0803 

.0977 

.1151 

.1325 

.1500 

.1674 

.1847 

36 

38 

.0111 

.0285 

.0460 

.0634 

.0808 

.0983 

.1157 

.1331 

.1505 

.1679 

.1853 

38 

40 

.0116 

.0291 

.0165 

.0640 

.0814 

.0989 

.1163 

.1337 

.1511 

.1685 

.1859 

40 

42 1 

.0122 

.0297 

.0171 

.0646 

.0820 

.0994 

.1169 

.1343 

.1517 

.1691 

.1865 

42 

41 

.0128 

.0303 

.0177 

.0651 

.0826 

.1000 

.1175 

.1349 

.152.3 

.1697 

.1871 

44 

46 

.0134 

.0308 

.0483 

.0657 

.0A32 

.1006 

.1180 

.1355 

.1529 

.1703 

.1876 

46 

4S 

.0110 

.0314 

.0489 

.066.3 

.0838 

.1012 

.1186 

.1360 

.15.34 

.1708 

.1882 

48 

50 

.0145 

.0320 

.0494 

.0669 

.0843 

.1018 

.1192 

.1366 

.1540 

.1714 

.1888 

50 

52 

.0151 

.0326 

.0500 

.0675 

.0849 

.1023 

.1198 

.1372 

.1546 

.1720 

.1894 

52 

54 

.0157 

.0332 

.0506 

.0681 

.0855 

.1029 

.1204 

.1378 

.1552 

.1726 

.1900 

54 

56 | 

.0163 

.03.37 

.0512 

.0686 

.0861 

.10.35 

.1209 

.1384 

.1558 

.1732 

.1905 

56 

58 ! 

.0169 

.0343 

.0518 

.0692 

.0807 

.1041 

.1215 

.1389 

.15:3 

.1737 

.1911 

58 

60 i 

.0175 

.0349 

.0524 

.0698 

.0872 

.1047 

.1221 

.1395 

.1569 

.1743 

.1917 

60 

































































































106 


TABLE OF CHORDS 


Table of Chords, in parts of a rad 1; for protracting'— Continued 


M. 

n° 

12° 

13° 

14° 

15° 

ir>° 

17° 

18° 

19° 

20° 

M. 

0 

.1017 

.2091 

.2264 

.2437 

.2611 

.2783 

.2956 

.3129 

.3301 

.3473 

O' 

2 

.1923 

.2096 

.2270 

.2443 

.2616 

.2789 

.2962 

.3134 

.3307 

.3479 

2 

l 

.1928 

.2102 

.2276 

.2449 

.2622 

.2795 

.2968 

.3140 

.3312 

.3484 

4 

6 

.1934 

.2108 

.2281 

.2455 

.2628 

.2801 

.2973 

.3146 

.3318 

.3490 

6 

8 

.1940 

.2114 

.2287 

.2460 

.2634 

.2807 

.2979 

.3152 

.3324 

.3496 

8 

10 

.1946 

.2119 

.2293 

.2466 

.2639 

.2812 

.2985 

.3157 

.3330 

.3502 

10 

12 

.1952 

.2125 

.2299 

.2472 

.2645 

.2818 

.2991 

.3163 

.3335 

.3507 

12 

14 

.1957 

.2131 

.2305 

.2478 

.2651 

.2824 

.2996 

.3169 

.3341 

.3513 

14 

16 

.1963 

.2137 

.2310 

.2484 

.2657 

.2830 

.3002 

.3175 

.3347 

.3519 

16 

18 

.1969 

.2143 

.2316 

.2489 

.2662 

.2835 

.3008 

.3180 

.3353 

.3525 

18 

20 

.1975 

.2148 

.2322 

.2495 

.2668 

.2841 

.3014 

.3186 

.3:158 

.3530 

20 

22 

.1981 

.2154 

.2328 

.2501 

.2674 

.2847 

.3019 

.3192 

.3364 

.3536 

22 

»4 

.1980 

.2160 

.2333 

.2507 

.2680 

.2853 

.3025 

.3198 

.3370 

.3542 

24 

2 (i 

.1992 

.2166 

.2339 

.2512 

.2685 

.2858 

.3031 

.3203 

.3376 

.3547 

26 

28 

.1998 

.2172 

.2345 

.2518 

.2691 

.2864 

.3037 

.3209 

.3381 

.3553 

28 

30 

.2004 

.2177 

.2351 

.2524 

.2697 

.2870 

.3042 

.3215 

.3387 

.3559 

30 

32 

.2010 

.2183 

.2357 

.2530 

.2703 

.2876 

.3048 

.3221 

.3393 

.3565 

32 

34 

.2015 

.2189 

.2362 

.2536 

.2709 

.2881 

.3054 

.3226 

.3398 

.3570 

34 

30 

.2021 

.2195 

.2368 

.2541 

.2714 

.2887 

.3060 

.3232 

.3404 

.3576 

36 

38 

.2027 

.2200 

.2374 

.2547 

.2720 

.2893 

.3065 

.3238 

.3410 

.3582 

38 

40 

.2033 

.2206 

.2380 

.2553 

.2726 

.2899 

.3071 

.3244 

.3416 

.3587 

40 

42 

.2038 

.2212 

.2385 

.2559 

.2732 

.2904 

.3077 

.3249 

.3421 

.3593 

42 

44 

.2044 

.2218 

.2391 

.2564 

.2737 

.2910 

.3083 

.3255 

.3427 

.3599 

44 

4I> 

.2050 

.2224 

.2397 

.2570 

.2743 

.2916 

.3088 

.3261 

.3433 

.3695 

46 

48 

.2056 

.2229 

.240.3 

.2576 

.2749 

.2922 

.3094 

.3267 

.3439 

.3610 

48 

50 

.2062 

.2235 

.2409 

.2582 

.2755 

.2927 

.3100 

.3272 

.3444 

.3616 

50 

52 

.2067 

.2241 

.2414 

.2587 

.2760 

.2933 

.3106 

.3278 

.3450 

.3622 

52 

54 

.2073 

.2247 

.2420 

.2593 

.2766 

.2939 

.3111 

.3284 

.3456 

.3628 

54 

56 

.2079 

.2253 

.2426 

.2599 

.2772 

.2945 

.3117 

.3289 

.3462 

.3653 

56 

58 

.2085 

.2258 

.2432 

.2605 

.2778 

.2950 

.3123 

.3295 

.3467 

.3639 

58 

60 


.2264 

• 2437 

.2611 

.2783 

.2956 

.3129 

.3301 

.3473 

.5645 

60 


M. 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

30° 

M. 

O' 

.3645 

.3816 

.3987 

.4158 

.4329 

.4499 

.4669 

.4838 

.5008 

.5176 

O' 

2 

.3650 

.3822 

.3993 

.4164 

.4334 

.4505 

.4675 

.4844 

.5013 

.5182 

2 

4 

.3656 

.3828 

.3999 

.4170 

.4340 

.4510 

.4680 

.4850 

.5019 

.5188 

4 

6 

.3662 

.3833 

.4004 

.4175 

.4346 

.4516 

.4686 

.4855 

.5024 

.5193 

6 

8 

.3668 

.3839 

.4010 

.4181 

.4352 

.4522 

.4692 

.4861 

.5030 

.5169 

8 

10 

.3673 

.3845 

.4016 

.4187 

.4357 

.4527 

.4697 

.4867 

.5036 

.5204 

10 

12 

.3679 

.3850 

.4022 

.4192 

.4363 

.4533 

.4703 

.4872 

.5041 

.5210 

12 

14 

.3685 

.3856 

.4027 

.4198 

.4369 

.4539 

.4708 

.4878 

.5047 

.5216 

14 

16 

.3690 

.3862 

.4033 

.4204 

.4374 

.4544 

.4714 

.4884 

.5053 

.5221 

16 

18 

.3696 

.3868 

.4039 

.4209 

.4:180 

.4550 

.4720 

.4889 

.5058 

.5227 

18 

20 

.3702 

.3873 

.4044 

.4215 

.4.186 

.4556 

.4725 

.4895 

.5064 

.52:13 

20 

22 

.3708 

.3879 

.4050 

.4221 

.4391 

.4561 

.4731 

.4901 

.5070 

.5238 

22 

24 

.3713 

.3885 

.4056 

.4226 

.4397 

.4567 

.4737 

.4906 

.5075 

.5244 

24 

26 

3719 

.3890 

.4061 

.4232 

.4403 

.4573 

.4742 

.4912 

.5081 

.5249 

26 

28 

.3725 

.3896 

.4067 

.4238 

.4408 

.4578 

.4748 

.4917 

.5086 

.5255 

28 

30 

.3730 

.3902 

.4073 

.4244 

.4414 

.4584 

.4754 

.4923 

.5092 

.5261 

30 

82 

.3736 

.3908 

.4079 

.4249 

.4420 

.4590 

.4759 

.4929 

.5098 

.5266 

32 

JR 

.3742 

.3913 

.4084 

.4255 

.4425 

.4595 

.4765 

.4934 

.5103 

.5272 

34 

36 

.3748 

.3919 

.4090 

.4261 

.4431 

.4601 

.4771 

.4940 

.5109 

.5277 

36 

38 

.3753 

.3925 

.4096 

.4266 

.4437 

.4607 

.4776 

.4946 

.5115 

.5283 

38 

40 

.3759 

.3930 

.4101 

.4272 

.4442 

.4612 

.4782 

.4951 

.5120 

.5289 

40 

42 

.3765 

.3936 

.4107 

.4278 

.4448 

.4618 

.4788 

.4957 

.5126 

.5294 

42 

44 

.3770 

.3942 

.4113 

.4283 

.4454 

.4624 

.4793 

.4263 

.5131 

.5300 

44 

46 

.3776 

.3947 

.4118 

.4289 

.4459 

.4629 

.4799 

.4968 

.5137 

.5306 

46 

48 

.3782 

.3953 

.4124 

.4295 

.4165 

.4635 

.4805 

.4974 

.5143 

.5311 

48 

50 

.3788 

.3959 

.413<l 

.4300 

.4471 

.4641 

.4810 

.4979 

.5148 

.5317 

50 

52 

.3793 

.3965 

.4135 

.4306 

.4476 

.4646 

.4816 

.4985 

.5154 

.5322 

52 

54 

.3799 

.3970 

.4141 

.4312 

.4482 

.4652 

.4822 

.4991 

.5160 

.5328 

54 

56 

.3805 

.3976 

.4147 

.4317 

.4488 

.4fi58 

.4827 

.4996 

.5165 


56 

58 

.3810 

.3982 

.4153 

.4323 

.4493 

.4663 

.4833 

.5002 

.5171 

. 53:49 

58 

60 

.3816 

.3987 

.4158 

.4329 

.4499 

.4669 

.4838 

.5008 

.5176 

.5345 

60 



































































































































































TABLE OF CHORDS. 


307 


Table of chords, in parts of a rad 1; for protracting: — Continued 


M. 

31° 

32° 

311° 

34° 

35° 

36° 

37° 

38° 

39° 

40° 

M. 

O' 

.5315 

.5513 

.5680 

.5847 

.6014 

.6180 

.6316 

.6511 

.6676 

.6810 

0 

2 

.5350 

.5518 

.5686 

.5853 

.6020 

.6186 

.6352 

.6517 

.6682 

.6816 

2 

4 

.5356 

.5524 

.5<i01 

.5859 

.6025 

.6101 

.6357 

.6522 

.6687 

.6851 

4 

6 

.5362 

.5530 

.5607 

.5864 

.6031 

.6197 

.6363 

.6528 

.6693 

.6857 

6 

8 

.5367 

.5535 

.5703 

.5870 

.6036 

.6202 

.6368 

.6533 

.6698 

.6862 

8 

10 

.5373 

.5511 

.5708 

.5875 

.6012 

.6208 

.6371 

.6539 

.6704 

.6868 

10 

12 

.5378 

.5516 

.5714 

.5881 

.6017 

.6214 

.6379 

.6544 

.6700 

.6873 

12 

14 

.5384 

.5552 

.5710 

.5886 

.6053 

.6210 

.6385 

.6550 

.6715 

.6879 

14 

16 

.5300 

.5557 

.5725 

.5892 

.6058 

.6225 

.6300 

.6555 

.6720 

.6881 

16 

18 

.5305 

.5563 

.5730 

.5807 

.6061 

.6230 

.6396 

.6561 

.6725 

.6890 

18 

•20 

.5101 

.5569 

.5736 

.5003 

.6070 

.6236 

.6101 

.6566 

.6731 

.0895 

20 

22 

.5106 

.5574 

.5742 

.5900 

.6075 

.6211 

.6107 

.6572 

.6736 

.6901 

22 

21 

.5112 

.5580 

•5147 

.5014 

.6081 

.6247 

.6112 

.6577 

.6742 

.6906 

24 

26 

.5118 

.5585 

.5(53 

.5920 

.6086 

.6252 

.6418 

.6583 

.6747 

.6911 

26 

28 

.5123 

.5501 

.5758 

.5025 

.6002 

.6258 

.6423 

.6588 

.6753 

.6917 

28 

30 

.5420 

.5597 

.5764 

.5031 

.6097 

.0263 

.6129 

.6594 

.6758 

.6922 

30 

32 

.5134 

.5602 

.5769 

.5936 

.6103 

.6260 

.6134 

.6500 

.6764 

.6928 

32 

31 

.5410 

.5608 

.5775 

.5942 

.6108 

.6271 

.6140 

.0005 

.6760 

.6933 

34 

36 

.5116 

.5613 

.5781 

.5047 

.6111 

.6280 

.6445 

6610 

.6775 

.6939 

36 

38 

.5451 

.5619 

.5786 

.5053 

.6119 

.6285 

.6151 

.6616 

.6780 

.6944 

38 

40 

.5457 

.5625 

.5702 

.5059 

.6125 

.6291 

.6456 

.6621 

.6786 

.6950 

40 

42 

.5162 

.5630 

.5797 

.5964 

.6130 

.6206 

.6162 

.6627 

.6791 

.6955 

42 

41 

.5468 

.5636 

.5803 

.5970 

.6136 

.6302 

.6167 

.6632 

.6797 

.6961 

44 

46 

.5471 

.5641 

.5808 

.5975 

.6142 

.6307 

.6473 

.6638 

.6802 

.6966 

46 

48 

.5170 

.5647 

.5814 

.5981 

.6147 

.6313 

.6478 

.6643 

.6808 

.6971 

48 

50 

.5485 

.5652 

.5820 

.5086 

.6153 

.6318 

.6184 

.6640 

.6813 

.6077 

50 

52 

.5400 

.5658 

.5825 

.5992 

.6158 

.6324 

.6489 

.0054 

.6819 

.6982 

52 

54 

.5106 

.5664 

.5831 

.5097 

.6164 

.6330 

.6495 

.6660 

.6824 

.6988 

54 

56 

.5502 

.5660 

.5836 

.6003 

.6169 

.6335 

.6500 

.0605 

.6829 

.6993 

56 

58 

.5507 

.5675 

.5842 

.6000 

.6175 

.6311 

.6506 

.6671 

.0835 

.6999 

58 

60 

.5513 

.5680 

.5847 

.6014 

.6180 

.6346 

.6511 

.t»676 

.6840 

.7004 

60 


M. 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

M. 

0' 

.7004 

.7167 

.7330 

.7492 

.7654 

.7815 

.7975 

.8135 

.8294 

.8152 

O' 

2 

.7010 

.7173 

.7335 

.7198 

.7659 

.7820 

.7980 

.8140 

.8299 

.8458 

2 

4 

.7015 

.7178 

.7311 

.7503 

.7664 

.7825 

.7986 

.8145 

.8304 

.8163 

4 

6 

.7020 

.7184 

.7316 

.7508 

.7670 

.7831 

.7991 

.8151 

.8310 

.8468 

6 

8 

.7026 

.7189 

.7352 

.7514 

.7675 

.7836 

.7996 

.8156 

.&315 

.8173 

8 

10 

.7031 

.7105 

.7357 

.7519 

.7681 

.7841 

.8002 

.8161 

.8320 

.8479 

10 

12 

.7037 

.7200 

.7362 

.7524 

.7686 

.7817 

.8007 

.8167 

.8326 

.8184 

12 

11 

.7012 

.7205 

.7368 

.7530 

.7691 

.7852 

.8012 

.8172 

.8331 

.8189 

11 

16 

.7048 

.7211 

.7373 

.7535 

.7697 

.7857 

.8018 

.8177 

.8336 

.8495 

16 

18 

.7053 

.7216 

.7379 

.7511 

.7702 

.7863 

.8023 

.8183 

.8341 

.8500 

18 

20 

.7059 

.7222 

.7381 

.7516 

.7707 

.7868 

.8028 

.8188 

.8347 

.8505 

20 

22 

.7061 

.7227 

.7390 

.7551 

.7713 

.7873 

.8031 

.8193 

.8352 

.8510 

22 

24 

.7069 

.7232 

.7395 

.7557 

.7718 

.7879 

.8039 

.8198 

.8357 

.8516 

21 

26 

.7075 

.7238 

.7100 

.7562 

.7723 

.7884 

.8041 

.8204 

.8363 

.8521 

26 

28 

.7080 

.7243 

.7406 

.7568 

.7720 

.7890 

.8050 

.8209 

.8368 

.8526 

28 

30 

.7086 

.7249 

.7411 

.7573 

.7731 

.7895 

.8055 

.8214 

.8373 

.8531 

30 

32 

.7001 

.7254 

.7117 

.7578 

.7710 

.7900 

.8060 

.8220 

,&378 

.8537 

32 

31 

.7097 

.7260 

.7422 

.7584 

.7715 

.7906 

.8066 

.8225 

.8384 

.8542 

34 

36 

.7102 

.7265 

.7427 

.7580 

.7750 

.7911 

.8071 

.8230 

.8389 

.8517 

36 

38 

.7108 

.7270 

.7433 

.7595 

.7756 

.7916 

.8076 

.8236 

.8394 

.8552 

38 

40 

.7113 

.7276 

.7438 

.7600 

.7761 

.7922 

.8082 

.8241 

.8400 

.8558 

40 

42 

.7118 

.7281 

.7443 

.7605 

.7766 

.7927 

.8087 

.8216 

.8405 

.8563 

42 

44 

.7121 

.7287 

.74 49 

.7611 

.7772 

.7932 

.8092 

.8251 

.8410 

.8568 

14 

46 

.7129 

.7292 

.7454 

.7616 

.7777 

.7938 

.8098 

.8257 

.8415 

.8573 

46 

48 

.7135 

.7298 

.7460 

.7621 

.7782 

.7913 

.810.3 

.8262 

.8421 

.8579 

48 

50 

.7140 

.7303 

.7465 

.7627 

.7788 

.7948 

.8108 

.8267 

4 

.8126 

.8584 

50 

52 

.7146 

.7308 

.7171 

.7632 

.7793 

.7951 

.8113 

.8273 

.8431 

.8589 

52 

54 

.7151 

.7314 

.7476 

.7638 

.7799 

.7959 

.8119 

.8278 

.8137 

.8594 

54 

56 

.7156 

.7319 

.7481 

.7613 

.7804 

.7961 

.8124 

.8283 

.8412 

.8600 

56 

58 

.7162 

.7325 

.7487 

.7618 

.7809 

.7970 

.8129 

.8289 

.8417 

.8605 

58 

60 

.7167 

.7330 

.7402 

.7654 

.7815 

.7975 

.8135 

.8294 

.8452 

.8610 

60 









































































































































108 


TABLE OF CHORDS 


Table of chords, in parts of a rad 1; for protracting' — Continued. 














M. 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

60° 

M. 

0 ’ 

.8610 

.8767 

.8924 

.9080 

.9235 

.9389 

.9543 

.9696 

.9848 

1.0000 

0 ' 

a 

.8015 

.8773 

.8929 

.9085 

.9240 

.9395 

.9548 

.9701 

.9854 

1.0005 

2 

4 

.8021 

.8778 

.8934 

.9090 

.9245 

.9400 

.9553 

.9706 

.9859 

1.0010 

4 

6 

.8026 

.8783 

.8940 

.9095 

.9250 

.9405 

.9559 

.9711 

.9864 

1.0015 

6 

8 

.8631 

.8788 

.8945 

.9101 

.9256 

.9410 

.9564 

.9717 

.9869 

1.0020 

8 

10 

.8036 

.8794 

.8950 

.9106 

.9261 

.9415 

.9569 

.9722 

.9874 

1.0025 

10 

12 

.8042 

8799 

.8955 

.9111 

.9266 

.9420 

.9574 

.9727 

.9879 

1.0030 

12 

14 

.8047 

.8804 

.8900 

.9116 

.9271 

.9425 

.9579 

.9732 

.9884 

1.0035 

14 

10 

.8052 

.8809 

.8966 

.9121 

.9276 

.9430 

.9584 

.9737 

.6889 

1.0040 

16 

18 

.8057 

.8814 

.8971 

.9126 

.9281 

.9436 

.9589 

.9742 

.9894 

1.0045 

18 

20 

.8603 

.8820 

.8976 

.9132 

.9287 

.9441 

.9594 

.9747 

.9899 

1.0050 

20 

22 

.8668 

.8825 

.8981 

.9137 

.9292 

.9446 

.9599 

.9752 

.9904 

1.0055 

22 

24 

.8673 

.8830 

.8986 

.9142 

.9297 

.9451 

.9604 

.9757 • 

.9909 

1.0060 

24 

20 

.8678 

.8835 

.8992 

.9147 

.9302 

.9456 

.9610 

.9762 

.9914 

1.0065 

26 

28 

.8084 

.8841 

.8997 

.9152 

.9307 

.9461 

.9615 

.9767 

.9919 

1.0070 

28 

30 

.8689 

.8846 

.9002 

.9157 

.9312 

.9466 

.9620 

.9772 

.9924 

1.0075 

30 

32 

.8694 

.8851 

.9007 

.9163 

.9317 

.9472 

.9625 

.9778 

.9929 

1.0080 

32 

34 

.8099 

.8856 

.9012 

.9168 

.9323 

.9477 

.9630 

.9783 

.9934 

1.0086 

34 

30 

.8705 

.8861 

.9018 

.9173 

.9328 

.9482 

.9635 

.9788 

.9939 

1.0091 

36 

38 

.8710 

.8867 

.9023 

.9178 

.9333 

.9487 

.9640 

.9793 

.9945 

1.0096 

38 

40 

.8715 

.8872 

.9028 

.9183 

.9338 

.9492 

.9645 

.9798 

.9950 

1.0101 

40 

42 

.8720 

.8877 

.9033 

.9188 

.9343 

.9497 

.9650 

.9803 

.9955 

1.0106 

42 

44 

.8726 

.8882 

.9038 

.9194 

.9348 

.9502 

.9655 

.9808 

.9960 

1.0111 

44 

40 

.8731 

.8887 

.9044 

.9199 

.9353 

.9507 

.9661 

.9813 

.9965 

1.0116 

46 

48 

.8736 

.8893 

.9049 

.9204 

.9359 

.9512 

.9666 

.9618 

.9970 

1.0121 

48 

50 

.8741 

.8898 

.9054 

.9209 

.9364 

.9518 

.9671 

.9823 

.9975 

1.0126 

50 

52 

.8747 

.8903 

.9059 

.9214 

.9369 

.9523 

.9676 

.9828 

.9980 

1.0131 

52 

54 

.8752 

.8908 

.9064 

.9219 

.9374 

.9528 

.9681 

.9833 

.9985 

1.0136 

54 

50 

.8757 

.8914 

.9069 

.9225 

.9379 

.9533 

.9686 

.9838 

.9990 

1.0141 

56 

58 

.8702 

.8919 

.9075 

.9230 

.9384 

.9538 

.9691 

.9843 

.9995 

1.0146 

58 

60 

.8767 

.8924 

.9080 

.9235 

.9389 

.9543 

.9696 

.9848 

1.0000 

1.0151 

60 

M. 

61° 

62° 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

70° 

M. 

0 

1.0151 

1.0301 

1.0450 

1.0598 

1.0746 

1.0893 

1.1039 

1.1184 

1.1328 

1.1472 

o- 

2 

1.0156 

1.0306 

1.0455 

1.0603 

1.0751 

1.0898 

1.1044 

1.1189 

1.1333 

1.1476 

2 

4 

1.0161 

1.0311 

1.0460 

1.0608 

1.0756 

1.0903 

1.1048 

1.1194 

1.1338 

1.1481 

4 

6 

1.0166 

1.0316 

1.0465 

1.0613 

1.0701 

1.0907 

1.1053 

1.1198 

1.1342 

1.1486 

6 

8 

1.0171 

1.0321 

1.0470 

1.0618 

1 0766 

1.0912 

1.1058 

1.1203 

1.1347 

1.1491 

8 

10 

1.0176 

1.0326 

1.0475 

1.0623 

1.0771 

1.0917 

1.1063 

1.1208 

1.1352 

1.1495 

10 

12 

1.0181 

1.0331 

1.0480 

1.0628 

1.0775 

1.0922 

1.1068 

1.1213 

1.1357 

1.1500 

12 

14 

1.0186 

1.0336 

1.0485 

1.0633 

1.0780 

1.0927 

1.1073 

1.1218 

1.1362 

1.1505 

14 

10 

1.0191 

1.0341 

1.0490 

1.0638 

1.0785 

1.0932 

1.1078 

1.1222 

1.1366 

1.1510 

16 

18 

1.0196 

1.0346 

1.0495 

1.0043 

1.0790 

1.0937 

1.1082 

1.1227 

1.1371 

1.1514 

18 

20 

-4 

1 0201 

1.0351 

1.0500 

1 0648 

1.0795 

1.0942 

1.1087 

1.1232 

1.1376 

1.1519 

20 

22 

1.0206 

1.0356 

1.05i4 

1.0653 

1.0800 

1.0946 

1.1092 

1.1237 

1.1381 

1.1524 

22 

24 

1.0211 

1.0361 

1.0509 

1.0658 

1.0805 

1.0951 

1.1097 

1.1242 

1.1386 

1.1529 

24 

26 

1.0216 

1.0366 

1.0514 

1.0662 

1.0810 

1.0956 

1.1102 

1.1246 

1.1390 

1.1533 

26 

28 

1.0221 

1.0370 

1.0519 

1.0667 

1.0815 

1.0961 

1.1107 

1.1251 

1.1395 

1.1538 

28 

30 

1.0226 

1.0375 

1.0524 

1.0072 

1.0820 

1.0966 

1.1111 

1.1256 

1.1400 

1.1543 

30 

32 

1.0231 

1.0380 

1.0529 

1.0677 

1.0824 

1.0971 

1.1116 

1.1261 

1.1405 

1.1548 

32 

34 

1.0236 

1.0385 

1.0534 

1.0682 

1.0829 

1.0976 

1.1121 

1.1266 

1.1409 

1.1552 

34 

36 

1.0241 

1.0390 

1.0539 

1.0687 

1.08114 

1.0980 

1.1126 

1.1271 

1.1414 

1.1557 

36 

38 

1.0246 

1.0395 

1.0544 

1.0692 

1.0839 

1.0985 

1.1131 

1.1275 

1.1419 

1.1562 

38 

40 

1.0251 

1.0400 

1.0549 

1.0697 

1.0844 

1.0990 

1.1136 

1.1280 

1.1424 

1.1567 

40 

42 

1.0256 

1.0405 

1.0554 

1.0702 

1.0849 

1.0995 

1.1140 

1.1285 

1.1429 

1.1571 

42 

44 

1.0201 

1.0410 

1.0559 

1.0707 

1.0854 

1.1000 

1.1145 

1.1290 

1.1433 

1.1576 

44 

40 

1.0266 

1.0415 

1 0564 

1.0712 

1.0859 

1.1005 

1.1150 

1.1295 

1.1438 

1.1581 

46 

48 

1.0271 

1.0420 

1.0569 

1.0717 

1.0863 

1.1010 

1.1155 

1.1299 

1.1443 

1.1586 

48 

50 

1.0276 

1.0425 

1.0574 

1.0721 

1.0868 

1.1014 

1.1160 

1.1304 

1.1448 

1.1590 

50 

52 

1.0281 

1.0430 

1.0579 

1.0726 

1.0873 

1.1019 

1.1165 

1.1309 

1.1452 

1.1595 

52 

54 

1.0286 

1.0435 

1.0584 

1.0731 

1.0878 

1.1024 

1.1169 

1.1314 

1.1457 

1.1600 

54 

56 

1.0291 

1.0440 

1.0589 

1.0736 

1.0883 

1.1029 

1.1174 

1.1819 

1.1462 

1.1605 

56 

68 

1.0296 

1.0445 

1.0593 

1.0741 

1.0888 

1.1034 

1.1179 

1.1.323 

1.1467 

1.1609 

58 

60 

1.0301 

1.0450 

1.0598 

1.0746 

1.0893 

1.1039 

1.1184 

1.1328 

1.1472 

1.1614 

60 



























































































































TABLE OF CHORDS. 109 

Table of Chords, in parts of a rad 1; for protracting — Continued. 


M. 

71° 

72° 

72° 

7-4° 

75° 

76° 

77° 

78° 

79° 

80° 

M. 

0 

1.1614 

1.1756 

1.1806 

1.2036 

1.2175 

1.2313 

1.2450 

1.2586 

1.2722 

1.2856 

0' 

2 

1.1810 

1.1760 

1.1001 

1.2041 

1.2180 

1.2318 

1.2455 

1.2501 

1.2726 

1.2860 

2 

4 

1.1624 

1.1765 

1.1006 

1.2046 

1.2184 

1.2322 

1.2459 

1.2595 

1.2731 

1.2865 

4 

6 

1.1828 

1.1770 

1.1910 

1.2050 

1.2180 

1.2327 

1.2464 

1.2600 

1.2735 

1.2869 

6 

8 

1.1833 

1.1775 

1.1015 

1.2055 

1.2101 

1.2332 

1.2468 

1.2604 

1.2740 

1.2874 

8 

10 

1.1638 

1.1770 

1.1920 

1.2060 

1.2198 

1.2336 

1.2473 

1.2609 

1.2744 

1.2878 

10 

12 

1.1642 

1.1784 

1.1924 

1.2064 

1.2203 

1.2341 

1.2478 

1.2614 

1.2748 

1.2882 

12 

11 

1.1647 

1.1789 

1.1920 

1.2069 

1.2208 

1.2345 

1.2482 

1.2618 

1.'2753 

1.2887 

14 

16 

1.1632 

1.1703 

1.1034 

1.2073 

1.2212 

1.2350 

1.2487 

1.2623 

1.2757 

1.2891 

16 

18 

1.1657 

1.1708 

1.1038 

1.2078 

1.2217 

1.2354 

1.2491 

1.2627 

1.2762 

1.2896 

18 

20 

1.1661 

1.1803 

1.1943 

1.2083 

1.2221 

1.2359 

1.7406 

1.2632 

1.2766 

1.2900 

20 

22 

1.1066 

1.1807 

1.1948 

1.2087 

1.2226 

1.2364 

1.2500 

1.2636 

1.2771 

1.2905 

22 

24 

1.1671 

1.1812 

1.1052 

1.2002 

1.2231 

1.2368 

1.2595 

1.2611 

1.2775 

1.2909 

24 

26 

1.1676 

1.1817 

1.1057 

1.2007 

1.2235 

1.2373 

1.2509 

1.2645 

1 2780 

1.2914 

26 

28 

1.1680 

1-1821 

1 1062 

1.2101 

1.2240 

1.2377 

1.2514 

1.2650 

1.2784 

1.2918 

28 

30 

1.1685 

1.1826 

1.1966 

1.2106 

1.2244 

1.2382 

1.2518 

1.2654 

1.2789 

1.2922 

30 

32 

1.1600 

1.1831 

1.1071 

1.2111 

1.2249 

1.2386 

1.2523 

1.2659 

1.2793 

1.2927 

32 

34 

1.1604 

1.1836 

1.1976 

1.2115 

1.2254 

1.2391 

1.2528 

1.2663 

1.2798 

1.2931 

34 

36 

1.1600 

1.1840 

1.1980 

1.2120 

1.2258 

1.2396 

1.2532 

1.2668 

1.2802 

1.2936 

36 

38 

1.1704 

1.1845 

1.1085 

1.2124 

1.2263 

1.2400 

1.2537 

1.2672 

1.2807 

1.2940 

38 

40 

1.1700 

1.1850 

1.1000 

1.2129 

1.2267 

1.2405 

1.2541 

1.2677 

1.2811 

1.2945 

40 

42 

1.1713 

1.1854 

1.1004 

1.2134 

1.2272 

1.2409 

1.2546 

1.2681 

1.2816 

1.2949 

42 

44 

1.1718 

1.1859 

1.1900 

1.2138 

1.2277 

1.2414 

1.2550 

1.2686 

1.2820 

1.2954 

44 

46 

1.1723 

1.1864 

1.2004 

1.2143 

1.2281 

1.2418 

1.2555 

1.2690 

1.2825 

1.2958 

46 

48 

1.1727 

1.1868 

1.2008 

1.2148 

1.2286 

1.2423 

1.2559 

1.2695 

1.2829 

1.2062 

48 

50 

1.1732 

1.1673 

1.2013 

1.2152 

1.2290 

1.2428 

1.2564 

1.2609 

1.2833 

1.2067 

50 

52 

1.1737 

1.1878 

1.2018 

1.2157 

1.2295 

1.2432 

1.2568 

1.2704 

1.2838 

1.2971 

52 

54 

1 1742 

1.1882 

1.2022 

1.2161 

1.2299 

1.2437 

1.2573 

1.2708 

1.2842 

1.2976 

54 

56 

1.1746 

1.1887 

1.2027 

1.2166 

1.2304 

1.2441 

1.2577 

1.2713 

1.2847 

1.2980 

56 

58 

1.1751 

1.1802 

1.2032 

1.2171 

1.2309 

1.2446 

1.2582 

1.2717 

1.2851 

1.2985 

58 

60 

1.1756 

1.1896 

1.2016 

1.2175 

1.2313 

1.2450 

1.2586 

1.2722 

1.2856 

1.2989 

60 


M. 

81° 

82° 

83° 

84° 

85° 

86° 

87° 

88° 

89° 

M. 

O' 

1.2980 

1.3121 

1.3252 

1.3383 

1.3512 

1.3640 

1.3767 

1.3893 

1.4018 

O' 

2 

1.2903 

1.3126 

1.3257 

1.3387 

1.3516 

1.3644 

1.3771 

1.3897 

1.4022 

2 

4 

1.2098 

1.3130 

1.3281 

1.3391 

1.3520 

1.3648 

1.3776 

1.3902 

1.4026 

4 

6 

1.3002 

1.3134 

1.3265 

1 -3396 

1.3525 

1 3653 

1.3780 

1.3906 

1.4031 

6 

8 

1.3007 

1.3139 

1.3270 

1.3400 

1.3529 

1.3657 

1.3784 

1.3910 

1 4035 

8 

10 

1.3011 

1.3143 

1.3274 

1.3404 

1.3533 

1.3661 

1.3788 

1.3914 

1.4039 

10 

12 

1.3015 

1.3147 

1.3279 

1.3409 

1.3538 

1.3665 

1.3792 

1.3918 

1.4043 

12 

14 

1.3020 

1.3152 

1.3283 

1.3413 

1.3512 

1.3670 

1.3797 

1.3922 

1.4017 

14 

16 

1.3024 

1 3156 

1.3287 

1.3417 

1.3546 

1.3674 

1.3801 

1.3927 

1.4051 

16 

18 

1.3029 

1.3161 

1.3292 

1 3421 

1.3550 

1.3678 

1.3805 

1 3931 

1.4055 

18 

20 

1.3033 

1.3165 

1.3296 

1.3426 

1.3555 

1.3682 

1.3809 

1.3935 

1.4060 

20 

22 

1.3038 

1.3160 

1.3300 

1.3430 

1.3559 

1.3687 

1.3813 

1.3939 

1.4061 

22 

24 

1.3012 

1.3174 

1.3305 

1.3434 

1.3563 

1.3691 

1.3818 

1.3943 

1.4068 

24 

26 

1.3016 

1.3178 

1.3309 

1.3439 

1.3567 

1.3695 

1.3822 

1.3947 

1.4072 

26 

28 

1.3051 

1.3183 

1.3313 

1.3443 

1.3572 

1.3699 

1.3826 

1.3952 

1.4076 

28 

30 

1.3055 

1.3187 

1.3318 

1.3447 

1.3576 

1.3704 

1.3830 

1.3956 

1.4080 

30 

32 

1.3060 

1.3191 

1.3322 

1.3452 

1.3580 

1.3708 

1.3834 

1.3960 

1.4084 

32 

31 

1 3064 

1.3196 

1.3326 

1.3456 

1.3585 

1.3712 

1.3839 

1.3964 

1.4089 

34 

36 

1.3068 

1.3200 

1.3331 

1.3460 

1.3589 

1 3716 

1.3843 

1.3968 

1.4093 

36 

38 

1.3073 

1.3204 

1.3335 

1.3165 

1.3593 

1.3721 

1.3847 

1.3972 

1.4097 

38 

40 

1.3077 

1.3200 

1.3339 

1.3169 

1.3597 

1.3725 

1.3851 

1.3977 

1.4101 

40 

42 

1.3082 

1.3213 

1.3344 

1.3473 

1.3602 

1.3729 

1.3855 

1.3981 

1.4105 

42 

41 

1.3086 

1.3218 

1.3348 

1.3477 

1.3606 

1.3733 

1.3860 

1.3985 

1.4109 

44 

46 

1.3090 

1.3222 

1.3352 

1.3482 

1.3610 

1.3738 

1.3864 

1.3989 

1.4113 

46 

48 

1.3095 

1.3226 

1.3357 

1.3486 

1.3614 

1.3742 

1.3868 

1.3993 

1.4117 

43 

60 

1.3090 

1.3231 

1.3361 

1.3490 

1.3619 

1.3746 

1.3872 

1.3997 

1.4122 

50 

~52* 

1.3104 

1.3235 

1.3365 

1.3495 

1 3623 

1.3750 

1.3876 

1.4002 

1.4126 

52 

54 

1.3108 

1.3230 

1.3370 

1.3499 

1.3627 

1.3754 

1.3881 

1.4006 

1.4130 

51 

56 

1.3112 

1.3244 

1.3374 

1.3503 

1.3631 

1.3759 

1.3885 

1.4010 

1.4134 

56 

58 

1.3117 

1.3248 

1.3378 

1.3508 

1.3636 

1.3763 

1.3889 

1.4014 

1.4138 

58 

60 

1.3121 

1.3252 

1.3383 

1.3512 

1.3640 

1.3767 

1.3893 

1.4018 

1.4142 

60 





































































































































110 


POLYGONS, 


POLYGON'S. 



0 a 


/ 

/p 


b 


o 


Pentagon. 

5 sides. 


Hexagon. 

6 sides. 


Heptagon. 

7 sides. 


Octagon. 

8 sides. 



Any straight-sided fig is called a polygon. If all the sides and angles are equal, it is a regular 
polygon ; if not, it is Irregular. Of course the number of polygons is infinite. 


Table of Regular Polygons. 


-j 

l 


Number 

of 

Sides. 

Name 

of 

Polygon. 

Area ~ 

(square of one 
side) mult by 

Radius of cir¬ 
cumscribing 
circle — side 
mult by 

Interior angle 

a h c contained 
between two 
sides. 

Angle at een, 

subtended 
by a side. 

3 { 

Equilateral 

triangle. 

| .433013 

.577350 

60° 

120° 

4 

Square. 

1.000000 

.707107 

90° 

90° 

5 

Pentagon. 

1.720477 

.850651 

108° 

72° 

6 

Hexagon. 

2.598076 

1.000000 

120° 

60° 

7 

Heptagon. 

3.633912 

1.152382 

128° 34 2857' 

51° 25.7143' 

8 

Octagon. 

4.828427 

1.306563 

135° 

45° 

9 

Non agon. 

6.181*24 

1.461902 

140° 

40° 

in 

Decagon. 

7.694209 

1.618034 

144° 

36° 

11 

Undeeagon. 

9 365640 

1.774733 

147° 16.3636' 

32° 43.6364' 

12 

Dodecagon 

11.196152 

1.931854 

150° 

30° 


Area of any regular polygon = length of one side, a b X perp p drawn from cen of fig to 

cen of side X half the number of sides. 


Sum of Interior angles, a b e, etc, of any polygon, regular or irregular = 180° X 

(number of sides — 2). 


Angle at cen subtended by a side, in any regular polygon : 360° -f- number of sides. 

TRIANGLES. 



B/ 


;\c 


jP 

i \ 

p 

s 

8 


E 


P 


We speak here of plane triangles only ; or those having straight sides. 

A triangle, is equilnternl when all its sides are equal, as a ; isosceles when only two sides 
are equal, as 15; scalene when all the sides are unequal, as C. I) and E . acute-angled when 
all its angles are acute, or each less than 90°, as A. B, aud C: right-angled when it contains a 
right angle, as I); obtuse-angled when it contains an ohtuse angle, or o„e greater than 90°, as E 
All the three angles of any triangle are equal to two right 
angles, or 180-'; therefore, if we know two oi them, we can find the thiru by 
subtracting their sum from 180°. AH triangles which have equal bases, C 
and equal perp heights, have also equal areas; t hus toe areas ot a w c, a w d. and *" 
a w e, are equal to each other. The area of any triangle is equal to half 
that of any parallelogram which has an equal base,aud an equal perp height. The 
areas of triangles which have equal bases, but diff perp heights, are to 
each other as, or iu proportion to. their perp heights; thus the triangle awn, 
with a perp height s n, equal to but one-half that (s e) of the three other trian¬ 
gles, but with the same base a w, has also but half the area of either of those 
others. 

Area of any triangle. Figs A, B, C, D, E, = half the base, S, X the height, or perp dist p to 



the opposite angle. Any side may he taken as the base of a triangle; hut the perp height must always 
be measured from the side so assumed; to do which, the side must sometimes be prolonged, as in 
Fig F.; but the prolongation is not to be considered as a part of the base. 

Area of any equilateral triangle = .433013 X square of one side. 


To fiml area, having the three sides. 

Add them together; div the sum by 2; from the half sum, subtract each side separately • mult the 
half sum and the three remainders continuously together; take the sq rt of the prod. ’ 

Ex.—The three sides = 20, 30, 40 ft. Here 20 + 30 -f- 40 rr 90 ; and ?? — 45. And 45 — 20 =r 25 

45 — 30 = 15 ; aud 45 — 40 ~ 5. And 45 X 25 X 15 X 5 ~ 84375 ; and the sq rt of 84375 is 290 47 sa ft 

area rend. H ’ 








































TRIANGLES. 


Ill 


T ° area ’ having: one side and the 2 angles at its ends. 

Afln thp *> r»n rrl r>a . a- i « . „ 


Find the nLslMofthlsa^le ^ also find^hp 0 ^ 18 ^ ^ Z™ "’»* be the angle opp the given side. 
Then as the nat sine of the fingll anglais to thenrod of ,‘h ® 0t , he - ang i e! V. and mult them to « etber - 
sguare of the given side to double the reqd area P h “ S '“ eS ° th ® ° 2 augles - 80 is the 


To hud area, having two sides, and ihe included angle. 

Mult tO£TPt.hpr t.ho ~ „ .3 . U - __ . _ .. 


MuU togetherthe two sides and the nat sine of the included angle ; div by 2. 

JSjX' oiuGs dou ft sod 980 ft i included aniflp aqo on' d_ , , . 

650 X 980 X 9356 lut - Juuea an 8 le «9 4 20 . By the table we find the nat sine .9356; 

therefore, ---- 297988 6 square ft area. 



To find area, having the three angles and the 
perp height, « 6. 

. f'a d th ®. nat ' sin ? s of Jbe three angles ; mult together the sines of the angles 
ti? ° ’ l .be sine of the angle 5 by the prod ; mult the quot by the square 
of the perp height a b; div by 2. J H 


To find any side, as d o, having the three 
angles, ft, 6 and o, and the area. 


Tho "V S * ne ° f ti V tw ? ee . the ar,,a ! »QH«re of d o. See Rem 2, p 119. 

its^ides 1 ??f , Ph* ° f a “. e 9« H at e ral triangle is equal to one side X .866025. Hence one of 
® f. ,? 8 ls et l u ‘ 1 |. to the perp height div by .866025 or to perp height X 1.1547. Or to find a side 
r*L be ;X n ° f HS area t>y 1 - 51967 - The side of aa equilateral triangle, mult bv’.658037 =r side ofa 
r ft 1 Same arCa : ° F mult by • 742517 11 8 ives the diam of a circle of the same area. 

I no iollnwinfr n.nnlv t.n nnv nlnno t ..»>i^ -.ui? ... . . 


m. _ . - 7 — F • ** 6 **” me umui oi a circle or m 

| h 9 following apply to any plane triangle, whether oblique or right-angled : 
1 . 1 he three angles amount to 180°, or two right angles. 

on?s A D and X B eri ° r aDgle ’ as A ^ **> is e 9 ual to the two interior and opposite 

8 . The greater side is opposite the greater angle. 

.. 4 - • J h A 8ide w a T as the sines of the,opposite angles. Thus, the side a is to 
the side b as the sine of A is to the sine of B. 

If any angle as s be bisected by a line s o, the two parts mo, oh of 
the opposite side m n will be to each other as the other two sides s m sn- 
or, m o: o n:: s m: 8 n. 

6. If lines be drawn from each angle r s t to the 
center of the opposite side, they will cross each 
other at one point, a, and the short part of each 
of the lines will be the third part of the whole line. 
Also, a is the cen of gray of the triangle 




* me ucu ui j^ruv me inaugie. 

7. If lines be drawn bisecting the three angles, they will meet at a point 
perpendicularly equidistant from each side, and consequently the center 
j of the greatest eircle that can be drawn in the triangle. 

. , * 8. If a line * n he drawn parallel to any side c a, 

the two triangles r s n, r c a, will be similar. » 

A ’ I a d « . * 7 d /v m ^ . A. Z , • _ _ _ _ V r 


.v, ~~ , or*-, / o u, win ue si in uar. 

9. T° divide any triangle a c r into two equal parts by alinesn parallel to 
iny one of its sides c a. On either one of the other sides, as a r, as a diam, 
lesenbe a semicircle a o r; and find its middle o. From r (opposite c a), with 
radius r o, describe the arc o n. From n draw n s, par¬ 
allel to c a. 

10 . To find the greatest parallelogram that can be 
drawn in any given triangle onb. Bisect the three sides at ace, and join 
a c, a e, e c. Then either a e b c, a e c o, orate*, each equal to half the 
triangle^ will be the reqd parallelogram. Any of these parallelograms can 




wo iuc mju paiaiieiogiam. any oi inese parallelograms can 
plainly be converted into a rectangle of equal area, and the greatest that can be 
drawn in the triangle. 

10’^. If a line a c bisects any two sides ob, o n, of a triangle, it will be nar- 

o. nnn half lfinor it 


* v * i <*■ nut; U b uiarbt 

allei to the third side n b, and half as long as it. 

11 . To find the greatest square that can be drawn in any triangle a x r. From 
an angle as a draw a perp a n to the opposite side xr, and find its length. Then 

«C T X ft 7t 

on, or a side v t of the square will = 



x r -\- a n 

Rem.— If the triangle is such that tw-o or three such perps can be drawn then 
two or three equal squares may be found. 
















112 


PLANE TRIGONOMETRY, 


Ri^lit-an^Iod Triangles. 

All the foregoing apply also to right-angled triangles; but what follow apply to them only. 

Call the right angle A, and the others B and C; and call the sides respectiv 
opposite to them a, b, and c. Then is 

“ = Sine C = c X Seo B — Cosine C = h X SeC C =V 

b ~ a X Sine li-«X Cos C = e X Cot C = c X Tang B. 
c = a X Sine C — a X Cos B ~ 6 X Tang C. 

c b e 

Also Sine of C = - ; Cos C = - ; Tang C — ^. 



be b 

And Sine of B = - ; Cos B = - ; Tang B = 


And Sine of A or 90° = 1. Cos A 0. Tang A rr infinity. Sec A = infinity. 
1. If from the right angle o a line o tv be drawn perp to the hypothec use or long side A g, then t 
two small triangles o to h, o to g, and the large one o h g, will Oe simil 
Or g tv : w o :: w o : to h ; and g to X w h — w o2. 

SS. A line drawn from the right angle to the center of the long side v 
be half as long as sa'd side. 

3. If on the three sides oft, o g, g ft we draw three squares t, u, v, 
three circles, or triaugles, or any other three figs that are similar, then t 
area of the largest one is equal to the sum of the areas of the two others. 

4. In a triangle whose sides are as 3, 4, and 5 (as are those of the i 
angle A B C), the angles are very approximately 90°; 53° 7'48.38"; a 
36° 52' 11.62". Their Sines, 1.; .8; and .6. Their Tangs, infinity ; 1.33 
and .75. 

£>. One whose sides are as 7, 7, and 9 9, has very approx one angle of 
and two of 45° each, near enough for all practical purposes. 



PLANE TRIGONOMETRY. 

Plans trigonometry teaches how to find certain unknown parts of plane, or straight - sided i 
angles, by means of other parts which are known ; and thus enables us to measure inaccessible < 
tauces, &c. A triangle consists of six parts, namely, three sides, and three angles; and if we kr 
any three of these, (except the three angles, and in the ambiguous case under “ Case 2,") we can f 
the other three. The following four cases include the whole subject; the student should committh 
to memory, „ 

C x 

Case 1. Having any two angles, and one side, A * 

to find the other sides and angle. 

Add the two angles together ; and subtract their sum from 180°; the rem 
will be the third angle. And for the sides, as 

Sine of the angle . Sine of the angle . . , .. . d {d 

opp the given side • opp the reqd side • • g»'en side . reqa side. 

Use the side thus found, as the given one; and in the same manner find 
the third side. 

Case 2. Having two sides, h a, a c, Fig X. and the angle a b 
opposite to one of them, to find the other side andangi 




Side a c opp 
the given an¬ 
gle a b c 


The other 
given side 
6 a 


Sine of the 
given angle 
a b c 


Sine of angle 6 d a or 
b c a opposite the other 
given side b a. 


Fig-. X. 


Having found the siue, take out the corresponding angle from the table 
nat sines, but, in doing so, if the side ac opp the given angli 

shorter than the other given side b a, bear in mind that an angle and its s 
plement have the same sine. Thus, in Fig X, the sine, as found above 

opp the angle 6 c a in the table. But a c, if shorter than b a, can evidently 
laid off in the opp direction, a d, in which case. 6 d a is the supplement of b 
If a c is as long as, or longer than, b a, there can be no doubt; for in that c 
it cannot be drawn toward b, but only toward n and the angle 6 c a will 
found at once in the table, opp the sine as found above. 




















PLANE TRIGONOMETRY, 


113 


When the two angles, a b e, b c a, have been found, find the remaining side by Case l. 

For the remaining angle, b a c, add together the angle a b c first given, and the one, b c a, found 

as above. Deduct their sum from 180°. 

Case 3. Having: two sides, and the angle included 

between them. 

Take the angle from 180°; the rem will be the sum of the two unknown angles. Div this sum by 
2; and find the nat tang of the quot. Theu as 

The sum of the . Their diff • • Tang of half the sum of . Tang of half 

two given sides • • • the two unknown augles • their diff. 

Take from the table or nat tang, the angle opposite this last tang. Add this angle to the half sum 
of the two unknown angles, and it will give the angle opp the longest given side; and subtract it 
from the same half sum, for the angle opp the shortest given side. Having thus fonnd the angles, 
find the third side by Case 1. 

As a practical example of the use of Case 3, we can ascertain the dist n to across a deep pond, by 
measuring two liues no and to o ; and the angle n o to. From these data we may calculate n to ; or 
by drawing the two sides, and the angle on paper, by a scale, we can afterward measure n to on 
the drawing. 


n 




Case 4. Having: the three sides. 

To find the three angles; upon one side a 5 as a base, draw (or suppose to be drawn) a perp c g from 
the opposite angle c. Find the diff between the other two sides, <ic and cb : also their sum. Then, as 

- Th b . Sum of the . . Diff of other . Diff of the two 

1 e • other two sides • • two sides • parts a g aud b g, of the base. 

Add half this diff of the parts, to half the base a 6 ; the sum will be the longest part ag ; which 
taken from the whole base, gives the shortest part <7 6 . By this means we get iu each of the small tri¬ 
angles a c g and egb, two sides, (namely, a c and a g; and c b and g b ;) and an angle (namely, the 
right angle eg a, or c g b) opposite to one of the given sides. Therefore, use Case 2 for fiuding the 
angles, a and b. When that is done, take their sum from 180°, for the angle a c b. 



Or, SSd mode : call half the sum of the three sides, s; and call the 
two sides which form either angle, m and n. Then the nat sine of 


half that angle will be equal to 


V (s — TO) X O 
to X n 




Ex. 1. To find the dist from a to an inac¬ 
cessible object c. 

Measure a line a b ; and from its ends measure the angles cab and 
cb a. Thus having found one side and two angles of the triangle a be, 
calculate a c by means of Case 1. Or if extreme accuracy is not reqd, 
draw the line a b on paper to any convenient scale; then by means of a 
protractor lay off the angles c ah, c b a; and draw a c and c b ; then 
measure a c by the same scale. 



Ex. 2. To find the height of a vertical 
object, na. 

Place the instrument for measuring angles, at any conve 
nietit spot o : also meas the dist o a : or if o a cannot be actually 
measd in consequence of gome obstacle, calculate it by the 
same process as a c in Fig 1. Then, first directing the instru¬ 
ment horizontally,* as os, measure the angle of depression. 
* o a, say 12°; also the angle son, say 30°. These two angles 
added together, give the angle a o n, 42°. Now, in the small 
triangle o s a we have the angle os a equal to 90°, because a » 
is vert, and os hor; and since the three angles of any triangle 
are equal to 180°, if we subtract the angles osa (90°), and soa 
(12°) from 180°, the rem (78°) will be the angle o a s or o a n. 
Therefore, in the triangle o n a, we have one side o a: and two 
angles a o n, and o a n, to calculate the side a n by Case I. 


* Angles and disfson sloping ground must be measured hor¬ 
izontally. The graduated hor 

circle of the instrument evidently meas¬ 
ures the angle between two objects hori¬ 
zontally, no matter how much higher one 
of them may be than the other; one per¬ 
haps requiring the telescope of the instru¬ 
ment to he directed upward toward it; 
and the other downward. If, therefore, 
the sides of triangles lying upon sloping 
ground, are not also measd hor, there can 
be no accordance between the two. Thu*, 



8 







































114 


PLANE TRIGONOMETRY 


Rkm. If, as In Fig 3T. it should he necessary to ascertain the vert height an from a point o, entirel 
above- it, then both the angles rneasd at o, namely, son, and s o a. will be aDgles of depression, c 
lelow the hor line o s assumed to measure them from. In this case we have the side o a as before 



Orff, as rn Fig 4, the observations are to be takeD from a point o, entirely below the object a n, the 
both the angles s o a, s o n, will be angles of elevation, or above the assumed hor line o*. Here w 


have in the triaugle o n a, the given side o a as 
before; the augle a o n~ a o n — a o a; and the 

angle on a = 18®° — |oj n (90°), and no *, j to 
calculate an by Case 1. 

If the object an, as in Fig 5, instead of being 
vert, is inclined ; and instead of its vert height, 
we wish to find its length a n, we must first as¬ 
certain its angle y t i of inclination to the hori¬ 
zon ; to which angle eaeh of the angles o an will 
be equal. To find this angle y ti, suspend a plumb- 
line i y, of any convenient kuown length, from the 
object an; and measure also y t horizontally. 
Then say as 

y t : i y : ; 1 : nat tang of angle y t i. 

From the table of nat tangs take out the angle 
y t i found opposite this nat tang; and use it for 
the angles o s n or o s a; instead of the 90° of Figs 
3 and 4. Also when the object inclines, the side 
a o of the triangle must be rneasd in line, or in 
range with the iuclination. If the object, as the 
rock a », Fig 6, is carved or irregular, a pole a s 
may be planted sloping in the direction a n ; and 



in the triangle a b c. npon sloping gronnd, the instrument at o, measures the hor angle io n ; and no, 
the angle h a c. Therefore, the side which corresponds with this hor angle t o », is the hor dist i n 
and not the sloping dist b c. In other words, when sides and angles are on sloping ground, we d 
not seek their actual measures; hut their hor ones. This remark applies to all surveying for farms 
railroads, triangulations of countries. Ac, Ac; and the want of a strict attention to it, is one cuus 
of the small errors, almost unavoidable, (and fortunately, of but trilling consequence in practice’ 
which occur in all ordinary field operations. See p 176. 

When St sextant is used, angles between objects at diff altitudes, as n ai 

q, may be rneasd hor, by first planting two vert rods r 

o aud s, in range with the objects; and then iakiug 
the hor angle o n s, subtended by the rods. 

Angles may be mestsd without 
any Inst, thus: Measure 100 1't toward 

each object, and drive stakes; measure the dist across 
from one stake to the other. Half this dist will be 
the sine of Aa?/the angle to a rad of 100 : and if we move 
the decimal point two places to the left, we get the nat 
sine of this one half of the angle to a rad of 1, as in the 
tables. Thus, suppose the dist to be 80.64 feet; then 
40.32 is the sine of half the angle; and .4032 will be 
the nat sine, opposite to which in the table of nat 
sines we find the angle 23° 47' ; which malt by 2. gives 

47° 34’, the reqd angle. If obstacles prevent measuring toward the objects, we may measure direct 
from them ; because, when two lines intersect, the opposite angles are eqnal. A rough measuremei 
may be made by sticking three pins vert, and a few ins apart, into a small piece of board, nailed h< 
to the top of a post. The pins would oecupy the positions nos, of the last figure. Pencil-lines m; 
then be drawn, connecting the pin-holes ; and the angle be rneasd with a protractor. By nailing 
piece of board vert to a tree, and then drawing upon it a short hor line, by means of a pocket ca 
penters’ spirit-level, vert angles of elevation and depression may be taken roughly in the same wa 
In this way the writer has at times availed himself of the outer door of a honse, by opening it until 
pointed toward some mountain-peak, the dist of which he knew approximately ; but of the height c 


























PLANE TRIGONOMETRY, 


115 



its angle ytiot inclination with the horizon found as before; 
in which case the dist a n is calculated. Or if the vert height c n 
is sought, the point c may first be found by sighting upward 
along a plumb-line held above the head. 


Ex. 3. To find the approximate height, 
s x, of a mountain, 


x 


Of which, perhaps, only the very summit, x, is visible above 
interposing forests, or other obstacies; but the dist, mi, of which 
is known. In this case, first direct the instrument hor, as m h ; 

and then measure the angle i m x. 
Then in the triangle im x we have 
one side mi; the rneasd angle imx, 
and the angle mix (90°), to find i * 
by Case 1. But to this i x we must 
add i o, equal to the height y m of the 
instrument above the ground; and 
also o s. Now, o s is apparently due 
entirely to the curvature of the earth, 
which is equal to very nearly 8 ins, or 
.667 ft in one mile: and increases as 
the squares of the dists; being 4 
times 8 ins in 2 miles; 9 times 8 ins 
in 3 miles, &c. But this is somewhat diminished by the refraction of the atmosphere ; which varies 
with temperature, moisture, &c; but always tends to make the object * appear higher than it 

actually is. At an average, this deceptive elevation amounts to about—th part of the curvature of 

7 1 

the earth; and like the latter.it varies with the squares of the dists. Consequently if we subtract — 

part from 8 ins, or .667 ft, we have at once the combined effect of curvature and refraction for one 
mile, equal to 6.857 ins, or .5714 ft; and for other dists, as shown in the following table, by the use 
of which we avoid the necessity of making separate allowances for curvature and refraction. 

Table of allowances to be added for curvature of the earth; 
and for refraction; combined. 



Dist. 
in yards. 

Allow. 

feet. 

Dist. 
in miles. 

Allow. 

feet. 

Dist. 
in miles. 

Allow. 

feet. 

Dist. 
in miles. 

Allow. 

feet. 

100 

.002 

% 

.036 

6 

20.6 

20 

223 

150 

.004 


.143 

7 

28.0 

22 

277 

200 

.007 

% 

.321 

8 

36.6 

25 

357 

300 

.017 

1 

.572 

9 

46.3 

30 

514 

400 

.030 

IV, 

.893 

10 

57.2 

35 

700 

500 

.046 


1.29 

11 

69.2 

40 

915 

600 

.066 

m 

1.75 

12 

82.3 

45 

1158 

700 

.090 

2 

2.29 

13 

96.6 

50 

1429 

800 

.118 

&A 

3.57 

14 

112 

55 

1729 

900 

.119 

3 

5.14 

15 

129 

60 

2058 

1000 

.185 

&A 

7.00 

16 

146 

70 

2801 

1200 

.266 

4 

9.15 

17 

165 

80 

3659 

1500 

.415 

4 ^ 

11.6 

18 

185 

90 

4631 

2000 

.738 

5 

14.3 

19 

206 

100 

5717 


Hence, if a person whose eye is 5.14 ft, or 112 ft above the sea, sees an object just at the sea's 
orizon, that object will be about 3 miles, or 14 miles distant from him. 

A horizontal line is not a level one, tor a straight line cannot be a 

;vel one. The curve of the earth, as exemplified in an expanse of quiet water, is level. In Fig 7, 
we suppose the curved line t y s g to represent the surface of the sea, then the points t y s and g are 
a a level with eacli other. T hey need not be equidistant from the center of the earth, for the sea at 
te poles is about 13 miles nearer it than at the equator; yet its surface is everywhere on a level. 

Jp, and down, refer to sea level. Eevcl means parallel to the curvature 
f the sea; and horizontal means tangential to a level. 


Ex. 4. If the inaccessible vert height c d, Fig 8, 

i so situated that we cannot reach it at all, then place the instrument for measuring angles, at any 
mvenient spotn; and in range between n and d, plant two staffs, whose tops o and i shall range 
recisely with n, though they need uot be on the same level or hor line with it. Measure n o: also 
om n measure the angles on d and o n c. Then move the instrument to the precise spot previously 


fhich he had no idea. For allowance for curvature and refraction see above Table. 
I triangle whose sides are as 3, 4, and 5, is right angled; and one 

hose sides are as 7; 7 ; and 9. 9; contains 1 right angle; and 2 angles of 45° each. As it is fre- 
uently necessary to lay down angles of 45° and 90° on the ground, these proportions may be used for 
te purpose, by shaping a portion of a tape-line or chain into such a triangle, and driving a stake at 
stch angle. See p. 58. 








































116 


PLANE TRIGONOMETRY 


•ccupied by the top o of the staff; and from o measure the angles iod and doc. This being done, sub 


d 



Fig-. 8. 


tract the angle i o c from 
180°; the rem will be the 
angle c o «. Consequent¬ 
ly in the triangle noc, we 
have one side no, and two 
angles, c n o and con, to 
And by Case 1 the side o c. 
Again, take the angle iod 
from 180°; the remainder 
will be the angle n o d, so 
that in the triangle d n o 
W'e have one side n o, and 
the two angles d n o and 
nod, to find by Case 1 
the side od. Finally, in 
the triangle cod, we have 
two sides c o and o d, and 
their included angle cod, 
to find c d, the reqd vert 
height. 



Fig-. 9. 


5 

Rem. If cd were in a valley, or on a hill, and the observations reqd to be made from either higher 
or lower ground, the operation would be precisely the same. 


Ex. 5. See Ex 10. 


To find the dist ao , FIs 9, between two entirely inaccessible 

objects. 

Measure a side nm; at n measure the angles anm and onm ; also at m measure the angles o m n. am! 
a vi n. This beiug done, we have in the triangle anm, one side n m, Fig 9, and the angles anm, auc 
nma; heuce, by Case l, we cau calculate the side an. 

Again, in the triangle o m n we have one side n m, and 
the two angles o mn, and mno; hence, by Case 1, we can 
calculate the side n o. This being doue, we have in the 
triangle a no, two sides an, and no; and their included 
angle a n o; hence, by Case 3, we can calculate the side 
ao, which is the reqd dist. It is plain that in this manner 
we may obtain also the position or direction of the inacces¬ 
sible line ao; for we can calculate the angle n ao ; and can 
therefrom deduce that of ao; and thus be enabled to run 
a line parallel to it, if required. By drawing n m on pa¬ 
per by a scale, and laying down the four measd angles, 
the dist a o may be measd upon the drawing by the same scale. 

If the position of the inaccessible dist c n. Fig 10, be such that 
we can place a stake p in line with it,we may proceed thus : Place 
the instrument at any suitable point s, and take the angles p s c 
andcsn. Also find the angle cps, and measure the dist ps. Then 
in the triangle p s c find s c by Case 1; again, the exterior angle 
n c *, being equal to the two interior and opposite angles cps, 
and p * c, we have in the triangle c s n, one side and two angles 
to find c n by Case 1. 



Fi<r. 10. 



Fig. 11. 


Ex. 6. To find a dist a b, Fig 11. of which 
the ends only sire accessible. 

From a and b, measure any two lines a c, b c. meeting at c; also 
measure the angle a c b. Then in the triangle ab c we have two 
sides, and the included angle, to find the third side a 6 by Case 3. 

Ex. 7. To find the vert height o m, of a 
hill, above a g-iven point i. 

Place the instrument at i; measure a m. Directing 
the instrument hor, as an, take the angle nnm. Then, 
since anm is 90° Fig 12, we have one side a m. and 
two angles, nam and anm. to find nm by Case 1. 

Add no, equal to at, the height of the instrument. 

Also, if the hill is a long one, add for curvature of the 
earth, and for refraction, as explained in Example 3, 

Fig 7. Or the instrument may be placed at the top of 
the hill; and an angle of depression measured ; instead 
of the angle of elevation n a m. 

Rem. 1. It is plain, that if the height o m be previously 
known, and we wish to ascertain the dist from its sum¬ 
mit m to any point i, the same measurement as before, 
of the angle n a m, will enable us to calculate a m by 
Case 1. So in Ex. 2, if the height n a be knowu, the angles measd in that example will enable us 

“ "«* *«■•—»>* •» or which ,he process is pS".7» 

®f the object. Then, as the length of the shadow of the stick is to the length of the stick abov< 



Fig-. 12. 


















PLANE TRIGONOMETRY, 


117 


cround, so is the length of the shadow of 
nust be equally inclined. 


the object, to its height. 


If the object is inclined, the stick 


Fig. 12 1 4 



'*• Or the height of a vert object mn, 

* whose distance r m is known, may be found by 

Its reflection in a vessel of water, or in a piece of 

looking glass placed perfectly horizontal at r; for as r ais to the height 
a i of the ere ahovp t)»o roflontna «. 


a 1 °I the eye above the reflector r, so is r m to 
tt' the height mnof the object above r. 

, . . . ^ Rem. 4. Or let o c, Fig 12^, be 

» planted pole, or a rod held vert by an assistant. Then 

tand at a proper dist back from it, and keeping the eyes steady, let marks be 
naae at o and c, where the lines of sight i n and i m strike the rod. Then as 
c is to c o, so is im to m n. 



Fig. 12% 




0 

- r 

ji 



The following examples may be regarded as substitutes for strict trigonome¬ 
try : and will at times be useful, in case a table of sines, &c, is not at hand for 
making trigonometrical calculations. 


Ex. 8. To find the dist a b, of which one end only 

is accessible. 

Drive a stake at any convenient point a ; from a lay off any angle b a c. In 
the line a c, at any convenient point c, drive a stake ; and from c lay off an angle 
ac d, equal to the angle &<ic. In the line c d, at any convenient point, as d, 
drive a stake. Then, standing at d, and looking at b, place a stake o in range 
with d b; and at the same time in the line a c. Measure ao, o c, and cd; then, 
from the principle of similar triangles, as 


Fig. 13. 


o c : c d : : a o : a b. 


Fig. 14. 



Or thus: 

Pig 14, n h being the dist, place a stake at n; and lay off the angle hnm 90°. 
At any convenient dist n m, place a stake m. Make the angle limy — 9 0°; and 
place a stake at y, in range with An. Measure ny and n m ; then, from the 
principle of similar triangles, as 

« y : n m :: n m : n h. 

Or thus. Fig 14. Lay off the angle hnm — 90°, placing a stake 

m, at any convenient dist n m. Measure n m. Also measure the angle n m A. 
Find nat tang of n m A by Table Mult this nat tang by n m. The prod 

will be n A. 

Or thus. Lay off angle hnm — 90°. From m measure the 

angle n m A, and lay off angle n m y equal to it, placing a stake at y in range 
with A n. Then is ny — n A. 



Or thus, without measuring 
any tingle; 

t u being the dist. Make u v of any convenient 
length, in range with t u. Measure any v o; and 
o x equal to it, in range. Measure no; and oy 
equal to it in range. Place a stake z in range with 
both x y, and t o. Then will y z be both equal to 
t u, and parallel to it. 



Or thus, without measuring any angle. 

Drive two stakes t and w, in range with the object s. From t lay off any 
convenient dist t x, in any direction. From w lay off m w parallel to t x, 
placing to in range with x s. Make u v equal to t x. Measure w v, v x, and 
x t. Then, as 

w v : v x : : x t: t s. 

Or thus. At a lay off angle oac = 5° 43'. Lay 
off oc at right angles to ao. Measure oc. Then 
no = 10 oc, too long only 1 part in 935.6, or 5.643 feet 
in a mile, or .1069 foot (full l£ inches) in 100 feet. 






























118 


PLANE TRIGONOMETRY 



n 

Fig-. 17. 


Ex. 9. To find the dist a b, of which the 
ends only are accessible. 

From a lay off the angle 6 a c; and from ft, the angle aid. each! 
90°. Make a c and b d equal to eaoh other. Then. cd~ab. Or J 
a b may he considered as the dist across the river in Figs 15, 13. or | 
14: and be ascertained in the same way. Or measure any dist. Fig 
17. no; and make o n in line and equal to it. Also measure ho; 
and make om in line and equal to it. Then will mn be both paral¬ 
lel to a b, and equal to it. 


Ex. 10. See Ex. 4. To find the entirely 
inaccessible dist y z> and also 
its direction. 

At any two convenient points a and b, from each of which 
y and z can be seen, drive stakes. Theu we have the four 
corners of a four sided figure, in which are given the directions 
of three of its sides, and of its two diags. These data euable us 
to lay out ou the ground, the small four sided fig acoi, exactly 
similar to the large one. Thus, iu the line a b place a stake 
c: and make co parallel to b z; o being at the same time in 
rauge of the diag a z. Also, from c make ct parallel to by; 
i being at the same time in rauge of a y. Then will to be in 
the same direction as y z. or parallel to it. Measure a c, a b, 
and io; then evidently, from the princiDle of similar figures, as 

a c : a b : : i o : y z. 

If y z were a visible line such as a fence or road, we could 
from a divide it iuto any required portions. Thus, if we wish 
to place a stake halfway between y aud z. first place one half¬ 
way between i and o; then standing at a, by means of signals, 
place a person in range on y z. Or, to find along a 6, a point t 
perp to y z at y, first make oi« - 90°; aud measure a s. Then, 
as , 

o t : a 8 : : y z : a t. 



Ex. 11. To find the position of a point, n, Fig 19, 


By meant of two angles a n b and b n c. taken from it to the three objects a b c, whose positions 

and diets apart are known. 



The use of this problem is more frequent in mariue than in land surveying. It is chiefly employed 
for determining the position n of a 
boat from which soundings are being 
taken along a coast. As the boat 
moves from point to point to take 
fresh soundings, it becomes necessary 
to make a fresh observation at each 
point, in order to define its position 
on the chart. An observation consists 
in the measurement by a sextant of 
the two angles an ft, b n c. to the sig¬ 
nals ab c. previously arranged on the 
shore. When practicable, this method 
shouM be rejected; and the observa- 
tions taken to the boat at the same 
instant, by two observers on shore, at 
two of the stations. The boat to show 
a signal at the proper moment. The 
most expeditious mode of fixing the 
point n upon the map, is to draw three 
lines, forming the two angles, and ex¬ 
tended indefinitely, on a piece of trans¬ 
parent paper. Place the paper upon the map, and move it about until the three lines passthrough 
the three stations ; then prick through the poiut n wherever it happens to come. 

Instead of the transparent paper, an instrument called a station pointer may be used when there 
are many, points to be fixed. 


Fig. 19. 


I 

I 

i 

i 

t 


t 

i 

t 

t 


But the position of the point « can be found more correctly by describing two circles, as in Fig 19, 
each of which shall pass through n and two of the station points. The question is to find the centers 
o and xof two such circles. This is very simple. We know that the angle a o ft at the center of a circle is 
twice as great as any angle a n b at the circumf of the same circle, when both are subtended by the 
same chord a ft. Consequently, if the angle a n ft, observed from the boat, is say. 50°, the angle a o 6 
must be 100 And, since the three angles of every plane triangle are equal to 180°, the two angles 
o (i ft and o 6 a are together equal to 180° — 100° 80°. And, since the two sides a o and 6 o are 

equal (being radii of the same circle), therefore, the angles oab and o ft a are equal; and each equal to 

~ 2 ~ ~ *® 0, Consequently, on the map we have only to laydown at a and ft, two angles of 40°; the 


point o of intersection will be the center of the circle a ft n. Proceed in the same wav with the angle 
ft n c, to find the center *. Then the intersection of the two circles at n will be the point sought. 





















PARALLELOGRAMS. 


119 


PARALLELOGRAMS. 


Square. Rectangle. 


a 


/ 



* 

/ 

/ 

p 


s 


s 


Rhombus. Rhomboid. 



A parallelogram is any figure of four straight sides, the opposite ones of which 
are parallel. There are but four, as in the above figs. The rhombus, like the rhom- 
boliedron, Fig 3, p 155, is sometimes called “ rhomb.” In the square and rhombus 
all the four sides are equal; in the rectangle and rhomboid only the opposite ones 
are equal. In any parallelogram the four angles amount to four right angles, or 
360°; and any two diagonally opposite angles are equal to each other; hence, having 
one angle given, the other three can readily be found. In a square, or a rhombus, a 
diag divides each of two angles into two equal parts; but in the two other parallel¬ 
ograms it does not. 


To find the area of any parallelogram. 

Multiply any side, as S, by the perp height, or dist p to the opposite side. Or, multiply together 
two sides and nat sine of their included angle. 

The diag a b of any square is equal to one side mult by L41421 ; and a side is equal to 
diagoual 

i 4142 T ’ or ’ t0 diag mult by • 7on ° 7 * 

The side of a square equal in area to a given circle, is equal to diam X .886227. 

The side of the greatest square, that can be inscribed in 
a given circle, is equal to diaiu X .707107. 

The side of a'square mult by 1.51967 gives the 6lde of an equi¬ 
lateral triangle of the same area. All parallelograms as A 
and C, which have equal bases, a c, and equal perp heights n 
c, have also equal areas ; aud the area of each is twice that of a tri¬ 
angle having the same base, and perp height. The area of a 
square Inscribed in a eirele is equal to twice the square of the « cm n 

fad. ^ 

In every pflTfi.llclogrfl.in* the 4 squares drawn on its sides have a united area equal to that of 
the two squares drawn ou its 2 diags. If a larger square be drawn on the diag ah of a smaller 
square, its area will be twice that of said smaller square. Hither diag of any parallelogram 
divides it into two equal triangles, and the 2 diags div it into 4 triangles of equal areas. The two 
diags of any parallelogram divide each other iu to two equal parts. Any line drawn through 
the center of a diag divides the parallelogram into two equal parts. 

Remark 1. — The area of any fig whatever as B that is enclosed by four straight 
lines, may be found thus : Mult together tue two diags a m, n b ; aud tiie nat sine of the least augle 
a o b ; or n o m, formed by their intersection. Div the product by 2. This is useful in land surveying, 
when obstacles, as is often the case, make it difficult to measure the sides of the fig or field; while it 
may be easy to measure the diags; and after finding their poiut of intersection o, to measure the re¬ 
quired angle. But If the fig is to be drawn, the parts o a, o b, o n, o m of the diags must also 

be measd. . 

Rem. S. —The sides of a parallelogram, triangle, and many other ligs may be 
found, when only the area and angles are given, thus: Assume some particular one of its 
sides to be of the leugth 1 ; and calculate what its area would be if that were the case. Then as the 
sq rt of the area thus found is to this side 1, so is the sq rt of the actual given area, to the corre¬ 
sponding actual side of the fig. 




On a given line wx,to draw a square,, 

w x n m. 

Prom w and x, with rad w x, describe the arcs xry and xv re. 
Prom their intersection r, and with rad equal to % of iv x, describe 
s s s. From to and x draw w n and x m tangential to s s s, and 
ending at the other arcs; join n m. 




















120 


TRAPEZOIDS AND TRAPEZIUMS, 


TRAPEZOIDS. 





A trapezoid at c n to, is any figure with four straight sides, ODly two of which, as ac and n w, zr 
parallel. . 

To find the area of any trapezoid. 

Add together the two parallel sides, a c and m n ; mult the sum by the perp dist s t betweei 
them; div the prod by 2. See the following rules for trapeziums, which are all equally applicabli 
to trapezoids; also see Remarks after Parallelograms. 


TRAPEZIUMS. 



A trapezium a b c o, is any fig with four straight sides, of which no two are parallel. 


To find the area of any trapezium, having given the diag 
bo, or a c, between either pair of opposite angles: and also 
the two perps, n, n, from tlie other two angles. 

Add together these two perps; mult the sum by the diag; div the prod by 2. 

Having the fonr sides; and either pair of opposite angles, 

as a b c, a o c; or b a o, ami b c o. 

Consider the trapezium as divided into two triangles, in each of which are given two sides and the 
included angle. Find the area of each of these triangles as directed under the preceding head “ Tri¬ 
angles.” and add them together. 

Having the fonr angles, and either pair of opposite sides. 

Begin with one of the sides, and the two angles at its ends. If the sum of these two angles exceeds 
180°, subtract each of them from 180°, and make use of the reins instead of the angles themselves. 
Then consider this side and its two adjacent angles (or the two rems, as the case may be) as those 
ef a triangle; and find its area as directed for that case under the preceding head “ Triangle.” Do 
the same with the other given side, and its two adjacent angles, (or their rems, as the case may be.) 
Subtract the least of the areas thus found, from the greatest; the rem will be the reqd area. 

Having tliroc sides; and flic two included angles. 

Mult together the middle side, and oue of the adjacent sides ; mult the prod by the nat sine of their 
included angle; call the result u. Do the same with the middle side and its other adjacent side, 
and the nat sine of the other iucluded angle; call the result b. Add the two angles together; find 
the ditf between their sum and 180°, whether greater or less; find the nat sine of this diff; mult 
together the two given sides which are opposite oue another; mult the prod by the nat sine just foand; 
call the result c. Add together the results a and b ; then, if the sum of the two given angles is lest 
than 180°, subtract c from the sum of a and b ; half the rem will be the area of the trapezium. But 
if the sum of the two given angles be greater than 180°, add together the three results a, b, and c; 
half their sum will be the area. 

Having the two diagonals, and either angle formed by their 

intersection. 

See Remarks after Parallelograms, p 119. 

In railroad measurements 

Of excavation and embankment, the trapezium 
l to n o frequently occurs; as well as the two 5-sided 
figures ( to n o t and l in n o s; in all of which m n 
represents the roadway ; rs.rc, and r t the center- 
depths or heights; l u and o v the side-depths or 
heights, as given by the level; I to and n o the side- 
slopes. 

The same general rule for area applies to all three 
of these figs; namely, mult the extreme hor width 
tt v by half the center depth r s, r c. or r t, as the 
case may be. Also mult one fourth of the width of 
roadway to n, by the sum of the two side-depths l u 
and o v. Add the two prods together; the sum is the 
reqd area. This rule applies whether the two side- 
slopes to i and n o have the same angle of inclination or not. In railroad work, etc., the mid¬ 
way hor width, center depth, and side depths of a prismoid are respectively = The half sums ot 
the corresponding end ones, and thus can be found without actual measurement. 




















POLYGONS. 


121 


To draw a hexagon, each side of which shall 
be equal to a given line, a b. 

From a and b, with rad a b, describe the two arcs; from their intersection, 
i, with the same rad, describe a circle; around the circumf of which, step off 
the same rad. 

Side of a hexagon = nn X .57735. 



i To draw an octagon, with each side 
equal to a given line, c e. 

From c and e draw two perps, cp, ep. Also prolong c e toward 
/ and g; and from c and e, with rad equal c e, draw the two 
quadrants ; and find their centers h h : join c h, and e h ; draw 
h s and h t parallel to cp; and make each of them equal to c e; 
make c o, and e o, each equal to h h ; join o o, o s, aud o t. 

Side of an octagon — nn x .41421354. 



To draw an octagon in a given square. 

, From each corner of the square, and with a rad equal to half its diag, 
> describe the four arcs; and join the points at which they cut the sides of the 
• square. 

To draw any regular polygon, with each side 
, equal to m w. 

Div 360 degrees by the number of sides; take the quot from 180°; div the 
, rem by 2. This will give the angle cm n, ore nm. At m and n lav down these 
! angles by a protractor: the sides of these angles will meet at a point, c, from 
which describe the circle mny, and around its circumf step off dists equal to 

m n. 

» 

i In any circle, m n y, to draw any regular 

polygon. 

® Div36fP by the number of sides ; thequot will betheangle men, at the center. 
, Lay off this’angle by a protractor; and its chord m n will be one side; which 
ite'p off around the circumf. 



To reduce any polygon, as abode fa, to a triangle of the 

same area. 



ir we produce the side fa toward w, and draw b g parallel to a c, and join g c, we get equal tri¬ 
angles ac b and a eg, both on the same base a c ; and both of the same perp height, inasmuch as 
tbev are between the two parallels a c and g b . But the part a c i forms a portion of both these tri- 
ancles or in other words, is common to both. Therefore, if it be taken away from both triangles, 
the remaining parts t c 6 of one of them, and iyaof the other, are also equal. Therefore, if the 
part i c 6 be left off from the polygon, and the part ijobe taken into it, the polygon g fed cig will 
have the same area as af e d c b a ; but it will have but live sides, while the other has six. Again, 
if e s be drawn parallel to d f, aud ds joined, we have upon the same, base es, and between the same 
parallels e s and d f the two equal triangles e s d, and e sf. with the part eos common to both ; and 
oouseaueutlv the remaining part e o d ot one, aud o sf of the other, are equal. Therefore, if o sf be 
left off from the polygon, aud e o d be taken into it, the new polygon g sd eg, Fig 2 will have the same 
area as a / e d c o; but it has but four sides, while the other has five. I-iually, if g s, Fig 2, be 
extended toward n ; and d n drawn parallel to c s; and c n joined, we have on the same base c *, and 
hetween the same parallels c s and d n, the two equal triangles cm.and c s d, with the part c s t 
common to both. Therefore, ir we leave out c d t, and take in « t n, we have the triangle gne equal 
to the polygou g s d c g. Fig 2; or to a f e d c b a, 1' ig 1. 

This simple method is applicable to polygons of auy number of sides. 
























122 


POLYGONS, 


fo reduce a large fig, abed e f g, to a smaller 
similar one. 

From any interior point o, which had better be near the center, draw lines 
to all the angles a, b, c, &c. Join these lines by others parallel to the sides 
of the fig. If it should be reqd to enlarge a small fig, draw, from any point 
o within it, lines extending beyond its auglcs; and join these lines by others 
parallel to the sides of the small fig. 


To reduce a map to one on a smaller 



scale. 


The best method is by dividing the large map into squares by faint lines, with 
pencil, and then drawing the reduced map upon a sheet of 
smaller squares. A pair of proportional dividers will assist 
much in fixing points intermediate of the sides of the squares. 

If the large map would be injured by drawing, and rubbing 
out the squares, threads may be stretched across it to form the 
squares. 

Maps, plans, and drawings of all kinds, are now copied, 
reduced, enlarged, and multiplied, cheaply and expeditiously, by 
photography. Most of the uewer illustrations in this work are 
from electrotypes made by the “wax process” of Struthers, 

Servoss & Co., 34 New Chambers St, New York, and American 
Bank Note Co, 78 Trinity Place, New York. 

Iii a rectangular fig, g h s d, 

Representing an open panel, to find the points o o o o in its 
sides; and at equal dists from the angles < 7 . and s; for inserting 
a diag piece o o o o, of a given width 11, measured at right 
angles to its length. From <7 and s as centers, describe several 
concentric arcs, as in the Fig. Draw upon transparent paper, 
two parallel lines a a. c c. at a distance apart equal to l l ; and 
placing these lines on top of the panel, move them about until it 
is shown by the arcs that the four dists g o. <7 o, s o, s o, are 
equal. Instead of the transparent paper, a strip of common 
paper, of the width l l may he used. 

Rem. Many problems which would otherwise be very difficult, 
may be thus solved with an accuracy sufficient for practical 

purposes, by means of transparent paper. 




To find tlie area of any irregular poly* 
gon. a 11 b c in. 

Div it into triangles, as a n 6 , a m c, and a b c; in each of 
which find the perp dist o, between its base a b, a c, or b c; and 
the opposite angle n. to, or a; mult each base by its perp dist; 
add all the prods together; div by 2 . 

To find approx the area of a long ir¬ 
regular fig, as abed. Between its ends a b, c d, 



space off equal dists, (the shorter they are the more accurate will be the result,) through whicl 
draw the intermediate parallel lines 1, 2, 3, &c, across the breadth of the fig. Measure the length 
of these intermediate lines ; add them together: to the sum add half the sum of the two end breadth 
a b and c d. Mult the entire sum by one of the equal spaces between the parallel lines. The proc 
will be the area. This rule answers as well if either one or both the ends terminate in points, as at t? 
and n. In th"? last of these cases, both a b and c d will be included in the intermediate lines; am 
half the two end breadths will be 0 , or nothiug. 

To find tlie area of a fig whose outline is extremely 

irregular. 

Draw lines around it which shall enclose withit 
them (as nearly as can be judged by eye) as muct! 
space not belonging to the fig, as they exclude spact i 
belonging to it. The area of the simplified fig thus 
formed,beingtn this manner rendered equal to thai 
of the com plicated one, may be calculated by dividing 
it into triangles, &c. By using a piece of fine thread 
the proper position for the new boundary lines may 
be found, before drawingthem in. Small irregulai 
areas may be found from a drawing, by laying upon 
it a piece of transparent paper carefully ruled intc 
small squares, each of a given area, say 10, 20. 01 i 
100 sq ft each; and by first counting the whole' 
squares, and then adding the fractions of squares. 


























CIRCLES 


123 


CIRCLES. 

A circle is the area included within a curved line of such a character that every point in it is 
squally distant from a certain point within it, called its center. The curved line itself is called the 
iircumference, or periphery of the circle; or very commonly it is called the circle. 

i To find the circumference. 

Mult diam by 3.1416, which gives too much by only .148 of an inch in a mile. Or, as 113 is to 355 
so is diam to circumf; too great 1 inch in 186 miles. Or, mult diam by 3y ; too great by about 1 
, part in 2485. Or, mult area by 12.566, and take sq root of prod. Or, use tables pp. 125 &c. The 
Sreek letter tt, also p, is used by writers to denote this 3.1416; and p 2 — 9.86960. 

> To find the diam. 

Div the Circumf by 3.1416 ; or, as 355 is to 113, so is circumf to diam ; or, mult the circumf. by 7: 
and div the prod by 22, wbioh gives the diam too small by only about one part in 2485; or, mult the 
area by 1.2732; and take the sq rt of the prod; or use tables of circles, pp 125, &c. 

The diam is to the circumf more exactly as 1 to 3.14159265. 

To find the urea of a circle. 

Square the diam; mult this square by .7854; or more accurately by .78539816; or square the cir- 
:utnf; mult this square by .07958: or more accurately by .07957747 ; or mult half the diam by half the 
jircumf; or refer to the following table of areas of circles. Also area — sq of rad X 3.1416. 

The area of a circle is to the area of any circumscribed straight sided fig, as the circumf of the 
:ircle is to the circumf or periphery' of the fig. The area of a square inscribed in a circle, is equal to 
; ,wice the square of the rad. Of a circle in a square, — square X .7854. 

It is convenient to remember, in rounding off a square corner a b c, by a quarter of j 
i circle, that the shaded area a b c is equal to about 1 part (correctly .2146) of the u 
vhole square abed. 5 

C 

For tables of circumferences and areas of cir¬ 
cles, see pages 125 to 140. 


To find the diam of a circle equal in area to a given square. 

Mult one side of the square by 1.12838. 

To find the rad of a circle to circumscribe a given square. 

Mult one side by .7071; or take % the diag. 

To find the side of a square equal in area to a given circle. 

Mult the diam by .88623. 

To find the side of the greatest square in a given circle. 

Mult diam by .7071. The area of the greatest square that can be inscribed in a circle is equal t* 
twice the square of the rad. The diam X by 1.3468 gives the side of an equilateral triangle of equal area. 






To find the center c, of a given circle. 

Draw any chord a b j and from the middle of it o, draw at right angles tc 
it, a diam d g ; find the center c of this diam. 



To describe a circle through any three 
points, abc, not in a straight line. 

Join the points by the lines ah , be; from the centers of these lines draw 
the dotted perps meeting, as at o, which will be the oenter of the circle. 
Or from b, with any convenient rad. draw the arc m n; and from a and c, 
with the same rad, draw arcs y and z ; then two lines drawn through the 
intersections of these arcs, will meet at the oenter o. 

To describe a circle to touch the three 
angles of a triangle is plainly the same as this. 

To inscribe a circle in a triangle draw two lines 

bisecting any two of the angles. Where these lines meet is the center of 
the circle. 













124 


CIRCLES, 







To draw a tangent, i e i, to a circle, from nnj 
given point, e, in its circumf. 

Through the center n, and the given point e, draw n o ; make e o equal t< * 
e n : from » and o, with any rad greater than half of o », describe the tw< j 
pairs of arc i i; join their intersections i i. 

Here, and in the following three figs, the tangents are ordinary or geo ' 
metrical ones; and may end where we please. But the trigonometrica 
tangent of a given angle , must end in a secant, as in the top fig of p 59. 

Or from e lay off two equal distances e a, e t; and draw i i 

parallel to c t. 


To draw a tang, n s h, to a circle, from a point, 
a, which is outside of the circle. 


Draw a c, and on it describe a semicircle; through the intersection, s, draw 
a s b. Here c is the center of the circle. 



To draw a tang, g 7i, from a circular arc, g a c , 

Of which n a is the rise. With rad g a, describe an arc, « a o. Make t a 
equal to s a. Through t draw g ft. 


y ns 


To draw a tang to two circles. 

First draw tne line m n, just touching the two 
circles; this gives the direction of the taug. Then 
from the centers of the circles draw the radii, o o, perp 
to m n. The points t t are the taug points. If the 
tang is in the position of the dotted line, s y, the ope¬ 
ration is the same. 

Rem. This empirical method is at 

least as accurate as the scientific ones, especially if 
a correct triangular ruler is used for the radii. 



C 



If any two chords, as a b, o c, cross each other, 

then as o n : n b :: a n : n c. Hence, nb X an = onXnc. That 
is, the product of the two parts of one of the lines, is = the pro¬ 
duct of the two parts of the other line. 






0 










CIRCLES 


125 


TABLE 1 OF CIRCLES. 

Diameters in units and eighths, &c. 

Circumferences or areas intermediate of those in this table, may be found by sim¬ 
ple arithmetical proportion. No errors. 


Diam 

Circumf. 

Area. 

Diam- 

Circumf. 

1 Area. 

Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area, 

1-64 

.019087 

.00019 

3. X 

10.9956 

9.6211 

10W 

31.8086 

80.516 

19 X 

60.4757 

291.04 

1-32 

.098175 

.00077 

9-16 

11.1919 

9.9678 

X 

32.2013 

82.516 

% 

60.8684 

294.83 

3-61 

.117262 

.00173 

% 

11.3883 

10.321 

% 

32.5940 

84.541 

X 

61.2611 

298.65 

1-16 

.196350 

.00307 

11-16 

11.5846 

10.680 

X 

32.9867 

86.590 

% 

61.6538 

302.49 

3-32 

.291524 

.00690 

X 

11.7810 

11.045 

% 

33.3794 

88.664 

X 

62.0435 

306.35 

X 

.392699 

.01227 

13-16 

11.9773 

11.416 

X 

33.7721 

90.763 

X 

62.4392 

310.24 

5-32 

.490874 

.01917 

% 

12.1737 

11.793 

% 

34.1618 

92.886 

20. 

62.8319 

314.16 

3-16 

589019 

.02761 

15-16 

12.3700 

12.177 

11. 

34.5575 

95.033 

X 

63.2246 

318.10 

7-32 

.687223 

.03758 

4. 

12.5664 

12.566 

X 

34.9502 

97.205 

X 

63.6173 

322.06 

X 

.785398 

.04909 

1-16 

12.7627 

12.962 

X 

35.3429 

99.402 

X 

64.0100 

326.05 

9-32 

.883573 

.06213 

X 

12 9591 

13.364 

% 

35.7356 

101.62 

k 

64.4026 

330.06 

5 16 

.981718 

.07670 

3-lrt 

13.1551 

13.772 

X 

36.1283 

103.87 

% 

64.7953 

334.10 

11-32 

1.07992 

.09281 

x 

13.3518 

14.186 

% 

36.5210 

106.14 

% 

65.1880 

338.16 

% 

1.17810 

.11015 

5-16 

13.5181 

11.607 

X 

36.9137 

108.43 

X 

65.5807 

342.25 

13-32 

1.27627 

.12952 

% 

13.7445 

15.033 

X 

37.3061 

110.75 

21. 

65.9734 

346.36 

7-16 

1.37415 

.15033 

7-16 

13.9108 

15.166 

12. 

37.6991 

113.10 

X 

66.3661 

350.50 

15 32 

1.47262 

.17257 

X 

14.1372 

15.901 

X 

38.0918 

115.47 

X 

66.7588 

354.66 

X 

1.57080 

.19635 

9-16 

14.3335 

16.349 

X 

38.4815 

117.86 

Vs 

67.1515 

358.84 

i7-32 

1.66897 

.22166 

% 

14.5299 

16.800 

X 

38.8772 

120.28 

k 

67.5442 

363.05 

9-16 

1.76715 

.21850 

11-16 

11.7262 

17.257 

X 

39.2699 

122.72 

% 

67.9369 

367.28 

19-32 

1.86532 

.27688 

X 

14.9226 

17.721 

% 

39.6626 

125.19 

X 

68.3296 

371.54 

% 

1.96350 

.30680 

13-16 

15.1189 

18.190 

X 

40.0553 

127.68 

X 

68.7223 

375.83 

21-32 

2.06167 

.33821 

x 

15.3153 

18.665 

X 

40.4480 

130.19 

22. 

69.1150 

380.13 

11-16 

2.15981 

.37122 

15-16 

15.5116 

19.147 

13. 

40.8107 

132.73 

X 

69.5077 

884.46 

23-32 

2.25802 

.40571 

5. 

15.7080 

19.635 

X 

41.2334 

135.30 

X 

69.9004 

388.82 

X 

2.35619 

.41179 

1-16 

15.9043 

20.129 

X 

41.6261 

137.89 

% 

70.2931 

393.20 

25-32 

2.15437 

.47937 

% 

16.1007 

20.629 

% 

42.0188 

110.50 

X 

70.6858 

397.61 

13-16 

2.55251 

.51819 

3-16 

16.2970 

21.135 

X 

42.4115 

113.14 

% 

71.0785 

402.04 

27-32 

2.65072 

.55911 

x 

16.4931 

21.618 

% 

42.8042 

145.80 

X 

71.4712 

406.49 

X 

2.74889 

.60132 

5-16 

16.6897 

22.166 

X 

43.1969 

148.49 

X 

71.8639 

410.97 

29-32 

2.81707 

.61501 

X 

16.8861 

22.691 

X 

43.5896 

151.20 

23. 

72.2566 

415.48 

15-16 

2.94524 

.69029 

7-16 

17.0821 

23.221 

14. 

43.9823 

153.94 

X 

72.6493 

420.00 

31-32 

3.01312 

.73708 

X 

17.2788 

23.758 

X 

44.3750 

156.70 

X 

73.0420 

424.56 

I. 

3.11159 

.78510 

9-16 

17.1751 

24.301 

X 

44.7677 

159.48 

% 

73.4347 

429.13 

1-16 

3.33791 

.88664 

% 

17.6715 

24.850 

X 

45.1604 

162.30 

X 

73.8274 

433.74 

M 

3.53429 

.99402 

11-16 

17.8378 

25.406 

X 

45.5531 

165.13 

% 

74.2201 

438.36 

3-16 

3.73061 

1.1075 

X 

18.0642 

25.967 

X 

45.9458 

167.99 

X 

74.6128 

443.01 

X 

3.92699 

1.2272 

13-16 

18.2605 

26.535 

X 

46.3385 

170.87 

X 

75.0055 

447.69 

5-16 

4.12334 

1.3530 

X 

18.1509 

27.109 

X 

46.7312 

173.78 

24. 

75.3982 

452.39 

% 

4 31969 

1.4819 

15-16 

18.6532 

27.688 

15. 

47.1239 

176.71 

X 

75.7909 

457.11 

7-16 

4.51604 

1.6230 

6. 

18.8196 

28.274 

X 

47.5166 

179.67 

X 

76-1836 

461.86 

X 

4.71239 

1.7671 

X 

19.2423 

29.465 

X 

47.9093 

182.65 

% 

76 5763 

466.64 

9-16 

4.90874 

1.9175 

X 

19.6350 

30.680 

% 

48.3020 

185.66 

X 

76.9690 

,71.44 

% 

5.10509 

2.0739 

% 

20.0277 

31.919 

X 

48.6947 

188.69 

% 

77.3617 

476.26 

11-16 

5.30144 

2.2365 

X 

20.4201 

33.183 

% 

49.0874 

191.75 

X 

77.7544 

481.11 

X 

5.49779 

2.4053 

% 

20.8131 

34.472 

X 

49.4801 

194.83 

X 

78.1471 

485.98 

13-16 

5 69414 

2.5802 

X 

21.2058 

35.785 

X 

49.8728 

197.93 

25. 

78.5398 

490.87 

x 

5.89049 

2.7612 

y* 

21.5984 

37.122 

16. 

50.2655 

201.06 

X 

78.9325 

495.79 

15-16 

6.08684 

2.9483 

7. 

21.9911 

38.485 

X 

50.6582 

204.22 

X 

79.3252 

500.74 

2. 

6.28319 

3.1416 

x 

22.3838 

39.871 

X 

51.0509 

207.39 

H 

79.7179 

505.71 

1-16 

6.17953 

3.3410 

X 

22.7765 

41.282 

% 

51.4436 

210.60 

X 

80.1106 

510.71 

X 

6.67588 

3.5466 

% 

23.1692 

42.718 

X 

51.8363 

213.82 

% 

80.5033 

515.72 

3-16 

6 87223 

3.7583 

X 

23.5619 

44.179 

% 

52.2290 

217-08 

X 

80.8960 

520.77 


7.06858 

3.9761 

% 

23.9546 

45 664 

X 

52.6217 

220.35 

X 

81.2887 

525.84 

5 16 

7.26193 

4.2000 

X 

24.3473 

47.173 

X 

53.0144 

223.65 

26. 

81.6814 

530.93 

y» 

7.16128 

4.4301 

y» 

21.7400 

48.707 

17. 

53.4071 

226.98 

X 

82.0741 

536.05 

7-16 

7.65763 

1.6661 

8. 

25.1327 

50.265 

X 

53.7998 

230.33 

X 

82.4668 

541.19 

X 

7.85398 

4.9037 

x 

25.5254 

51.849 

X 

51.1925 

233.71 

X 

82.8595 

546.35 

9-16 

8.05033 

5.1572 

X 

25 9181 

53.456 

% 

54.5852 

237.10 

X 

83.2522 

551.55 

% 

8.21668 

5.4119 

x 

26.3108 

55.088 

X 

54.9779 

240.53 

% 

83.6449 

556.76 

11-16 

8.11303 

5.6727 

X 

26.7035 

56.745 

% 

55.3706 

243.98 

X 

84.0376 

562.00 

X 

8.63938 

5.9396 

% 

27.0962 

58.426 

X 

55.7633 

247.45 

X 

84.4303 

567.27 

13-16 

8.83573 

6.2126 

X 

27.4889 

60.132 

X 

56.1560 

250.95 

27. 

84.8230 

572.56 

y» 

9.03208 

6.4918 

y» 

27.8816 

61.862 

18. 

56.5487 

254.47 

X 

85.2157 

577.87 

15-16 

9.22813 

6.7771 

9. 

28.2713 

63.617 

X 

56.9414 

258.02 

a 

85.6084 

583.21 


9.42478 

7.0686 

x 

28.6670 

65.397 

X 

57.3341 

261.59 

h 

86.0011 

588.57 

1-16 

9.62113 

7.3662 

x 

29.0597 

67.201 

X 

57.7268 

265.18 


86 3938 

593.98 

M 

9.81748 

7.6699 

% 

29.4524 

69.029 

k 

58.1195 

208.80 

X 

86.7865 

599.37 

3-16 

10.0138 

7.9798 

X 

29.8451 

70.882 

% 

58.5122 

272.45 

X 

87.1792 

604.81 

y. 

10.2102 

8.2958 

% 

30.2378 

72.760 

X 

58.9049 

276.12 

X 

87.5719 

610.27 

5-1G 

10.4065 

8.6179 

X 

30.6305 

74 662 

X 

59.2976 

279.81 

28. 

87.9646 

615.75 

3/„ 

10 6029 

8.9462 

y* 

31.0232 

76.589 

19. 

59.6903 

283.5.3 

X 

88.3573 

621.26 

7-16jl0.7992 

9.2806 

10. 

31.4159 

78,540 

X 

60.0830 

287.27 

X 

88.7500 

626.80 

































126 


CIRCLES 


TABLE 1 OF CIRCLES— (Continued). 
Diameters in units and eighths, Ac. 


Diatn. 

Circuraf. 

Area. 

Diam. 

Circuraf. 

Area. 

Diam- 

Circumf. 

Area. 

Diam- 

Circumf. 

Area 

-- . 


89.1427 

632.36 

38. 

119 381 

1134.1 

47^ 

149.618 

1781.4 

b7X 

179.856 

2574. 

X 

89.5354 

637.94 

X 

X 

119.773 

1141.6 

X 

150.011 

1790.8 

X 

180.249 

2585-1 

X 

89.9281 

643 55 

120.166 

1149.1 

X 

150.404 

1800.1 

X 

180.642 

2596. 

X 

90.3208 

649.18 

X 

120.559 

1156.6 

48. 

150.796 

1809.6 

X 

181.034 

2608. 

% 

90.7135 

654.84 

X 

120.951 

1164.2 

X 

151.189 

1819.0 

X 

181.427 

2619. 

29. 

91.1062 

660.52 

X 

121.344 

1171.7 

X 

151.582 

1828.5 

X 

181.820 

2630. 

X 

91.4989 

666.23 

X 

121.737 

1179.3 

X 

151.975 

1837.9 

58. 

182.212 

2642. 

x 

91.8916 

671.96 

X 

122.129 

1186.9 

X 

152.367 

1847.5 

X 

182.605 

2653.• 

% 

92.2843 

677.71 

39. 

122.522 

1194.6 

X 

152.760 

1857.0 

X 

182.998 

2664. 

X 

92.6770 

683.49 

X 

122.915 

1202.3 

X 

153.153 

1866.5 

X 

183.390 

2676. i 

% 

93.0697 

689.30 

X 

123.308 

1210.0 

X 

153.545 

1876.1 

X 

183.783 

2687. 

x 

93.4624 

695.13 

X 

123.700 

1217.7 

49. 

153.938 

1885.7 

X 

184.176 

2699.: 

% 

93.8551 

700.98 

X 

124.093 

1225.4 

X 

154.331 

1895.4 

X 

184.569 

2710.: 

so. 

91.2478 

706.86 

X 

124.486 

123.3.2 

X 

154.723 

1905.0 

X 

184.961 

2722. 

X 

94.6405 

712.76 

X 

124.878 

1241.0 

X 

155.116 

1914.7 

59. 

185.354 

2734.1 

X 

95.0332 

718.69 

X 

125.271 

1248.8 

X 

155.509 

1924.4 

X 

185.747 

2745.1 

% 

95.4259 

724.64 

40. 

125.664 

1256.6 

X 

155 902 

1934.2 

X 

186.139 

2757.: 

X 

95.8186 

730.62 

X 

126.056 

1264.5 

X 

156.294 

1943.9 

X 

186.532 

2768.! 

% 

96.2113 

736.62 

X 

126.449 

1272.4 

X 

156.687 

1953.7 

X 

186.925 

27»O.I 

X 

96.6010 

742.64 

X 

126.842 

1280.3 

50. 

157.080 

1963.5 

X 

187.317 

2792.:j 

y» 

96.9967 

748.69 

X 

127.235 

1288.2 

X 

157.472 

1973.3 

X 

187.710 

2803.! J 

31. 

97.3894 

754.77 

% 

127.627 

1296.2 

X 

157.865 

1983.2 

X 

188.103 

2815.' 

X 

97.7821 

760.87 

X 

128.020 

1304.2 

X 

158.258 

1993.1 

60. 

188.496 

2827.-i 

X 

98.1748 

766.99 

X 

128.413 

1312.2 

X 

158.650 

2003-0 

X 

188.888 

2839.! 

x 

98.5675 

773.14 

41. 

128.805 

1320.3 

X 

159.043 

2012.9 

X 

189.281 

2851.1 

X 

98.9602 

779.31 

X 

129 198 

1328.3 

X 

159.436 

2022.8 

X 

189.674 

2862.! 

% 

99.3529 

785.51 

X 

129.591 

1336.4 

X 

159.829 

2032.8 

X 

190.066 

2874.1 

X 

99.7456 

791.73 

X 

129.983 

1344.5 

51. 

160.221 

2042.8 

X 

190.459 

2886.1 

X 

100.138 

797.98 

X 

130.376 

1352 7 

X 

160.614 

2052.8 

X 

190.852 

2898.1 

32. 

100.531 

804.25 

X 

130.769 

1360.8 

X 

161.007 

2062.9 

X 

191.244 

2910.! | 

X 

100.924 

810.54 

X 

131.161 

1369.0 

X 

161.399 

2073.0 

61. 

191.637 

2922.! 

x 

101.316 

816.86 

X 

131.554 

1377.2 

X 

161.792 

2083.1 

X 

192.030 

2934.! 

X 

101.709 

823.21 

42. 

131.947 

1385.4 

X 

162.185 

2093.2 

X 

192.423 

2946.!' 

X 

102.102 

829.58 

X 

132.310 

1393 .'7 

X 

162.577 

2103.3 

X 

192.815 

2958.! 

% 

102.494 

835.97 

X 

132.732 

1402.0 

X 

162.970 

2113.5 

X 

193.208 

2970.1 

X 

102.887 

842.39 

X 

133.125 

1410.3 

52. 

163.363 

2123.7 

X 

193.601 

2982.1 

X 

103.280 

848.83 

X 

133.518 

1418.6 

X 

163.756 

2133.9 

X 

193.993 

2994.f 

33. 

103.673 

855.30 

X 

133.910 

1427.0 

X 

164.148 

2144.2 

X 

194.386 

3006.5 

X 

104.065 

861.79 

X 

134.303 

1435.4 

X 

164.541 

2154.5 

62. 

194.779 

3019.1 

x 

104.458 

868.31 

X 

134.696 

1443.8 

X 

164.934 

2164.8 

X 

195.171 

303 U 

X 

104.851 

874.85 

43. 

135.088 

1452.2 

X 

165.326 

2175.1 

X 

195.564 

3043.J 

X 

105.243 

881.41 

X 

135.481 

1460.7 

X 

165.719 

2185.4 

X 

195.957 

3055.7 

X 

X 

105.636 

888.00 

X 

135.874 

1469.1 

X 

166.112 

2195.8 

X 

196.350 

3068.C 

106.029 

894.62 

X 

136.267 

1477.6 

53. 

166.504 

2206.2 

X 

196.742 

3080.3 

X 

106.421 

901.26 

X 

136.659 

1486.2 

X 

166.897 

2216.6 

X 

197.135 

3092.6 

34. 

106.814 

907.92 

% 

137.052 

1494.7 

X 

167.290 

2227.0 

X 

197.528 

3104.S 

X 

107.207 

914.61 

X 

137.445 

1503.3 

X 

167.683 

2237.5 

63. 

197.920 

3117.2 

X 

107.600 

921.32 

X 

137.837 

1511.9 

X 

168.075 

2248.0 

X 

198.313 

3129.6 

X 

107.992 

928.06 

44. 

138.230 

1520.5 

X 

168.468 

2258.5 

X 

198.706 

3142.0 

X 

108.385 

934.82 

X 

138.623 

1529.2 

X 

168.861 

2269.1 

X 

199.098 

3154.5 

X 

108.778 

941.61 

X 

139.015 

1537.9 

X 

169.253 

2279.6 

X 

199.491 

3166.9 

X 

109.170 

948.42 

X 

139.408 

1546.6 

54. 

169.646 

2290.2 

X 

199.884 

3179.4 

X 

109.563 

955.25 

X 

139.801 

1555.3 

X 

170.039 

2300.8 

X 

200.277 

3191.9 

ib. 

109.956 

962.11 

X 

140.194 

1564.0 

X 

170.431 

2311.5 

X 

200.669 

3204.4 

X 

110.348 

969.00 

X 

140.586 

1572.8 

X 

170.824 

2322.1 

64. 

201.062 

3217.0 

X 

110.741 

975.91 

X 

140.979 

1581.6 

X 

171.217 

2332.8 

X 

201.455 

3229.6 

X 

111.134 

982-84 

45. 

141.372 

1590.4 

X 

171.609 

2343.5 

X 

201.847 

3242.1 

X 

111.527 

989.80 


141.764 

1599.3 

X 

172.002 

2354.3 

X 

202.240 

3254.3 

X 

111.919 

996.78 

X 

142.157 

1608.2 

X 

172.395 

2365.0 

X 

202.633 

3267.5 

X 

112.312 

1003.8 

X 

142.550 

1617.0 

55. 

172.788 

2375.8 

X 

203.025 

3280.1 

X 

112.705 

1010.8 

X 

142.942 

1626.0 

X 

173.180 

2386.6 

X 

203.418 

3292.8 

86. 

113.097 

1017.9 

X 

143.335 

1634.9 

X 

173.573 

2397.5 

X 

203.811 

3305.6 

X 

113.490 

1025 0 

X 

143.728 

1643.9 

% 

173.966 

2408.3 

65. 

204.204 

3318.3 

X 

113.883 

1032 1 

X 

144.121 

1652.9 

X 

174.358 

2419.2 

X 

204.596 

3331.1 

X 

114.275 

1039.2 

46. 

144.513 

1661.9 

X 

174.751 

2430.1 

X 

204.989 

3343.9 

X 

114.668 

1016.3 

X 

144.906 

1670.9 

X 

175.144 

2441.1 

% 

205.382 

3356.7 

X 

115.061 

1053.5 

X 

145.299 

1680.0 

X 

175.536 

2452.0 

X 

205.774 

3369.6 

X 

115.454 

1060.7 

X 

145.691 

1689.1 

56. 

175.929 

2463.0 

X 

206.167 

3382.4 

X 

115.846 

1068 0 

X 

146.084 

1698.2 

X 

176.322 

2474.0 

X 

206.560 

3395.3 

37. 

116.239 

1075.2 

% 

146.477 

1707.4 

X 

176.715 

2485.0 

X 

206.952 

3408.2 

X 

116.632 

1082.5 

X 

146.869 

1716.5 

X 

177.107 

2496.1 

66. 

207.345 

3421.2 

X 

117.024 

1089.8 

X 

147.262 

1725.7 

X 

177.500 

2507.2 

X 

207.738 

3434.2 

X 

117.417 

1097.1 

47. 

147.655 

1734.9 

X 

177.893 

2518.3 

X 

208.131 

3447.2 

X 

117810 

1104.5 

X 

148.048 

1744.2 

X 

178.285 

2529.4 

X 

208.523 

3460.2 

X 

118.202 

1111.8 

X 

148.440 

1753.5 

X 

178.678 

2540.6 

X 

208.916 

3473.2 

X 

118.596 

1119.2 

X 

148.833 

1762.7 

57. 

179.071 

2551.8 

X 

209.309 

3486.3 

X 

118.988 

1126.7 

X 

149.226 

1772.1 

X 

179.463 

2563.0 

X 

209.701 

3499.4 








































CIRCLES 


127 


TABLE 1 OF CIRCLES— (Continued). 

Diameters in units and eighths, Ac. 


iara. 


Circumf. 


7 . 


; 9 . 


ro. 


71 . 


73 . 


73 . 


X 

X 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

X 

X 

% 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

% 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 


74 . 


75 . 


•210.094 

210.487 

•210.879 

211.272 

211.665 

212.058 

212.450 

212.843 

213.236 

213.628 

214.021 

214.414 

214.806 

215.199 

215.592 

215.984 

216.377 

216.770 

217.163 

217.555 

217.948 

218.341 

218.733 

219.126 

219.519 

219.911 

220.304 

220.697 

221.090 

221.482 

221.875 

222.'268 

222.660 

223.053 

223.446 

223.838 

224.231 

224.624 

225.017 

225.409 

225.802 

226.195 

226.587 

226.980 

227.373 

227.765 

228.158 

228.551 

228 944 

229.336 

229.729 

230.122 

230.514 

230.907 

231.300 

231.692 

232.085 

232.478 

232.871 

233 263 

233.656 

234.049 

234.441 

234.834 

235.227 

235.619 

236.012 


Area. 

Diam. 

Circumf. 

Area. 

Ham. 

Dircumf. 

Area. 

liam. 

Dircumf. 

Area. 

3512.5 

75X 

236.405 

4447.4 

83 X 

262.716 

5492.4 

92. 

289.027 

6647.6 

3525.7 

X 

236.798 

4462.2 

X 

263.108 

5508.8 

X 

289.419 

6665.7 

3538.8 

*X 

237.190 

4477.0 

X 

263.501 

5525.3 

X 

289.812 

6683.8 

3552.0 

X 

237.583 

4491.8 

84. 

263.894 

5541.8 

X 

290.205 

6701.9 

3565.2 

X 

237.976 

4506.7 

X 

264.286 

5558.3 

X 

290.597 

6720.1 

3578.5 

X 

238.368 

4521.5 

X 

264.679 

5574.8 

X 

290.990 

6738.2 

3591.7 

76. 

238.761 

4536.5 

X 

265.072 

5591.4 

X 

291.383 

6756.4 

3605.0 

X 

239.154 

4551.4 

X 

265.465 

5607.9 

X 

291.775 

6774.7 

3618.3 

X 

239.546 

4566.4 

x. 

265.857 

5624.5 

93. 

292.168 

6792.9 

3631.7 

X 

239.939 

4581.3 

X 

266.250 

5641.2 

X 

292.561 

6811.2 

3645.0 

X 

240.332 

4596.3 

X 

266.643 

5657.8 

X 

292.954 

6829.5 

3658.4 

X 

240.725 

4611.4 

85. 

267.035 

5674.5 

X 

293.346 

6847.8 

3671.8 

X 

241.117 

4626.4 

X 

267.428 

5691.2 

X 

293.739 

6866.1 

3685.3 

X 

241.510 

4641.5 

X 

267.821 

5707.9 

X 

294.132 

6884.5 

3698.7 

77. 

241.903 

4656.6 

X 

268.213 

5724.7 

X 

294.524 

6902.9 

3712.2 

X 

242.295 

4671.8 

X 

268.606 

5741.5 

X 

294.917 

6921.3 

3725.7 

X 

242.688 

4686.9 

X 

268.999 

5758.3 

94. 

295.310 

6939.8 

3739.3 

X 

243.081 

4702.1 

X 

269.392 

5775.1 

X 

295.702 

6958.2 

3752.8 

X 

243.473 

4717.3 

X 

269.784 

5791.9 

X 

296.095 

6976.7 

3766.4 

X 

243.866 

4732.5 

86. 

270.177 

5808.8 

X 

296.488 

6995.3 

3780.0 

X 

244.259 

4747.8 

X 

270.570 

5825.7 

X 

296.881 

7013.8 

3793.7 

X 

244.652 

4763.1 

X 

270.962 

5842.6 

X 

297.273 

7032.4 

3807.3 

CO 

4 — 

245.044 

4778.4 

X 

271.355 

5859.6 

X 

297.666 

7051.0 

3821.0 

X 

245.437 

4793.7 

X 

271.748 

5876.5 

X 

298.059 

7069.6 

3834.7 

X 

245.830 

4809.0 

X 

272.140 

5893.5 

95. 

298.451 

7088.2 

3848.5 

X 

246.222 

4824.4 

X 

272 533 

5910.6 

X 

298.844 

7106.9 

3862.2 

X 

246.615 

4839.8 

X 

272.926 

5927.6 

X 

299.237 

7125.6 

3876.0 

X 

247.008 

4855.2 

87. 

273.319 

5944.7 

X 

299.629 

7144.3 

3889.8 

X 

247.400 

4870.7 

X 

273.711 

5961.8 

X 

300.022 

7163.0 

3903.6 

X 

247.793 

4886.2 

X 

274.104 

5978.9 

X 

300.415 

7181.8 

3917.5 

79. 

248.186 

4901.7 

X 

274.497 

5996.0 

X 

300.807 

7200.6 

3931.4 

X 

248.579 

4917.2 

X 

274.889 

6013.2 

X 

301.200 

7219.4 

3945.3 

X 

248.971 

4932.7 

X 

275.282 

6030.4 

96. 

301.593 

7238.2 

3959.2 

X 

249.364 

4948.3 

X 

275.675 

6047.6 

X 

301.986 

7257.1 

3973.1 

X 

249.757 

4963.9 

X 

276.067 

6064.9 

X 

302.378 

7276 0 

3987.1 

X 

250.149 

4979.5 

88. 

276.460 

6082.1 

X 

302.771 

7294.9 

4001.1 

X 

250.542 

4995.2 

X 

276.853 

6099.4 

Vi 

303.164 

7313.8 

4015.2 

X 

250.935 

5010.9 

X 

277.246 

6116.7 

X 

303.556 

733*2.8 

4029.2 

80. 

251.327 

5026.5 

X 

277.638 

6134.1 

X 

303.949 

7351.8 

4043.3 

X 

251.720 

5042.3 

X 

278.031 

6151.4 

, y» 

304.342 

7370.8 

4057.4 

X 

252.113 

5058.0 

X 

278.424 

6168.8 

97. 

304.734 

7389.8 

4071.5 

X 

252.506 

5073.8 

X 

278.816 

6186.2 

X 

305.127 

7408.9 

4085.7 

X 

252.898 

5089.6 

X 

279.209 

6203.7 

X 

305.520 

7428 0 

4099.8 

X 

253.291 

5105.4 

89. 

279.602 

6221.1 

X 

305.913 

7447.1 

4114.0 

X 

253.684 

5121.2 

X 

279.994 

6238.6 

X 

306.305 

7466.2 

4128.2 

X 

254.076 

5137.1 

X 

280.387 

6256.1 

X 

306.698 

7485.3 

4142.5 

81. 

254.469 

5153.0 

X 

280.780 

6273.7 

X 

307.091 

7504.5 

4156.8 

X 

254.862 

5168.9 

X 

281.173 

6291.2 

X 

307.463 

7523.7 

4171.1 

X 

1 255.254 

5184.9 

X 

281.565 

6308.8 

98. 

307.876 

7543.0 

4185.4 

x 

255.647 

5200.8 

X 

281.958 

6326.4 

X 

308.269 

7562.2 

4199.7 

X 

256.040 

5216.8 

X 

282.351 

6344.1 

X 

308.661 

7581.5 

4214 1 

X 

256.433 

5232.8 

90. 

282.743 

6361.7 

X 

309.054 

7600.8 

4228.5 

X 

256.825 

5248.9 

X 

283.136 

6379.4 

X 

309.447 

4 6*20.1 

4242.9 

X 

257.218 

5264.9 

X 

283.529 

6397.1 

X 

309.840 

7639.5 

4257.4 

82. 

257.611 

5281.0 

X 

283.921 

6414.9 

X 

310.232 

7658.9 

4271.8 

X 

258.003 

5297.1 

X 

284.314 

6432.6 

X 

310.625 

7678.3 

4286.3 

X 

1 258.396 

5313.3 

X 

284.707 

6450.4 

99. 

311.018 

7697.7 

4300.8 

X 

258.789 

5329.4 

X 

285.100 

6468.2 

X 

311.410 

7717.1 

4315.4 

X 

259.181 

5345.6 

X 

285.492 

6486.0 

X 

311.803 

7736.6 

4329.9 

X 

259.574 

5361.8 

91. 

285.885 

6503.9 

X 

312.196 

7756.1 

4344.5 

X 

259.967 

5378.1 

X 

286.278 

6521.8 

X 

312.588 

7775.6 

4359.2 

X 

260.359 

5394.3 

X 

286.670 

6539.7 

X 

312.981 

7795.2 

4373.8 

83. 

260.752 

5410.6 

X 

287.063 

6557.6 

X 

313.374 

7814.8 

4388.5 

X 

261.145 

5426.9 

X 

287.456 

6575.5 

X 

313.767 

7834.4 

4403.1 

X 

261.538 

5443.3 

X 

287.848 

6593.5 

100. 

314.159 

7854.6 

4117.9 

X 

261 930 

5459.6 

X 

288.241 

6611.5 




4432.6 

K 

262.323 

5476.0 

X 

288.634 

6629.6 





















































128 


CIRCLES, 


TABLE 2 OF CIRCLES. 


Diameters in units and tenths. 


Dia. 

Circunif. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

0.1 

.314159 

.007854 

6.3 

19.79203 

31.17245 

12.5 

39.26991 

122.718; 

.2 

.628319 

.031416 

.4 

20.10619 

32.16991 

.6 

39.58407 

124.689? 

.3 

.942478 

.070686 

.5 

20.42035 

33.18307 

.-7 

39.89823 

126.6761 

.4 

1.256637 

.125664 

.6 

20.73451 

34.21194 

.8 

40.21239 

128.6791 

.5 

1.570796 

.196350 

.7 

21.04867 

35.25652 

.9 

40.52655 

130.6981 

.6 

1.884956 

.282743 

.8 

21.36283 

36.31681 

13.0 

40.84070 

132.732: 

.7 

2.199115 

.384845 

.9 

21.67699 

37.39281 

.1 

41.15486 

1:34.7821 

.8 

2.513274 

.502655 

7.0 

■21 99115 

38.48451 

.2 

41.46902 

136.847? 

.9 

2.827433 

.636173 

.1 

22.30531 

39.59192 

.3 

41.78318 

138.9291 

1.0 

3.141593 

.785398 

.2 

22.61947 

40.71504 

.4 

42.09734 

141.0261 

.1 

3.455752 

.950332 

.3 

22.93363 

41.85387 

.5 

42.411.50 

143.138? 

.2 

3.769911 

1.13097 

.4 

23.24779 

43.00840 

.6 

42.72566 

145.2672 

.3 

4.084070 

1.32732 

.5 

23.56194 

44.17865 

.7 

43.03982 

147.4114 

.4 

4.398230 

1.53938 

.6 

23.87610 

45.36460 

.8 

43.35398 

149.5712 

.5 

4.712389 

1.76715 

.7 

24.19026 

46.56626 

.9 

43.66814 

151.7468 

.6 

5.026548 

2.01062 

.8 

24.50442 

47.78362 

14 0 

43.98230 

153.9380 

.7 

5.340708 

2.26980 

.9 

24.81858 

49.01670 

.1 

44.29646 

156.1450 

.8 

5.654867 

2.54469 

8.0 

25.13274 

50.26548 

.2 

44.61062 

158.3677 

.9 

5.969026 

2.83529 

.1 

25.44690 

51.52997 

.3 

44.92477 

160.6061 

2.0 

6.283185 

3.14159 

.2 

25.76106 

52.81017 

.4 

45.23893 

162.8602 

.1 

6.597345 

3.46361 

.3 

26.07522 

54.10608 

.5 

45.55309 

165.1300 

.2 

6.911504 

3.80133 

.4 

26.38938 

55.41769 

.6 

45.86725 

167.4155 

.3 

7.225663 

4.15476 

.5 

26.70354 

56.74502 

.7 

46.18141 

169.7167 

.4 

7.539822 

4.52389 

.6 

27.01770 

58.08805 

.8 

46.49557 

172.0336 

.5 

7.853982 

4.90874 

.7 

27.33186 

59.44679 

.9 

46.80973 

174.3662 

.6 

8.168141 

5.30929 

.8 

27.64602 

60.82123 

15.0 

47.12389 

176.7146 

.7 

8.482300 

5.72555 

.9 

27.96017 

62.21139 

.1 

47.43805 

179.0786 

.8 

8.796159 

6.15752 

9.0 

28.27433 

63.61725 

.2 

47.75221 

181.4584 

.9 

9.110619 

6.60520 

.1 

28.58849 

65.03882 

.3 

48.06637 

183.8539 

8.0 

9.424778 

7.06858 

.2 

28.90265 

66.47610 

.4 

48.38053 

186.2650 

.1 

9.738937 

7.54768 

.3 

29.21681 

67.92909 

.5 

48.69469 

188.6919 

.2 

10.05310 

8.04248 

.4 

29.53097 

69.39778 

.6 

49.00885 

191.1345 

.3 

10.36726 

8.55299 

.5 

29.84513 

70.88218 

.7 

49.32300 

193.5928 

.4 

10.68142 

9.07920 

.6 

30.15929 

72.38229 

.8 

49.63716 

196.0668 

.5 

10.995)7 

9.62113 

.7 

30.47345 

73.89811 

.9 

49.95132 

198.5565 

.6 

11.309/3 

10.17876 

.8 

30.78761 

75.42964 

16.0 

50.26.548 

201.0619 

.7 

11.62389 

10.75210 

.9 

31.10177 

76.97687 

.1 

50.57964 

203.5831 

.8 

11.93805 

11.34115 

10.0 

31.41593 

78.53982 

.2 

50.89380 

206.1199 

.9 

12.25221 

11.91591 

.1 

31.73009 

80.11847 

.3 

51.20796 

208.6724 

4.0 

12.56637 

12.56637 

.2 

32.04425 

81.71282 

.4 

51.52212 

211.2407 

.1 

12.88053 

13.20254 

.3 

32.35840 

83.32289 

.5 

51.83628 

213.8246 

.2 

13.19469 

13.85442 

.4 

32.67256 

84.94S67 

.6 

52.15044 

216.4243 

.3 

13.50885 

14.52201 

.5 

32.98672 

86.59015 

.7 

52.46460 

219.0397 

.4 

13.82301 

15.20531 

.6 

33.30088 

88.24734 

.8 

52.77876 

221.6708 

•5 

14.13717 

15.90431 

.7 

33.61504 

89.92024 

.9 

53.09292 

224.3176 

.6 

14.45133 

16.61903 

.8 

33.92920 

91.60884 

17.0 

53.40708 

226.9801 

.7 

14.76549 

17.34945 

.9 

34.24336 

93.31316 

.1 

53.72123 

229 6583 

.8 

15.07964 

18.09557 

11.0 

34.55752 

95.03318 

.2 

54.03539 

232 35‘>2 

.9 

15.39380 

18.85741 

.1 

34.87168 

96.76891 

.3 

54.34955 

235.0618 

5.0, 

15.70796 

19.63495 

.2 

35.18584 

98.52035 

.4 

54.66371 

237.7871 

.1 

16.02212 

20.42821 

.3 

35.50000 

100.2875 

.5 

54.97787 

240 5282 

.2 

16.33628 

21.23717 

.4 

35.81416 

102.0703 

.6 

55.29203 

243 2849 

.3 

16.65044 

22.06183 

.5 

36.12832 

103.8689 

.7 

55.60619 

246 0574 

.4 

16.96160 

22.90221 

.6 

36.44247 

105.6832 

.8 

55.92035 

248.8456 

•5 

17.27876 

23.75829 

.7 

36.75663 

107.5132 

.9 

56.23451 

251 6494 

.6 

17.59292 

24.63009 

.8 

37.07079 

109.3588 

18.0 

56.54867 

254 4690 

.7 

17.90708 

25.51759 

.9 

37.38495 

111.2202 

.1 

56.86283 

257 3043 

.8 

18.22124 

26.42079 

12.0 

37.69911 

113.0973 

.2 

57.17699 

260 1553 

.9 

18.53540 

27.3:1971 

.1 

38.01327 

114.9901 

.3 

57.49115 

263 0220 

6.0 

18.84956 

28.27433 

.2 

38.32743 

116.8987 

.4 

57.80530 

265 9044 

.1 

19.16372 

29.22467 

.3 

38.64159 

118.8229 

A 

58.11946 

268 8025 

.2 

19.47787 

30.19071 

.4 

38.95575 

120.7628 

.6 

58.43362 

271.7163 




































CIRCLES, 


129 


TABLE 2 OF CIRCLES—(Continued). 
Diameters in units and tenths. 


Dia 

Circuuif. 

Area. 

Dia 

Circuuif. 

Area. 

Dia. 

Circumf. 

Area. 

18.7 

58.74778 

274.6459 

24.9 

78.22566 

486.9547 

31.1 

97.70353 

759 6450 

.8 

59.06194 

277.5911 

25.0 

78.53982 

490.8739 

.2 

98.01769 

764 5380 


59.37610 

280.5521 

.1 

78.85398 

494 8087 

.3 

98.33185 

769 4467 

19.0 

59.69026 

283.5287 

.2 

79.16813 

498.7592 

.4 

98.64601 

774 3712 

.1 

60.00442 

286.5211 

.3 

79.48229 

502.7255 

.5 

98.96017 

779.3113 

.2 

60.31858 

289.5292 

.4 

79.79645 

506.7075 

.6 

99.27433 

784.2672 

.3 

60.632/4 

292.5530 

.5 

80.11061 

510.7052 

.7 

99.58849 

789 2388 

.4 

60.94690 

295.5925 

.6 

80.42477 

514.7185 

.8 

99.90265 

794 2260 

J) 

61.26106 

298.6477 

.7 

80.73893 

518.7476 

.9 

100.2168 

799.2290 

.6 

61.57522 

301.7186 

.8 

81.05309 

522.7924 

32.0 

100.5310 

804.2477 

.7 

61.88938 

304.8052 

.9 

81.36725 

526.8529 

.1 

100.8451 

809 2821 

.8 

62.20353 

307.9075 

26.0 

81.68141 

530.9292 

.2 

101.1593 

814.3322 

.9 

62.51769 

311.0255 

.1 

81.99557 

535.0211 

.3 

101.4734 

819 3980 

20.0 

62.83185 

314.1593 

.2 

82.30973 

539.1287 

.4 

101.7876 

824.4796 

.1 

63.14601 

317.3087 

.3 

82.62389 

543.2521 

.5 

102.1018 

829.5768 

.2 

63.46017 

320.4739 

.4 

82.93805 

547.3911 

.6 

102.4159 

834.6898 

.3 

63.77433 

323.6547 

.5 

83.25221 

551.5459 

.7 

102.7301 

839.8184 

.4 

64.08849 

326.8513 

.6 

83.56636 

555.7163 

.8 

103.0442 

844.9628 

•5 

64.40265 

330.0636 

.7 

83.88052 

559.9025 

.9 

103.3584 

850.1228 

.6 

64.71681 

333.2916 

.8 

84.19468 

564.1044 

33.0 

103.6726 

855.2986 

.7 

65.03097 

336.5353 

.9 

84.50884 

558.3220 

.1 

103.9867 

860.4901 

.8 

65.34513 

339.7947 

27.0 

84.82300 

572.5553 

.2 

104.3009 

865.6973 

.9 

65.65929 

343.0698 

.1 

85.13716 

576.8043 

.3 

104.6150 

870.9202 

21.0 

65.97345 

346.3606 

.2 

85.45132 

581.0690 

.4 

104.9292 

876.1588 

.1 

66.28760 

349.6671 

.3 

85.76548 

585.3494 

.5 

105.2434 

881.4131 

2 

66.60176 

352.9894 

.4 

86.0796-1 

589.6455 

.6 

105.5575 

886.6831 

.3 

66.91592 

356.3273 

.5 

86.39380 

593.9574 

.7 

105.8717 

891.9688 

.4 

67.23008 

359.6809 

.6 

86.70796 

598.2849 

.8 

106.1858 

897.2703 

.5 

67.54424 

363.0503 

.7 

87.02212 

602.6282 

.9 

106.5000 

902.5874 

.6 

67.85840 

366.4354 

.8 

87.33628 

606.9871 

34.0 

106.8142 

907.9203 

.7 

68.17256 

369.8361 

.9 

87.65044 

611.3618 

.1 

107.1283 

913.2688 

.8 

68.48672 

373.2526 

28.0 

87.96459 

615.7522 

.2 

107.4425 

918.6331 

.9 

68.80088 

376.6848 

.1 

88.27875 

620.1582 

.3 

107.7566 

924.0131 

'2.0 

69.11504 

380.1327 

.2 

88.59291 

624.5800 

.4 

108.0708 

929.4088 

.1 

69.42920 

383.5963 

.3 

88.90707 

629.0175 

.5 

108.3849 

934.8202 

.2 

69.74336 

387.0756 

A 

89.22123 

633.4707 

.6 

108.6991 

940.2473 

.3 

70.05752 

390.5707 

.5 

89.53539 

637.9397 

.7 

109.0133 

945.6901 

.4 

70.37168 

394.0814 

.6 

89.84955 

642.4243 

.8 

109.3274 

951.1486 

.5 

70.68583 

397.6078 

.7 

90.16371 

646.9246 

.9 

109.6416 

956.6228 

.6 

70.99999 

401.1500 

.8 

90.47787 

651.4407 

35.0 

109.9557 

962.1128 

.7 

71.31415 

404.7078 

.9 

90.79203 

655.9724 

.1 

110.2699 

967.6184 

.8 

71.62831 

408.2814 

29.0 

91.10619 

660.5199 

.2 

110.5841 

973.1397 

.9 

71.94247 

411.8707 

.1 

91.42035 

665.0830 

.3 

110.8982 

978.6768 

3.0 

72.2.5663 

415.4756 

.2 

91.73451 

669.6619 

.4 

111.2124 

984.2296 

.1 

72.57079 

419.0963 

.3 

92.04866 

674.2565 

.5 

111.5265 

989.7980 

.2 

72.88495 

422.7327 

.4 

92.36282 

678.8668 

.6 

111.8407 

995.3822 

.3 

73.19911 

426.3848 

.5 

92.67698 

683.4928 

.7 

112.1549 

1000.9821 

.4 

73.51327 

430.0526 

.6 

92.99114 

688.1345 

.8 

112.4690 

1006.5977 

.5 

73.82743 

433.7361 

.7 

93.30530 

692.7919 

.9 

112.7832 

1012.2290 

.6 

74.14159 

437.4354 

.8 

93.61946 

697.4650 

36.0 

113.0973 

1017.8760 

.7 

74.45575 

441.1503 

.9 

93.93362 

702.1538 

.1 

113.4115 

1023.5387 

.8 

74.76991 

444.8809 

30.0 

94.24778 

706.8583 

.2 

113.7257 

1029.2172 

.9 

75.08406 

448.6273 

.1 

94.56194 

711.5786 

.3 

114.0398 

1034.9113 

1.0 

75.39822 

452.3893 

.2 

94.87610 

716.3145 

.4 

114.3540 

1040.6212 

.1 

75.71238 

456.1671 

.3 

95.19026 

721.0662 

.5 

114.6681 

1046.3467 

.2 

76.02654 

459.9606 

.4 

95.50442 

725.8336 

.6 

114.9823 

1052.0880 

.3 

76.34070 

463.7698 

.5 

95.81858 

730.6166 

.7 

115.2965 

1057.8449 

.4 

76.65486 

467.5947 

.6 

96.13274 

735.4154 

.8 

115.6106 

1063.6176 

.5 

76.96902 

471.4352 

.7 

96.44689 

740.2299 

.9 

115.9248 

1069.1060 

.6 

77.28318 

475.2916 

.8 

96.76105 

745.0601 

;;7.o 

116.2389 

1075.2101 

.7 

77.59734 

479.1636 

.9 

97.07521 

749.9060 

.1 

116.5531 

1081.0299 

.8 

77.91150 

483.0513 

11.0 

97.38937 

754.7676 

.2 

116.8672 

1086.8654 


9 


















































130 


CIRCLES 


TABLE 2 OF CIRCLES—(Continued). 
Diiiineters in units ami ten ills. 


Dia. 

Circuinf. 

Area. 

87.3 

117.1814 

1092.7166 

.4 

117.4956 

1098.5835 

.5 

117.8097 

1104.4662 

.6 

118.1239 

1110.3645 

.7 

118.4380 

1116.2786 

.8 

118.7522 

1122.2083 

.9 

119.0664 

1128.1538 

38.0 

119.3805 

1134.1149 

.1 

119.6947 

1140.0918 

.2 

120.0088 

1146.0814 

.3 

120.3230 

1152.0927 

.4 

120.6372 

1158.1167 

.5 

120.9513 

1164.1564 

.6 

121.2655 

1170.2118 

.7 

121.5796 

1176.2830 

.8 

121.8938 

1182.3698 

.9 

122.2080 

1188.4724 

39.0 

122.5221 

1194.5906 

.1 

122.8363 

1200.7246 

.2 

123.1504 

1206.8742 

.3 

123.4646 

1213.0396 

.4 

123.7788 

1219.2207 

.5 

124.0929 

122514175 

.6 

124.4071 

1231.6300 

.7 

124.7212 

1237.8582 

.8 

125.0354 

1244.1021 

.9 

125.3495 

1250.3617 

40.0 

125.6637 

1256.6371 

.1 

125.9779 

1262.9281 

.2 

126.2920 

1269.2348 

.3 

126.6062 

1275.5573 

.4 

126.9203 

1281.8955 

.5 

127.2345 

1288.2493 

.6 

127.5487 

1294.6189 

.7 

127.8628 

1301.0042 

.8 

128.1770 

1307.4052 

.9 

128.4911 

1313.8219 

41.0 

128.8053 

1320.2543 

.1 

129.1195 

1326.7024 

.2 

129.4336 

1333.1663 

.3 

129.7478 

1339.6458 

.4 

130.0619 

1346.1410 

.5 

130.3761 

1352.6520 

.6 

130.6903 

1359.1786 

.7 

131.0044 

1365.7210 

.8 

131.3186 

1372.2791 

.9 

131.6327 

1378.8529 

42.0 

131.9469 

1385.4424 

.1 

132.2611 

1392.0476 

.2 

132.5752 

1398.6685 

.3 

132.8894 

1405.3051 

.4 

133.2035 

1411.9574 

.5 

133.5177 

1418.6254 

.6 

133.8318 

1425.3092 

.7 

134.1460 

1432.0086 

.8 

134.4602 

1438.7238 

.9 

134.7743 

1445.4546 

43.0 

135.0885 

1452.2012 

.1 

135.4026 

1458.9635 

.2 

135.7168 

1465.7415 

.3 

136.0310 

1472.5352 

.4 

136.3451 

1479.3446 


Dia. 

Cireumf. 

Area. 

43.5 

136.6593 

1486.1697 

.6 

136.9734 

1493.0105 

.7 

137.2876 

1499.8670 

.8 

137.6018 

1506.7393 

.9 

137.9159 

1513.6272 

44.0 

138.2301 

1520.5308 

.1 

138.5442 

1527.4502 

.2 

188.8584 

1534.3853 

.3 

139.1726 

1541.3360 

.4 

139.4867 

1548.3025 

.5 

139.8009 

1555.2847 

.6 

140.1150 

1562.2826 

.7 

140.4292 

1569.2962 

.8 

140.7434 

1576.3255 

.9 

141.0575 

1583.3706 

45.0 

141.3717 

1590.4313 

.1 

141.6858 

1597.5077 

.2 

142.0000 

1604.5999 

.3 

142.3141 

1611.7077 

.4 

142.6283 

1618.8313 

.5 

142.9425 

1625.9705 

.6 

143.2566 

1633.1255 

.7 

143.5708 

1640.2962 

.8 

143.8849 

1647.4826 

.9 

144.1991 

1654.6847 

46.0 

144.5133 

1661.9025 

.1 

144.8274 

1669.1360 

.2 

145.1416 

1676.3853 

.3 

145.4557 

1683.6502 

.4 

145.7699 

1690.9308 

.5 

146.0841 

1698.2272 

.6 

146.3982 

1705.5392 

.7 

146.7124 

1712.8670 

.8 

147.0265 

1720.2105 

.9 

147.3407 

1727.5697 

47.0 

147.6549 

1734.9445 

.1 

147.9690 

1742.3351 

2 

148.2832 

1749.7414 

.3 

148.5973 

1757.1635 

.4 

148.9115 

1764.6012 

.5 

149.2257 

1772.0546 

.6 

149.5398 

1779.5237 

.7 

149.8540 

1787.0086 

.8 

150.1681 

1794.5091 

.9 

150.4823 

1802.0254 

48.0 

150.7964 

1809.5574 

.1 

151.1106 

1817.1050 

.2 

151.4248 

1824.6684 

.3 

151.7389 

1832.2475 

.4 

152.0531 

1839.8423 

.5 

152.3672 

1847.4528 

.6 

152.6814 

1855.0790 

.7 

152.9956 

1862.7210 

.8 

153.3097 

1870.3786 

.9 

153.6239 

1878.0519 

40.0 

153.9380 

1885.7410 

.1 

154.2522 

1893.4457 

.2 

154.5664 

1901.1662 

.3 

154.8805 

1908.9024 

.4 

155.1947 

1916.6543 

.5 

155.5088 

1924.4218 

.6 

155.8230 

1932.2051 


Dia. 

Circuinf. 

Area. 

49.7 

156.1372 

1940.0041 

.8 

156.4513 

1947.8189 

.9 

156.7655 

1955.6493] 

50.0 

157.0796 

1963.4954 

.1 

157.3938 

1971.3572 

.2 

157.7080 

1979.2348 

.3 

1-58.0221 

1987.1280 

.4 

158.3363 

1995.0370- 

.5 

158.6504 

2002.9617 

.6 

158.9646 

2010.9020 

.7 

159.2787 

2018.8581 

.8 

159.5929 

2026.8299 

.9 

159.9071 

2034.8174 

51.0 

160.2212 

2042.8206 

.1 

160.5354 

2050.8395 

.2 

160.8495 

2058.8742 

.3 

161.1637 

2066.9245 

.4 

161.4779 

2074.9905 

.5 

161.7920 

2083.0723 

.6 

162.1062 

2091.1697 

.7 

162.4203 

2099.2829 

.8 

162.7345 

2107.4118 

.9 

163.0187 

2115.5563 1 

52.0 

163.3628 

2123.7166 

.1 

163.6770 

2131.8926 

.2 

163.9911 

2140.0843 

.3 

164.3053 

2148.2917' 

.4 

164.6195 

2156.5149 

.5 

164.9336 

2164.7537 

.6 

165.2478 

2173.0082 

.7 

165.5619 

2181.2785 

.8 

165.8761 

2189.5644 

.9 

166.1903 

2197.8661 

53.0 

166.5044 

2206.1834 

.1 

166.8186 

2214.51651 

2 

167.1327 

2222.8653 

.3 

167.4469 

2231.2298 

.4 

167.7610 

2239.6100 

.5 

168.0752 

2248.0059 

.6 

168.3894 

2256.41751 

.7 

168.7035 

2264.8448 

.8 

169.0177 

2273.2879 

.9 

169.3318 

2281.7466 

54.0 

169.6460 

2290.2210 

.1 

169.9602 

2298.711: 

.2 

170.2743 

2307.217: 

.3 

170.5885 

2315.7386 

.4 

170.9026 

2324.2759 

.5 

171.2168 

2332 8289 

.6 

171.5310 

2341.3976 

.7 

171.8451 

2349.9820 

.8 

172.1593 

2358.5821 

.9 

172.4734 

2367.1979 

55.0 

172.7876 

2375.8294 

.1 

173.1018 

2384.4767 

.2 

173.4159 

2393.1396 

.3 

173.7301 

2401.8183 

.4 

174.0442 

2410.5126 

.5 

174.3584 

2419.2227 

.6 

174.6726 

2427.9485 

.7 

174.9867 

2436.6899 

.8 

175.3009 

2445.4471 


























CIRCLES 


131 


TABLE 2 OF CIRCLES—(Continued). 


Diameters in units and tenths. 


Dia. 

Circunif. 

Area. 

Dia. 

Circunif. 

Area. 

Dia. 

Circumf. 

55.9 

175.6150 

2454.2200 

62.1 

195.0929 

3028.8173 

68.3 

214.5708 

56.0 

175.9292 

2463.0086 

.2 

195.4071 

3038.5798 

.4 

214.8849 

.1 

176.2433 

2471.8130 

.3 

195.7212 

3048.3580 

.5 

215.1991 

.2 

176.5575 

2480.6330 

.4 

196.0354 

3058.1520 

.6 

215.5133 

.3 

176.8717 

2489.4687 

.5 

196.3495 

3067.9616 

.7 

215.8274 

.4 

177.1858 

2498.3201 

.6 

196.6637 

3077.7869 

.8 

216.1416 

.5 

177.5000 

2507.1873 

.7 

196.9779 

3087.6279 

.9 

216.4557 

.6 

177.8141 

2516.0701 

.8 

197.2920 

3097.4847 

69.0 

216.7699 

.7 

178.1283 

2524.9687 

.9 

197.6062 

3107.3571 

.1 

217.0841 

.8 

178.4-125 

2533.8830 

63.0 

197.9203 

3117.2453 

.2 

217.3982 

.9 

178.7566 

2542.8129 

.1 

198.2345 

3127.1492 

.3 

217.7124 

57.0 

179.0708 

2551.7586 

.2 

198.5487 

3137.0688 

.4 

218.0265 

.1 

179.3849 

2560.7200 

.3 

198.8628 

3147.0040 

.5 

218.3407 

.2 

179.6991 

2569.6971 

.4 

199.1770 

3156.9550 

.6 

218.6548 

.3 

180.0133 

2578.6899 

.5 

199.4911 

3166.9217 

.7 

218.9690 

.4 

180.3274 

2587.6985 

.6 

1S9.8053 

3176.9042 

.8 

219.2832 

.5 

180.6416 

2596.7227 

.7 

200.1195 

3186.9023 

.9 

219.5973 

.6 

180.9557 

2605.7626 

.8 

200,1336 

3196.9161 

70.0 

219.9115 

.7 

181.2699 

2614.8183 

.9 

200.7478 

3206.9456 

.1 

220.2256 

.8 

181.5841 

2623.8896 

64.0 

201.0619 

3216.9909 

.2 

220.5398 

.9 

1 181.8982 

2632.9767 

.1 

201.3761 

3227.0518 

.3 

220.8540 

>8.0 

182.2124 

2642.0794 

.2 

201.6902 

3237.1285 

.4 

221.1681 

.1 

182.5265 

2651.1979 

.3 

202.0044 

3247.2209 

.5 

221.4823 

.2 

182.8407 

2660.3321 

.4 

202.3186 

3257.3289 

.6 

221.7964 

.3 

183.1549 

2669.4820 

.5 

202.6327 

3267.4527 

.7 

222.1106 

.4 

183.4690 

2678.6476 

.6 

202.9469 

3277.5922 

.8 

222.4248 

.5 

183.7832 

2687.8289 

.7 

203.2610 

3287.7474 

.9 

222.7389 

.6 

184.0973 

2697.0259 

.8 

203.5752 

3297.9183 

71.0 

223.0531 

.7 

184.4115 

2706.2386 

.9 

203.8894 

3308.1049 

.1 

223.3672 

.8 

184.7256 

2715.4670 

65.0 

204.2035 

3318.3072 

.2 

223.6814 

.9 

185.0398 

2724.7112 

.1 

204.5177 

3328.5253 

.3 

223.9956 

19.0 

185.3540 

2733.9710 

.2 

204.8318 

3338.7590 

.4 

224.3097 

.1 

185.6681 

2743.2466 

.3 

205.1460 

3349.0085 

.5 

224.6239 

.2 

185.9823 

2752.5378 

.4 

205.4602 

3359.2736 

.6 

224.9380 

.3 

186.2964 

2761.8448 

.5 

205.7743 

3369.5545 

.7 

225.2522 

.4 

186.6106 

2771.1675 

.6 

206.0885 

3379.8510 

.8 

225.5664 

.5 

186.9248 

2780.5058 

.7 

206.4026 

3390.1633 

.9 

225.8805 

.6 

187.2389 

2789.8599 

.8 

206.7168 

3400.4913 

72.0 

226.1947 

.7 

187.5531 

2799.2297 

.9 

207.0310 

3410.8:350 

.1 

226.5088 

.8 

187.8672 

2808.6152 

66.0 

207.3451 

3421.1944 

.2 

226.8230 

.9 

188.1814 

2818.0165 

.1 

207.6593 

3431.5695 

.3 

227.1371 

0.0 

188.4956 

2827.4334 

.2 

207.9734 

3441.9603 

.4 

227.4513 

.1 

188.8097 

2836.8660 

.3 

208.2876 

3452.3669 

.5 

227.7655 

jy 

189.1239 

2846.3144 

.4 

208.6018 

3462.7891 

.6 

228.0796 

.3 

189.4380 

2855.7784 

.5 

208.9159 

3473.2270 

.7 

228.3938 

.4 

189.7522 

2865.2582 

.6 

209.2301 

3483.6807 

.8 

228.7079 

.5 

190.0664 

2874.7536 

.7 

209.5442 

3494.1500 

.9 

229.0221 

.6 

190.3805 

2884.2648 

.8 

209.8584 

3504.6351 

73.0 

229.3363 

.7 

190.6947 

2893.7917 

.9 

210.1725 

3515.1359 

.1 

229.6504 

.8 

191.0088 

2903.3343 

67.0 

210.4867 

3525.6524 

.2 

229.9646 

.9 1 

191.3230 

2912.8926 

.1 

210.8009 

3536.1845 

.3 

230.2787 

1.0 

191.6372 

2922.4666 

.2 

211.1150 

3546.7324 

.4 

230.5929 

.1 

191.9513 

2932.0563 

.3 

211.4292 

3557.2960 

.5 

230.9071 

.2 

192.2655 

2941.6617 

.4 

211.7433 

3567.8754 

.6 

231.2212 

.3 

192.5796 

2951.2828 

.5 

212.0575 

3578.4704 

.7 

231.5354 

.4 

192.8938 

2960.9197 

.6 

212.3717 

3589.0811 

.8 

231.8495 

.5 

193.2079 

2970.5722 

.7 

212.6858 

3599.7075 

.9 

232.1637 

.6 

193.5221 

2980.2405 

.8 

213.0000 

3610.3497 

74.0 

232.4779 

.7 

193.8363 

2989.9244 

.9 

213.3141 

3621.0075 

.1 

232.7920 

.8 

194.1504 

2999.6241 

68.0 

213.6283 

3631.6811 

.2 

233.1062 

.9 

194.4646 

3009.3395 

.1 

213.9425 

3642.3704 

.3 

233.4203 

2.0 | 

[ 

194.7787 

3019.0705 

.2 

214.2566 

3653.0754 

.4 

233.7345 

i 


Area. 


3663.7960 

3674.5324 

3685.2845 

3696.0523 

3706.8359 

3717.6331 

3728.4500 

3739.2807 

3750.1270 

3760.9891 

3771.8668 

3782.7603 

3793.6695 

3804.5944 

3815.5350 

3826.4913 

3837.4633 

3848.4510 

3859.4544 

3S70.4736 

3881.5084 

3892.5590 

3903.6252 

3914.7072 

3925.8049 

3936.9182 

3948.0473 

3959.1921 

3970.3526 

3981.5289 

3992.7208 

4003.9284 

4015.1518 

4026.3908 

4037.6456 

4048.9160 

4060 2022 

4071.5041 

4082.8217 

4094.1550 

4105.5040 

4116.8687 

4128.2491 

4139.6452 

4151.0571 

4162.4846 

4173.9279 

4185.3868 

4196.8615 

4208.3519 

4219.8579 

4231.3797 

4242.9172 

4254.4704 

4266.0394 

4277.6240 

4289.2243 

4300.8403 

4312.4721 

4324.1195 

4335.7827 

4347.4616 











































132 


CIRCLES, 




TABLE 2 OF CIRCLES—(Continued). 


Diameters in units and tenths. 


Din. 

Circumf. 

Area. 

)ia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. « 

74.5 

234.0487 

4359.1562 

80.7 

253.5265 

5114.8977 

86.9 

273.0044 

5931.020 ■} 

.6 

234.3628 

4370.8664 

.8 

253.8407 

5127.5819 

87.0 

273.8186 

5944.678 li 

.7 

234.6770 

4382.5924 

.9 

254.1518 

5140.2818 

.1 

273.6327 

5958.3521 

.8 

234.9911 

4394.3341 

81.0 

251.4690 

5152.9974 

.2 

273.9469 

5972.0421 

.9 

235.3053 

4406.0916 

.1 

254.7832 

5165.7287 

.3 

274.2610 

5985.747 

75.0 

235.6194 

4417.8647 

.2 

255.0973 

5178.4757 

.4 

274.5752 

5999.468 

.1 

235.9336 

4429.6535 

.3 

255.4115 

5191.2384 

.5 

271.8894 

6013.204. 

.2 

236.2478 

4411.4580 

.4 

255.7256 

5204.0168 

.6 

275.2035 

6026.957 

.3 

236.5619 

4453.2783 

.5 

256.0398 

5216.8110 

.7 

275.5177 

6010.725 

.4 

236.8761 

4465.1142 

.6 

256.3540 

5229.6208 

.8 

275.8318 

6054.508 

.5 

237.1902 

4476.9659 

.7 

256.6681 

5242.4463 

.9 

276.3460 

6068.308 

.6 

237.5044 

4488.8332 

.8 

256.9823 

5255.2876 

88.0 

276.4602 

6082.123 

.7 

237.8186 

4500.7163 

.9 

257.2964 

5268.1446 

.1 

276.7743 

6095.954 

.8 

238.1327 

4512.6151 

8*2.0 

257.6106 

5281.0173 

.2 

277.0885 

6109.800 

.9 

238.4469 

4524.5296 

.1 

257.9248 

5293.9056 

.3 

277.4026 

6123.663 

76.0 

238.7610 

4536.4598 

.2 

258.2389 

5306.8097 

•4 

277.7168 

6137.541 

.1 

239.0752 

4548.4057 

.3 

258.5531 

5319.7295 

.5 

278.0309 

6151.434 

jy 

239.3894 

4560.3673 

.4 

258.8672 

5332.6650 

.6 

278.3451 

6165.344 

i 

239.7035 

4572.3446 

.5 

259.1814 

5345.6162 

.7 

278.6593 

6179.269 

.4 

240.0177 

4584.3377 

.6 

259.4956 

5358.5832 

.8 

278.9734 

6193.210 

.5 

240.3318 

4596.3464 

.7 

259.8097 

5371.5658 

.9 

279.2876 

6207.166 

.6 

240 6460 

4608.3708 

.8 

260.1239 

53S4.5641 

89.0 

279.6017 

6221.138 

.7 

240.9602 

4620.4110 

.9 

260.4380 

5397.5782 

.1 

279.9159 

6235.126 

.8 

241.2743 

4632.4669 

83.0 

260.7522 

5410.6079 

.2 

280.2301 

6249.130 

.9 

241.5885 

4644.5384 

.1 

261.0663 

5423.6534 

.3 

280.5442 

6263.149 

77.0 

241.9026 

4656.6257 

.2 

261.3805 

5436.7146 

.4 

280.8584 

6277.184 

.1 

242.2168 

4668.7287 

.3 

261.6947 

5449.7915 

.5 

281.1725 

6291 235 

2 

242.5310 

4680.8474 

.4 

262.0088 

5-162.8840 

.6 

281.4867 

6305.302 

.3 

242.8451 

4692.9818 

.5 

262.3230 

5475.9923 

.7 

281.8009 

6319.384 

.4 

243.1593 

4705.1319 

.6 

262.6371 

5489.1163 

'.8 

282.1150 

6333.482 

.5 

243.4734 

4717.2977 

.7 

262:9513 

5502.2561 

.9 

282.4292 

6347.595 

.6 

243.7876 

4729.4792 

.8 

263.2655 

5515.4115 

90.0 

282.7433 

6361.725 

.7 

244.1017 

4741.6765 

.9 

263.5796 

5528.5826 

.1 

283.0575 

6375.870 

.8 

244.4159 

4753.8894 

84.0 

263.8938 

5541.7694 

.2 

283.3717 

6390.030 

.9 

244.7301 

4766.1181 

.1 

264.2079 

5554.9720 

.3 

283.6858 

6404.207 

78.0 

245.0442 

4778.3624 

.2 

264.5221 

5568.1902 

.4 

284.0000 

6418.399 

.1 

245.3584 

4790.6225 

.3 

264.8363 

5581.4242 

.5 

284.3141 

6432.607 

.2 

245.6725 

4802.8983 

.4 

265.1504 

5594.6739 

.6 

284.6283 

6446.830 

.3 

245.9867 

4815.1897 

.5 

265.4646 

5607.9392 

.7 

284.9425 

6461.070 

.4 

246.3009 

4827.4969 

.6 

265.7787 

5621.2203 

.8 

285.2566 

6475.325 

.5 

246.6150 

4839.8198 

.7 

266.0929 

5634.5171 

.9 

285.5708 

6489.595 

.6 

246.9292 

4852.1584 

.8 

266.4071 

5647.8296 

91.0 

285.8849 

6503.882 

.7 

247.2433 

4864.5128 

.9 

266.7212 

5661.1578 

.1 

286.1991 

6518.184 

.8 

247.5575 

4876.8828 

85.0 

267.03:54 

5674.5017 

.2 

286.5133 

6532.502 

.9 

247.8717 

4889.2685 

.1 

267.3495 

5687.8614 

.3 

286.8274 

6546.835 

79.0 

248.1858 

4901.6699 

.2 

267.6637 

5701.2367 

.4 

287.1416 

6561.184 

.1 

248.5000 

4914.0871 

.3 

267.9779 

5714.6277 

.5 

287.4557 

6575.549 

.2 

248.8141 

4926.5199 

.4 

268.2920 

5728.0345 

.6 

287.7699 

6589.930 

.3 

249.1283 

4938.9685 

.5 

268.6062 

5741.4569 

.7 

288.0840 

6604.326 

.4 

249.4425 

4951.4328 

.6 

268.9203 

5754.8951 

.8 

288.3982 

6618.738 

.5 

249.7566 

4963.9127 

.7 

269.2345 

5768.3490 

.9 

288.7124 

6633.166 

.6 

250.0708 

4976.4084 

.8 

269.5186 

5781.8185 

9*2.0 

289.0265 

6647.610 

.7 

250.3849 

4988.9198 

.9 

269.8628 

5795.3038 

.1 

289.3407 

6662.069' 

.8 

250.6991 

5001,1469 

86.0 

270.1770 

5808.8048 

.2 

289.6548 

6676.514 

.9 

251.0133 

5013.9897 

.1 

270.4911 

5822.3215 

.3 

289.9690 

6691.034' 

80.0 

251.3274 

5026.5482 

.2 

270.8053 

5835.8539 

.4 

290.2832 

6705.54H 

.1 

251.6416 

5039.1225 

.3 

271.1194 

5849.4020 

.5 

290.5973 

6720.0631 


251.9557 

5051.7124 

.4 

271.4336 

5862.9659 

.6 

290.9115 

6734.600: 

.3 

252.2699 

5064.3180 

.5 

271.7478 

5876.5454 

.7 

291.2256 

6749.154! 

.4 

252.5840 

5076.9394 

.6 

272.0619 

5890.1407 

.8 

291.5398 

6763.723: 

.5 

252.8982 

5089.5764 

.7 

272.3761 

5903.7516 

.9 

291.8540 

6778.308' 

.6 

253.2124 

5102.2292 

.8 

272.6902 

5917.3783 

93.0 

292.1681 

' 6792.908' 












































CIRCLES, 


133 


TABLE 2 OF CIRCLES— (Continued). 


Diameters in units and tenths. 


Dia. 

ICircumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

M.l 

292.4823 

6807.5250 

95.5 

300.0221 

7163.0276 

97.8 

307.2478 

7512.2078 

.2 

292.7964 

6822.1569 

.6 

300.3363 

7178.0366 

.9 

307.5619 

7527.5780 

.3 

293.1106 

6836.8046 

.7 

300.6504 

7193.0612 

98.0 

307.8761 

7542.9640 

.4 

293.4248 

6851.4680 

.8 

300.9646 

7208.1016 

.1 

308.1902 

7558.3656 

.5 

293.7389 

6866.1471 

.9 

301.2787 

7223.1577 

j2 

308.5014 

7573.7830 

.6 

294.0531 

6880.8419 

96.0 

301.5929 

7238.2295 

.3 

308.8186 

7589.2161 

.7 

294.3672 

6895.5524 

.1 

301.9071 

7253.3170 

A 

309.1327 

7604.6648 

.8 

294.6814 

6910.2786 

.2 

302.2212 

7268.4202 

.5 

309.4469 

7620.1293 

.9 

294.9956 

6925.0205 

.3 

302.5354 

7283.5391 

.6 

309.7610 

7635.6095 

J4.0 

295.3097 

6939.7782 

.4 

302.8495 

7298.6737 

.7 

310.0752 

7651.1054 

.1 

295.6239 

6954.5515 

.5 

303.1637 

7313.8240 

.8 

310.3894 

7666.6170 

.2 

295.9380 

6969.3406 

.6 

303.4779 

7328.9901 

.9 

310.7035 

7682.1444 

.3 

296.2522 

6984.1453 

.7 

303.7920 

7344.1718 

99.0 

311.0177 

7697.6874 

.4 

296.5663 

6998.9658 

.8 

304.1062 

7359.3693 

.1 

311.3318 

7713.2461 

.5 

296.8805 

7013.8019 

.9 

304.4203 

7374.5824 

.2 

311.6460 

7728.8206 

i .6 

297.1947 

7028.6538 

97.0 

304.7345 

7389.8113 

.3 

311.9602 

7744.4107 

i ./ 

297.5088 

7043.5214 

.1 

305.0486 

7405.0559 

.4 

312.2743 

7760.0166 

.8 

297.8230 

7058.4047 

.2 

305.3628 

7420.3162 

.5 

312.5885 

7775.6382 

.9 

298.1371 

7073.3037 

.3 

305.6770 

7435.5922 

.6 

312.9026 

7791.2754 

>5.0 

298.4513 

7088.2184 

.4 

305.9911 

7450.8839 

.7 

313.2168 

7806.9284 

.1 

298.7655 

7103.1488 

.5 

306.3053 

7466.1913 

.8 

313.5309 

7822.5971 

.2 

299.0796 

7118.0950 

.6 

306.6194 

7481.5144 

.9 

313.8451 

7838.2815 

.3 

.4 

299.3938 

299.7079 

7133.0568 

7148.0343 

.7 

306.9336 

7496.8532 

100.0 

314.1593 

7853.9816 


Circumferences when the diameter has more than one 

place of decimals. 


iam. 

Circ. 

Diam. 

Circ. 

Diam. 

Circ. 

Diam. 

Circ. 

Diam. 

Circ. 

.i 

.314159 

.01 

.031416 

.001 

.003142 ' 

.0001 

.000314 

.00001 

.000031 

.2 

.628319 

.02 

.062832 

.002 

.006283 

.0002 

.000628 

.00002 

.000063 

.3 

.942478 

.03 

.094248 

.003 

.009425 

.0003 

.000942 

.00003 

.000094 

.4 

1.256637 

.04 

.125664 

.004 

.012566 

.0004 

.001257 

.00004 

.000126 

.5 

1.570796 

.05 

.157080 

.005 

.015708 

.0005 

.001571 

.00005 

.000157 

.6 

1.884956 

.06 

.188496 

.006 

.018850 

.0006 

.001885 

.00006 

.000188 

.7 

2.199115 

.07 

.219911 

.007 

.021991 

.0007 

.002199 

.00007 

.000220 

.8 

2.513274 

.08 

.251327 

.008 

.025133 

.0008 

.002513 

.00008 

.000251 

.9 

2.827433 

.09 

.282743 

.009 

.028274 

.0009 

.002827 

.00009 

.000283 


Examples. 


iameter = 3.12699 


ircu inference = 

Sum of 

Circ for dia of 

3.1 

= 9.738937 


.02 

= .062832 

U 

.006 

= .018850 

it 

.0009 

== .002827 

u 

.00009 

= .000283 



9.823729 


Circumfce = 
Diameter = 

9.823729 

Sum of 

Dia for circ of 

9.73S937 = 

3.1 

U 

.084792 
.062832 == 

.02 

it 

.021960 
.018850 = 

.006 

it 

.003110 
.002827 = 

.0009 

ti 

.000283 
.000283 = 

.00009 

3.12099 



































































134 


CIRCLES, 


TABLE 3 OF CIRCLES. 


Diams in units and twelfths* as in feet ami inches. 


Din. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area - 

Ft.In. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft 1 




5 

0 

15.70796 

19.63495 

10 0 

31.41593 

78.539 

0 1 

.261799 

.005454 


1 

15.96976 

20.29491 

1 

31.67773 

79.854 

2 

.523599 

.021817 


2 

16.23156 

20.96577 

2 

31.93953 

81.179| 

3 

.785398 

.049087 


3 

16.49336 

21.64754 

3 

32.20132 

82.51,' 

4 

1.047198 

.087266 


4 

16.75516 

22.34021 

4 

32.46312 

83.8631 

5 

1.308997 

.136354 


5 

17.01696 

23.04380 

5 

32.72492 

85.2211 

6 

1.570796 

.196350 


6 

17.27876 

23.75829 

6 

32.98672 

86.591 

7 

1.832596 

.267254 


7 

17.54056 

24.48370 

7 

33.24852 

87.9711 

8 

2.094395 

.349066 


8 

17.80236 

25.22001 

8 

33.51032 

89.361 

9 

2.356195 

•441786 


9 

18.06416 

25.96723 

9 

33.77212 

90.761 

10 

2.617994 

.545415 


10 

18.32596 

26.72535 

10 

34.03392 

92.175. 

11 

2.879793 

.659953 


11 

18.58776 

27.49439 

11 

34.29572 

93.598 1 

1 0 

3.14159 

.785398 

6 

0 

18.84956 

28.27433 

11 0 

34.55752 

95.03: 

1 

3.40339 

.921752 


1 

19.11136 

29.06519 

1 

34.81932 

96.478 

2 

3.66519 

1.06901 


2 

19.37315 

29.86695 

2 

35.08112 

97.934 

3 

3.92699 

1.22718 


3 

19.63495 

30.67962 

3 

35.34292 

99.401 

4 

4.18879 

1.39626 


4 

19.89675 

31.50319 

4 

35.60472 

100.881 

5 

4.45059 

1.57625 


5 

20.15855 

32.33768 

5 

35.86652 

102.361 

6 

4.71239 

1.76715 


6 

20.42035 

33.18307 

6 

36.12832 

103.868 

7 

4.97419 

1.96895 


7 

20.68215 

34.03937 

7 

36.39011 

105.379 

8 

5.23599 

2.18166 


8 

20.94395 

34.90659 

8 

36.65191 

106.901 

9 

5.49779 

2.40528 


9 

21.20575 

35.78470 

9 

36.91371 

108.434 

10 

5.75959 

2.63981 


10 

21.46755 

36.67373 

10 

37.17551 

109.977 

11 

6.02139 

2.88525 


11 

21.72935 

37.57367 

11 

37.43731 

111.53:1 

2 0 

6.28319 

3.14159 

7 

0 

21.99115 

38.48451 

12 0 

37.69*911 

113.097 

1 

6.54498 

3.40885 


1 

22.25295 

39.40626 

1 

37.96091 

114.67: 

2 

6.80678 

3.68701 


o 

22.51475 

40.33892 

2 

38.22271 

116.261 

3 

7.06858 

3.97608 


3 

22.77655 

41.28249 

3 

38.48451 

117.858 

4 

7.33038 

4.27606 


4 

23.03835 

42.23697 

4 

38.74631 

119.467 

5 

7.59218 

4.58694 


5 

23.30015 

43.20235 

5 

39.00811 

121.087 

6 

7.85398 

4.90874 


6 

23.56194 

44.17865 

6 

39.26991 

122.718 

7 

8.11578 

5.24144 


7 

23.82374 

45.16585 

7 

39.53171 

124.361 

8 

8.37758 

5.58505 


8 

24.08554 

46.16396 

8 

39.79351 

126.01: 

9 

8.63938 

5.93957 


9 

24.34734 

47.17298 

9 

40.05531 

127.671 

10 

8.90118 

6.30500 


10 

24.60914 

48.19290 

10 

40.31711 

129.351 

11 

9.16298 

6.68134 


11 

24.87094 

49.22374 

11 

40.57891 

131.031 1 

3 0 

9.42478 

7.06858 

8 

0 

25.13274 

50.26548 

13 0 

40.84070 

132.73: 

1 

9.68658 

7.46674 


1 

25.39454 

51.31813 

1 

41.10250 

134.439 

2 

9.94838 

7.87580 


2 

25.65634 

52.38169 

2 

41.36430 

136.157 

3 

10.21018 

8.29577 


3 

25.91814 

53.45616 

3 

41.62610 

137.S81 

4 

10.47198 

8.72665 


4 

26.17994 

54.54154 

4 

41.88790 

139.621 

5 

10.73377 

9.16843 


5 

26.44174 

55.63782 

5 

42.14970 

141.37- 

6 

10.99557 

9.62113 


6 

26.703,54 

56.74502 

6 

42.41150 

143.13, 

7 

11.25737 

10.08473 


7 

26.96534 

57.86312 

7 

42.67330 

144.91, 

8 

11.51917 

10.55924 


8 

27.22714 

58.99213 

8 

42.93510 

146.694 

9 

11.78097 

11.04466 


9 

27.48894 

60.13205 

9 

43.19690 

148.489 

10 

12.04277 

11.54099 


10 

27.75074 

61.28287 

10 

43.45870 

150.29 i 

11 

12.30457 

12.04823 


11 

28.01253 

62.44461 

11 

43.72050 

152. J111 

4 0 

12.56637 

12.56637 

9 

0 

28.27433 

63.61725 

14 0 

43.98230 

153.938 

1 

12.82817 

13.09542 


1 

28.53613 

64.80080 

1 

44.24410 

155.771 

2 

13.08997 

13.63538 


2 

28.79793 

65.99526 

2 

44.50590 

157.627 

3 

13.35177 

14.18625 


3 

29 05973 

67.20063 

3 

44.76770 

159.481 

4 

13.61357 

14.74803 


4 

29.32153 

68.41691 

4 

45.02949 

161.357 

5 

13.87537 

15.32072 


5 

29.58333 

69.64409 

5 

15.29129 

163.237 

6 

14.13717 

15.90431 


6 

29.84513 

70.88218 

6 

45.55309 

165.131 

7 

11.39897 

16.49882 


7 

30.10693 

72.13119 

7 

45.81489 

167.03: 

8 

14.66077 

17.10423 


8 

30.36873 

73.39110 

8 

46.07669 

168.947 

9 

14.92257 

17.72055 


9 

30.63053 

74.66191 

9 

46 33849 

170.87: 

10 

15.18436 

18.34777 


10 

30.89233 

75.94364 

10 

46.60029 

172.809 

11 

115.44616 

18.98591 


11 

31.15413 

77.23627 

11 

46.86209 

174.751 



























CIRCLES. 


135 


TABLE 3 OF CIRCLES—(Continued). 


I)iams In units ami twelfths; as in feet and inelies. 


Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Ft.In 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.Iu. 

Feet. 

Sq. ft. 

15 0 

47.12389 

176.7146 

20 0 

62.83185 

314.1593 

25 0 

78.53982 

490.8739 

1 

47.38569 

178.6835 

1 

63.09365 

316.7827 

1 

78.80162 

494.1518 

2 

47.64749 

180.6634 

2 

63.35545 

319.4171 

2 

79.06342 

497.4407 

8 

47.90929 

182.6542 

3 

63.61725 

322.0623 

3 

79.32521 

500.7404 

4 

48.17109 

184.6558 

4 

63.87905 

324.7185 

4 

79.58701 

504.0511 

5 

48.43289 

186.6684 

5 

64.14085 

327.3856 

5 

79.84881 

507.3727 

6 

48.69469 

188.6919 

6 

64.40265 

330.0636 

6 

80.11061 

510.7052 

7 

48.95649 

190.7263 

7 

64.66445 

332.7525 

7 

80.37241 

514.0486 

8 

49.21828 

192.7716 

8 

64.92625 

335.4523 

8 

80.63421 

517.4029 

9 

49.48008 

194.8278 

9 

65.18805 

338.1630 

9 

80.89601 

520.7681 

10 

49.74188 

196.8950 

10 

65.44985 

340.8846 

10 

81.15781 

524.1442 

11 

50.00368 

198.9730 

11 

65.71165 

343.6172 

11 

81.41961 

527.5312 

IG 0 

50.26548 

201.0619 

21 0 

65.97345 

346.3606 

26 0 

81.68141 

530.9292 

1 

50.52728 

203.1618 

1 

66.23525 

349.1149 

1 

81.94321 

534.3380 

2 

50.78908 

205.2725 

2 

66.49704 

351.8802 

2 

82.20,501 

537.7578 

8 

51.05088 

207.3942 

3 

66.75884 

354.6564 

3 

82.46681 

541.1884 

4 

51.31268 

209.5268 

4 

67.02064 

357.4434 

4 

82.72861 

544.6:400 

5 

51.57448 

211.6703 

5 

67.28244 

360.2414 

5 

82.99041 

548.0825 

6 

51.83628 

213.8246 

6 

67.54424 

363.0503 

6 

83.25221 

551.5459 

7 

52.09808 

215.9899 

7 

67.80604 

365.8701 

7 

83.51400 

555.0202 

8 

52.35988 

218.1662 

8 

68.06784 

368.7008 

8 

83.77580 

558.5054 

9 

52.62168 

220.3533 

9 

68.32964 

371.5424 

9 

84.03760 

562.0015 

10 

52.88348 

222.5513 

10 

68.59144 

374.3949 

10 

84.29940 

565.5085 

11 

.53.14528 

224.7602 

11 

68.85324 

377.2584 

11 

84,56120 

569.0264 

17 0 

53.40708 

226.9801 

22 0 

69.11504 

380.1327 

27 0 

84.82300 

572.5553 

1 

53.66887 

229.2108 

1 

69.37684 

383.0180 

1 

85.08480 

576.0950 

2 

53.93067 

231.4525 

2 

69.63864 

385.9141 

2 

85.34660 

579.6457 

3 

54.19247 

233.7050 

3 

69.90044 

388 8212 

3 

85.60810 

583.2072 

4 

54.45427 

235.9685 

4 

70.16224 

391.7392 

4 

85.87020 

586.7797 

5 

54.71607 

238.2429 

5 

70.42404 

394.6680 

5 

86.13200 

590.3631 

6 

54.97787 

240.5282 

6 

70.68583 

397.6078 

6 

86.39380 

593.9574 

7 

55.23967 

242.8244 

7 

70.94763 

400.5585 

7 

86.65560 

597.5626 

8 

55.50147 

245.1315 

8 

71.20943 

403.5201 

8 

86.91740 

604.1787 

9 

55.76327 

247.4495 

9 

71.47123 

406.4926 

9 

87.17920 

604.8057 

10 

56.02507 

249.7784 

10 

71.73303 

409.4761 

10 

87.44100 

608.4436 

11 

56.28687 

252.1183 

11 

71.99483 

412.4704 

11 

87.70279 

612.0924 

.8 0 

56.54867 

254.4690 

23 0 

72.25663 

415.4756 

28 0 

87.96459 

615.7522 

1 

56.81047 

256.8307 

1 

72.51843 

418.4918 

1 

88.22639 

619.4228 

2 

57.07227 

259.2032 

2 

72.78023 

421.5188 

2 

88.48819 

623.1044 

3 

57.33407 

261.5867 

3 

73.04203 

424,5.568 

3 

88.74999 

626.7968 

4 

57.59587 

263.9810 

4 

73.30383 

427.6057 

4 

89.01179 

630.5002 

5 

57.85766 

266.3863 

5 

73.56563 

430.6654 

5 

89.27359 

634.2145 

6 

58.11946 

268.8025 

6 

73.82743 

433.7361 

6 

89.53539 

687.9397 

7 

58.38126 

271.2296 

7 

74.08923 

436.8177 

7 

89.79719 

641.6758 

8 

58.64306 

273.6676 

8 

74.35103 

439.9102 

8 

90.05899 

645.4228 

9 

58.90486 

276.1165 

9 

74.61283 

443.0137 

9 

90.32079 

649.1807 

10 

59.16666 

278.5764 

10 

74.87462 

446.1280 

10 

90.58259 

652.9495 

11 

59.42846 

281.0471 

11 

75.13642 

449.2532 

11 

90.84439 

656.7292 

9 0 

59.69026 

283.5287 

24 0 

75.39822 

452.3893 

29 0 

91.10619 

660.5199 

1 

59.95206 

286.0213 

1 

75.66002 

455.5364 

1 

91.36799 

664.3214 

2 

60.21386 

288.5247 

2 

75.92182 

458.6943 

2 

91.62979 

668.1339 

3 

60.47566 

291.0391 

3 

76.18362 

461.8632 

3 

91.89159 

671.9572 

4 

60.73746 

293.5644 

4 

76.44542 

465.0430 

4 

92.15338 

675.7915 

5 

60.99926 

296.1006 

5 

76.70722 

468.2337 

5 

92.41518 

679.6367 

6 

61.26106 

298.6477 

6 

76.96902 

471.4352 

6 

92.67698 

683.4928 

7 

61.52286 

301.2056 

7 

77.23082 

474.6477 

7 

92.93878 

687.3597 

8 

61.78466 

303.7746 

8 

77.49262 

477.8711 

8 

93.200.58 

691.2377 

9 

62.04645 

306.3544 

9 

77.75442 

481.1055 

9 

93.46238 

695.1265 

10 

62.30825 

308.9451 

10 

78.01622 

484.3507 

10 

93.72418 

699.0262 

11 

62.57005 

311.5467 

11 

78.27802 

487.6068 

11 

93.98598 

702.9368 































136 


CIRCLES, 


TABLE 3 OF CIRCLES— (Continued). 

IHams in nnits and twelfths; as in feet and inches. 


Dia. 

jcircumf. 

Area. 

Dia. 

Circuinf. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

30 0 

94.24778 

706.8583 

3o 0 

109.9557 

1 

94.50958 

710.7908 

1 

110.2175 

2 

94.77138 

714.7341 

2 

110.4793 

3 

95.03318 

718.6884 

3 

110.7411 

4 

95.29498 

722.6536 

4 

111.0029 

5 

95.55678 

726.6297 

5 

111.2647 

6 

95.81858 

730.6166 

6 

111.5265 

7 

96.08038 

734.6145 

7 

111.7883 

8 

96.34217 

738.6233 

8 

112.0501 

y 

96.60397 

742.6431 

9 

112.3119 

10 

96.86577 

7466737 

10 

112.5737 

n 

97.12757 

750.7152 

11 

112.8355 

31 0 

97.38937 

754.7676 

36 0 

113.0973 

1 

97.65117 

758.8310 

1 

113.3591 

2 

97.91297 

762.9052 

2 

113.6209 

3 

98.17477 

766.9904 

3 

113.8827 

4 

98.43657 

771.0865 

4 

114.1445 

5 

98.69837 

775.1934 

5 

114.4063 

6 

98.96017 

779.3113 

6 

114.6681 

7 

99.22197 

783.4401 

7 

114.9299 

8 

99.48377 

787.5798 

8 

115.1917 

9 

99.74557 

791.7304 

9 

115.4535 

10 

100.0074 

795.8920 

30 

115.7153 

11 

100.2692 

800.0644 

11 

115.9771 

32 0 

100.5310 

804.2477 

37 0 

116.2389 

1 

100.7928 

808.4420 

1 

116.5007 

2 

101.0546 

812.6471 

2 

116.7625 

3 

101.3164 

816.8632 

3 

117.0213 

4 

101.5782 

821.0901 

4 

117.2861 

5 

101.8400 

825.3280 

5 

117.5479 

6 

102.1018 

829.5768 

6 

117.8097 

7 

102.8636 

833.8365 

7 

118.0715 

8 

102.6254 

838.1071 

8 

118.3333 

y 

102.8872 

842.3886 

9 

118.5951 

10 

103.1490 

846.6810 

10 

118.8569 

n 

103.4108 

850.9844 

11 

119.1187 

S3 0 

103.6726 

855.2986 

38 0 

119.3805 

l 

103.9344 

859.6237 

1 

119.6423 

2 

104.1962 

863.9598 

2 

119.9041 

3 

104.4580 

868.3068 

3 

120.1659 

4 

101.7198 

872.66-16 

4 

120.4277 

5 

101.9816 

877.0334 

5 

120.6895 

6 

105.2434 

881.4131 

6 

120.9513 

7 

105.5052 

885.8037 

7 

121.2131 

8 

105.7670 

890.2052 

8 

121.4749 

y 

106.0288 

894.6176 

9 

121.7367 

10 

106.2906 

899.0409 

10 

121.9985 

li 

106.5524 

903.4751 

11 

122.2603 

84 0 

106.8142 

907.9203 

39 0 

122.5221 

1 

107.0759 

912.3763 

1 

122.7839 

2 

107.3377 

916.8433 

2 

123.0457 

3 

107.5995 

921.3211 

3 

123.3075 

4 

107.8613 

925.8099 

4 

123.5693 

5 

108.1231 

930.3096 

5 

123.8311 

6 

108.3849 

934.8202 

6 

124.0929 

7 

108.6467 

939.3417 

7 

124.3547 

8 

108.9085 

943.8741 

8 

124.6165 

9 

109.1703 

948.4174 

9 

124.8783 

10 

109.4321 

952.9716 

10 

125.1401 

11 

109.6939 

957.5367 

11 

125.4019 


Area. 

Dia. 

Circuinf. 

Area. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 1 

962.1128 

40 0 

125.6637 

1256.6371 

966.6997 

1 

125.9255 

3261.8785 

971.2975 

2 

126.1873 

1267.1309 

975.9063 

3 

126.4491 

1272.3941 

980.5260 

4 

126.7109 

1277.6683 

985.1566 

5 

126.9727 

1282.9531 

989.7980 

6 

127.2345 

1288.2493 

994.4504 

7 

127.4963 

1293.5562 

999.1137 

8 

127.7581 

1298.8740 

1003.7879 

9 

128.0199 

1304.2027 

1008.4731 

10 

128.2817 

1309.5424 

1013.1691 

11 

128.5435 

1314.8929 

1017.8760 

41 0 

128.8053 

1320.2543 

1022.5939 

1 

129.0671 

1325.6267 

1027.3226 

2 

129.3289 

1331.0099 

1032.0623 

3 

129.5907 

1336.4041 

1036.8128 

4 

129.8525 

1341.8091 

1041.5743 

5 

130.1113 

1347.2251 

1046.3467 

6 

130.3761 

1352.6520 

1051.1300 

7 

130.6379 

1358.0898 

1055.9242 

8 

130.8997 

1363.53S5 

1060.7293 

9 

131.1615 

1368.9981 

1065.5453 

10 

131.4233 

1374.4686 

1070.3723 

11 

131.6851 

1379.9501 

1075.2101 

42 0 

131.9469 

1385.4424 

10S0.05S8 

1 

132.2087 

1390.9456 

1084.9185 

2 

132.4705 

1396.459* 

1089.7890 

3 

132.7323 

1101.9848 

1091.6705 

4 

132.9941 

1107.5208 

1099.5629 

5 

133.2559 

1413.0676 

1104.4662 

6 

133.5177 

1-118.6254 

1109.3804 

7 

133.7795 

1424.1941 

1114.3055 

8 

134.0413 

1429.7737 

1119.2415 

9 

134.3031 

1435.3642 

1124.1884 

10 

131.5649 

1-140.9656 

1129.1462 

11 

134.8267 

1446.578C 

1134.1149 

43 0 

135.0885 

1452.2012 . 

1139.0946 

1 

135.3503 

1457.8353" 

1144.0S51 

2 

135.6121 

1463.4804 

1149.0866 

3 

135.8739 

1469.1364 

1154.0990 

4 

136.1357 

1474.8032 

1159.1222 

5 

136.3975 

1480.4810 

1164.1564 

6 

136.6593 

1486.1697 

1169.2015 

7 

136.9211 

1491.8692 

1174.2575 

8 

337.1829 

1497.5791 

1179.3244 

9 

137.4447 

1503.3012 

1184.4022 

10 

137.7065 

1509.0335 

1189.4910 

11 

137.9683 

1514.7767 

1194.5906 

44 0 

138.2301 

1520.53081 

1199.7011 

1 

138.4919 

1526.2959™ 

1204.8226 

2 

138.7537 

1532.0718 

1209.9550 

3 

139.0155 

1537.8587 

1215.0982 

4 

139.2778 

1543.6565 

1220.2524 

5 

139.5391 

1549.4651 

1225.4175 

6 

139.8009 

1555.2847 

1230.5935 

7 

140.0627 

1561.1152 

1235.7804 

8 

140.3245 

1566.9566 

1240.9782 

9 

340.5863 

1572 8089 

1246.1869 

10 

140.8481 

1578.6721 

1251.4065 

11 

141.1099 

1584.5462 


































CIRCLES 


13? 


TABLE 3 OF CIRCLES —(Continued). 


I>iams in units and twelfths; as in feet and inehes. 


I)ia. 

Circumi 

Area. 

Dia. 

Circumi 

Area. 

Dia. 

Circumi 

Area. 

Ft.In 

• Feet. 

Sq. ft. 

Ft.In 

. Feet. 

Sq. ft. 

Ft .In 

Feet. 

Sq. ft. 

45 0 

141.3717 

1590.4313 

50 0 

157.0796 

1963.4954 

do 0 

172.7876 

2375.8294 

l 

141.6335 

1596.3272 

1 

157.3414 

1970.0458 

1 

173.0494 

2383.0344 

2 

141.8953 

1602.2341 

2 

157.6032 

1976.6072 

2 

173.3112 

2390.2502 

3 

142.1571 

1608.1518 

3 

157.8650 

1983.1794 

3 

173.5730 

2397.4770 

i 

142.4189 

1614.0805 

4 

158.1268 

1989.7626 

4 

173.8348 

2404.7146 

£ 

142.6807 

1620.0201 

5 

158.3886 

1996 3567 

5 

174.0966 

2411.9632 

6 

142.9425 

1625.9705 

6 

158.6504 

2002.9617 

6 

174.3584 

2419.2227 


143.2043 

1631.9319 

7 

158.9122 

2009.5776 

7 

174.6202 

2426.4931 

o 

8 

143.4661 

1637.9042 

8 

159.1740 

2016.2044 

8 

174.8820 

2433.7744 


143.7279 

1643.8874 

9 

159.4358 

2022.8421 

9 

175.1438 

2441.0666 

JO 

143.9897 

1649.8816 

10 

159.6976 

2029.4907 

10 

175.4056 

2448.3697 

11 

144.2515 

1655.8866 

11 

159.9594 

2036.1502 

11 

175.6674 

2455.6837 

IG 0 

144.5133 

1661.9025 

51 0 

160.2212 

2042.8206 

56 0 

175.9292 

2463.0086 


i 144.7751 

1667.9294 

1 

160.4830 

2049.5020 

1 

176.1910 

2470.3445 

2 

145.0369 

1673.9671 

2 

160.7448 

2056.1942 

2 

176.4528 

2477.6912 

3 

145.2987 

1680.0158 

3 

161.0066 

2062.8974 

3 

176.7146 

2485.0489 

4 

145.5605 

1686.0753 

4 

161.2684 

2069.6114 

4 

176.9764 

2492.4174 

5 

1 145.8223 

1692.1458 

5 

161.5302 

2076.3364 

5 

177.2382 

2499.7969 

6 

146.0841 

1698.2272 

6 

161.7920 

2083.0723 

6 

177.5000 

2507.1873 

7 

j 146.3459 

1704.3195 

7 

162.0538 

2089.8191 

7 

177.7618 

2514.5886 

8 

346.6077 

1710.4227 

8 

162.3156 

2096.5768 

8 

178.0236 

2522.0008 

9 

146.8695 

1716.53(58 

9 

162.5774 

2103.3454 

9 

178,2854 

2529.4239 

10 

147.1313 

1722.6618 

10 

162.8392 

2110.1249 

10 

178.5472 

2536.8579 

11 

147.3931 

1728.7977 

11 

163.1010 

2116.9153 

11 

178.8090 

2544.3028 

17 0 

147.6549 

1734.9445 

52 0 

163.3628 

2123.7166 

57 0 

179.0708 

2551. / 086 

1 

147.9167 

1741.1023 

1 

163.6246 

2130.5289 

1 

179.3326 

2559.2254 

2 

148.1785 

1747.2709 

2 

163.8864 

2137.3520 

2 

179.5944 

2566.7030 

3 

148.4403 

1753.4505 

3 

164.1482 

2144.1861 

3 

179.8562 

2574.1916 

4 

148.7021 

1759.6410 

4 

164.4100 

2151.0310 

4 

180.1180 

2581.6910 

5 

148.9639 

1765.8423 

5 

164.6718 

2157.8869 

5 

180.3798 

2589.2014 

6 

149.2257 

1772.0546 

6 

164.9336 

2164.7537 

6 

180.6416 

2596.7227 

7 

149.4875 

1778.2778 

7 

165.1954 

2171.6314 

7 

180.9034 

2604.2549 

8 

149.7492 

1784.5119 

8 

165.4572 

2178.5200 

8 

181.1652 

2611.7980 

9 

150.0110 

1790.7569 

9 

165.7190 

2185.4195 

9 

181.4270 

2619.3520 

10 

150.2728 

1797.0128 

10 

165.9808 

2192.3299 

10 

181.6888 

2626.9169 

11 

150.5346 

1803.2796 

11 

166.2426 

2199.2512 

11 

181.9506 

2634.4927 

8 0 

150.7964 

1809.5574 

53 0 

166.5044 

2206.1834 

58 0 

182.2124 

2642.0794 

1 

151.0582 

1815.8460 

1 

166.7662 

2213.1266 

1 

182.4742 

2649.6771 

2 

151.3200 

1822.1456 

2 

167.0280 

2220.0806 

2 

182.7360 

2657.2856 

3 

151.5818 

1828.4560 

3 

167.2898 

2227.0456 

3 

182.9978 

2664.9051 

4 

151.8436 

1834.7774 

4 

167.5516 

2234.0214 

4 

183.2596 

2672.5354 

5 

152.1054 

1841.1096 

5 

167.8134 

2241.0082 

5 

183.5214 

2680.1767 

6 

152.3672 

1847.4528 

6 

168.0752 

2248.0059 

6 

183.7832 

2687.8289 

7 

152.6290 

1853.8069 

7 

168.3370 

2255.0145 

7 

184.0450 

2695.4920 

8 1 

152.8908 

1860.1719 

8 

168.5988 

2262.0340 

8 

184.3068 

2703.1659 

9 

153.1526 

1866.5478 

9 

168.8606 

2269.0644 

9 

184.5686 

2710.8508 

10 

153.4144 

1872.9346 

10 

169.1224 

2276.1057 

10 

184.8304 

2718.5467 

11 

153.6762 

1879.3324 

11 

169.3842 

2283.1579 

11 

185.0922 

2726.2534 

9 0 

153.9380 

1885.7410 

54 0 

169.6460 

2290.2210 

59 0 

185.3540 

2733.9710 

1 

154.1998 

1892.1605 

1 

169.9078 

2297.2951 

1 

185.6158 

2741.6995 

2 

154.4616 

1898.5910 

2 

170.1696 

2304.3800 

2 

185.8776 

2749.4390 

3 

154.7234 

1905.0323 

3 

170.4314 

2311.4759 

3 

186.1394 

2757.1893 

4 

154.9852 

1911.4846 

4 

170.6932 

2318.5826 

4 

186.4012 

2764.9506 

5 

155 2470 

1917.9478 

5 

170.9550 

2325.7003 

5 

186.6630 

2772.7228 

6 

155.5088 

1924.4218 

6 

171.2168 

2332.8289 

6 

186.9248 

2780.5058 

7 

155.7706 

1930.9068 

7 

171.4786 

2339.9684 

7 

187.1866 

2788.2998 

8 

156.0324 

1937.4027 

8 

171.7404 

2347.1188 

8 

187.4484 

2796.1047 

9 

156.2942 

1943.9095 

9 

172.0022 

2354.2801 

9 

187.7102 

2803.9205 

10 

156.5560 

1950.4273 

10 

172.2640 

2361.4523 

10 

187.9720 

2811.7472 

11 

156.8178 

1956.9559 

11 

172.5258 

2368.6354 

11 

188.2338 

2819.5849 

























































• 

[!). 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

n 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 


CIRCLES 


Feet. 

188.4956 

188.7574 

189.0192 

189.2810 

189.5428 

189.8046 

190.0664 

190.3282 

190.5900 

190.8518 

191.1136 

191.3754 

191.6372 

191.8990 

192.1608 

192.4226 

192.6843 

192.9461 

193.2079 

193.4697 

193.7315 

193.9933 

194.2551 

194.5169 

194.7787 

195.0405 

195.3023 

195.5641 

195.8259 

196.0877 

196.3495 

196.6113 

196.8731 

197.1349 

197.3967 

197.6585 

197.9203 

198.1821 

198.4439 

198.7057 

198.9675 

199.2293 

199.4911 

199.7529 

200.0147 

200.2765 

200.5383 

200.8001 

201.0619 

201.3237 

201.5855 

201.8473 

202.1091 

202.3709 

202.6327 

202.8945 

203.1563 

203.4181 

203.6799 

203.9417 


TABLE 3 OF CIRCLES—(Continued). 

t units sinti twelfths; as in feet and inches. 


Area. 

Dia. 

Circunif. 

Area. 

Dia. 

Circunif. 

Area. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.Iu. 

Feet. 

Sq. ft. 

2827.4334 

65 0 

204.2035 

3318.3072 

70 0 

219.9115 

3848.4510 

2835.2928 

1 

204.4653 

3326.8212 

1 

220.1733 

3857.6194 

2843.1632 

2 

204.7271 

3335.3460 

2 

220.4351 

3866.7988 

2851.0444 

3 

204.9889 

3343.8818 

3 

220.6969 

3875.9890 1 

2858.9366 

4 

205.2507 

3352.4284 

4 

220.9587 

3885.1902 

2866.8397 

5 

205.5125 

3360.9860 

5 

221.2205 

3894.4022 

2874.7536 

6 

205.7743 

3369.5545 

6 

221.4823 

3903.6252 ] 

2882.6785 

7 

206.0361 

3378.1339 

7 

221.7441 

3912.8591 

2890.6143 

8 

206.2979 

3386.7241 

8 

222.0059 

3922.1039 j 

2898.5610 

9 

206.5597 

3395.3253 

9 

222.2677 

3931.3596 

2906.5186 

10 

206.8215 

3403.9375 

10 

222.5295 

3940.6262 \ 

2914.4871 

11 

207.0833 

3412.5605 

11 

222.7913 

3949.9037 

2922.4666 

66 0 

207.3451 

3421.1944 

71 0 

223.0531 

3959.1921 

2930.4569 

1 

207.6069 

3429.8392 

1 

223.3149 

3968.4915 | 

2938.4581 

2 

207.8687 

3438.4950 

2 

223.5767 

3977.8017 

2946.4703 

3 

208.1305 

3447.1616 

3 

223.8385 

3987.1229 

2954.4934 

4 

208.3923 

3455.8392 

4 

224.1003 

3996.4549 

2962.5273 

5 

208.6541 

3464.5277 

5 

224.3621 

4005.7979 

2970.5722 

6 

208.9159 

3473.2270 

6 

224.6239 

4015.1518 

2978.6280 

7 

209.1777 

3481.9373 

7 

224.8857 

4024 5165 

2986.6947 

8 

209.4395 

3490.6585 

8 

225.1475 

4033.8922 

2994.7723 

9 

209.7013 

3499.3906 

9 

225.4093 

4043.2788 

3002.8608 

10 

209.9631 

3508.1336 

10 

225.6711 

4052.6763 

3010.9602 

11 

210.2249 

3516.8875 

11 

225.9329 

4062.0848 

3019.0705 

67 0 

210.4867 

3525.6524 

72 0 

226.1947 

4071.5041 

3027.1918 

1 

210.7485 

3534.4281 

1 

226.4565 

4080.9343 

3035.3239 

2 

211.0103 

3543.2147 

2 

226.7183 

4090.3755 

3043.4670 

3 

211.2721 

3552.0123 

3 

226.9801 

4099.8275 

3051.6209 

4 

211.5339 

3560.8207 

4 

227.2419 

4109.2905 

3059.7858 

5 

211.7957 

3569.6401 

5 

227.5037 

4118.76431 

3067.9616 

6 

212.0575 

3578.4704 

6 

227.7655 

4128.2491 

3076.1483 

7 

212.3193 

3587.3116 

7 

228.0273 

4137.7448 

3084.3459 

8 

212.5811 

3596.1637 

8 

228.2891 

4147.2514 

3092.5544 

9 

212.8429 

3605.0267 

9 

228.5509 

4156.7689' 

3100.7738 

10 

213.1047 

3613.9006 

10 

228.8127 

4166.2973 

3109.0041 

11 

213.3665 

3622.7854 

11 

229.0745 

4175.83661 

3117.2453 

68 0 

213.6283 

3631.6811 

73 0 

229.3363 

4185.3868 

3125.4974 

1 

213.8901 

3640.5877 

1 

229.5981 

4194.9479 

3133.7605 

2 

214.1519 

3649.5053 

2 

229 8599 

4204.5200 

3142.0344 

3 

214.4137 

3658.4337 

3 

230.1217 

4214.10291 

3150.3193 

4 

214.6755 

3667.3731 

4 

230.3835 

4223.6968 

3158.6151 

5 

214.9373 

3676.3234 

5 

230.6453 

4233.3016 

3166.9217 

6 

215.1991 

3685.2845 

6 

230.9071 

4242.91721| 

3175.2393 

7 

215.4609 

3694.2566 

7 

231.1689 

4252.5438 

3183.5678 

8 

215.7227 

3703.2396 

8 

231.4307 

4262.1815 

3191.9072 

9 

215.9845 

3712.2335 

9 

231.6925 

4271.8297 

3200.2575 

10 

216.2463 

3721.2383 

10 

231.9543 

4281.4890? 

3208.6188 

11 

216.5081 

3730.2540 

11 

232.2161 

4291.1592 

3216.9909 

69 0 

216.7699 

3739.2807 

74 0 

232.4779 

4300.8403 

3225.3739 

1 

217.0317 

3748.3182 

1 

232.7397 

4310.5324 

3233.7679 

2 

217.2935 

3757.3666 

2 

233.0015 

4320.2353 

3242.1727 

3 

217.5553 

3766.4260 

3 

233.2633 

4329.9492 

3250.5885 

4 

217.8171 

3775 4962 

4 

233.5251 

4389.6739 

3259.0151 

5 

218.0789 

3784.5774 

5 

233.7869 

4349.4090 

3267.4527 

6 

218.3407 

3793.6695 

6 

234.0487 

4359.1562 

3275.9012 

7 

218.6025 

3802.7725 

7 

234.3105 

4368.9130 

3284.3606 

8 

218.8643 

3811.8864 

8 

234.5723 

4378.6820 

3292.8309 

9 

219.1261 

3821.0112 

9 

234.8341 

4388.4613 

3301.3121 

10 

219.3879 

3830.1469 

10 

235.0959 

4398.2513 

3309.8042 

11 

219.6497 

3839.2935 

11 

235.3576 

4408.0520 































CIRCLES, 


139 


TABLE 3 OF CIRCLES— (Continued). 

Diams In units and twelfths; as in feet and inches. 


I)ia. 

Circumf 

Area. 

I)ia. 

Circumf 

Area. 

Dia. 

Circumf. 

Area. 

Ft.In 

Feet. 

Sq. ft. 

Ft.In 

Feet. 

Sq. ft. 

Ft.In 

1 Feet. 

Sq. ft. 

75 0 

235.6194 

4417.8647 

80 0 

251.3274 

5026.5482 

85 0 

267.0354 

5674.5017 

1 

235.8812 

4427.6876 

1 

251.5892 

5037.0257 

1 

267.2972 

5685 6337 

2 

236.1430 

4437.5214 

2 

251.8510 

5047.5140 

2 

267.5590 

5696 77(15 

3 

236.4048 

4447.3662 

3 

252.1128 

5058.0133 

3 

267.8208 

5707 9302 

4 

236.6666 

4457.2218 

4 

252.3746 

5068.5234 

4 

268.0826 

5719 0949 

5 

236.9284 

4467.0884 

5 

252.6364 

5079.0445 

5 

268.3444 

5730 2705 

6 

237.1902 

4476.9659 

6 

252.8982 

5089.5764 

6 

268.6062 

5741.4569 

7 

237.4520 

4486.8543 

7 

253.1600 

5100.1193 

7 

268.8680 

5752.6543 

8 

9 

237.7138 

237.9756 

4496.7536 

4506.6637 

8 

9 

253.4218 

253.6836 

5110.6731 

5121.2378 

8 

9 

269.1298 

269.3916 

5763.8626 

5775.0818 

10 

238.2374 

4516.5849 

10 

253.9454 

5131.8134 

10 

269.6534 

5786.3119 

11 

238.4992 

4526.5169 

11 

254.2072 

5142.3999 

11 

269.9152 

5797.559.9 

76 0 

238.7610 

4536.4598 

81 0 

254.4690 

5152.9974 

86 0 

270.1770 

5808.8048 

1 

239.0228 

4546.4136 

1 

254.7308 

5163.6057 

1 

270.4388 

5820.0676 

2 

239.28-16 

4556.3784 

2 

254.9926 

5174.2249 

2 

270.7006 

5831 3414 

3 

239.5464 

4566.3540 

3 

255.2544 

5184.8551 

3 

270.9624 

5842.6260 

4 

239.8082 

4576.3406 

4 

255.5162 

5195.4961 

4 

271.2242 

5853.9216 

5 

240.0700 

4586.3380 

5 

255.7780 

5206.1481 

5 

271.4860 

5865.2280 

6 

240.3318 

4596.3464 

6 

256.0398 

5216.8110 

6 

271.7478 

5876.5454 

7 

210.5936 

4606.3657 

7 

256.3016 

5227.4847 

7 

272.0096 

5887.8737 

8 

• 240.8554 

4616.3959 

8 

256.5634 

5238.1694 

8 

272.2714 

5899.2129 

9 

241.1172 

4626.4370 

9 

256.8252 

5248.8650 

9 

272.5332 

5910.5630 

10 

241.3790 

4636.4890 

10 

257.0870 

52595715 

10 

272.7950 

5921.9240 

11 

241.6408 

4646.5519 

11 

257.3488 

5270.2889 

11 

273.0568 

5933.2959 

77 0 

241.9026 

4656.6257 

82 0 

257.6106 

5281.0173 

87 0 

273.3186 

5944.6787 

1 

242.1644 

4666.7104 

1 

257.8724 

5291.7565 

1 

273.5804 

5956.0724 

2 

242.4262 

4676.8061 

2 

258.1342 

5302.5066 

2 

273.8422 

5967.4771 

3 

242.6880 

4686.9126 

3 

258.3960 

5313.2677 

3 

274.1040 

5978.8926 

4 

242.9498 

4697.0301 

4 

258.6578 

5324.0396 

4 

274.3658 

5990.3191 

5 

243.2116 

4707.1584 

5 

258.9196 

5334.8225 

5 

274.6276 

6001.7564 

6 

243.4734 

4717.2977 

6 

259.1814 

5345.6162 

6 

274.8894 

6013.2047 

7 

213.7352 

4727.4479 

7 

259.4432 

5356.4209 

7 

275.1512 

6024.6639 

8 

243.9970 

4737.6090 

8 

259.7050 

5367.2365 

8 

275.4130 

6036.1:340 

9 

244.2588 

4747.7810 

9 

259.9668 

5378.0630 

9 

275.6748 

6047.6149 

10 

244.5206 

4757.9639 

10 

260.2286 

5388.9004 

10 

275.9366 

6059.1068 

11 

244.7824 

4768.1577 

11 

260.4904 

5399.7487 

11 

276.1984 

6070.6097 

78 0 

245.0442 

4778.3624 

83 0 

260.7522 

5410.6079 

88 0 

276.4602 

6082.1234 

1 

245.3060 

4788.5781 

1 

261.0140 

5421.4781 

1 

276.7220 

6093.6480 

2 

215.5678 

4798.8046 

2 

261.2758 

5432.3591 

2 

276.9838 

6105.1835 

3 

245.8296 

4809.0420 

3 

261.5376 

5443.2511 

3 

277.2456 

6116.7300 

4 

246.0914 

4819.2904 

4 

261.7994 

5454.1539 

4 

277.5074 

6128.2873 

5 

246.3532 

4829.5497 

5 

262.0612 

5465.0677 

5 

277.7692 

6139.8556 

6 

246.6150 

4839.8198 

6 

262.3230 

5475.9923 

6 

278.0309 

6151.4348 

7 

246.8768 

4850.1009 

7 

262.5848 

5486.9279 

7 

278.2927 

6163.0248 

8 

247.1386 

4860.3929 

8 

262.8466 

5497.8744 

8 

278.5545 

6174.6258 

9 

247.4004 

4870 6958 

9 

263.1084 

5508.8318 

9 

278.8163 

6186.2377 

10 

247.6622 

4881.0096 

10 

263.3702 

5519.8001 

10 

279.0781 

6197.8605 

11 

247.9240 

4891.3343 

11 

263.6320 

5530.7793 

11 

279.3399 

6209.4942 

79 0 

248.1858 

4901.6699 

84 0 

263.8938 

5541.7694 

89 0 

279.6017 

6221.1389 

1 

248.4476 

4912.0165 

1 

264.1556 

5552.7705 

1 

279.8635 

6232.7944 

2 

248.7094 

4922.3739 

2 

264.4174 

5563.7824 

2 

280.1253 

6244.4608 

3 

248.9712 

4932.7423 

3 

264.6792 

5574.8053 

3 

280.3871 

6256.1382 

4 

249.2330 

4943.1215 

4 

264.9410 

5585.8390 

4 

280.6489 

6267.8264 

5 

249.4948 

4953.5117 

5 

265.2028 

5596.8837 

5 

280.9107 

6279.5256 

6 

249.7566 

4963.9127 

6 

265.4646 

5607.9392 

6 

281.1725 

6291.2356 

7 

250,0184 

4974.3247 

7 

265.7264 

5619.0057 

7 

281.4343 

6302.9566 

8 

250.2802 

4984.7476 

8 

265.9882 

5630.0831 

8 

281.6961 

6314.6885 

9 

250.5420 

4995.1814 

9 

266.2500 

5641.1714 

9 

281.9579 

6326.4313 

10 

2.50.8038 

5005.6261 

10 

266.5118 

5652.2706 

10 

282.2197 

6338.1850 

11 

251.0656 

5016.0817 

11 

266.7736 | 

5663.3807 

11 

282.4815 

6349.9496 











































140 


CIRCLES 


TABLE 3 OF CIRCLES— (Continued). 


Diams in units and twelfths; as in feet and inches. 


Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

90 0 

282.7433 

6361.7251 

93 5 

293.4771 

6853.9134 

96 9 

303.9491 

7351.7686 

1 

283.0051 

6373.5116 

6 

293.7389 

6866.1471 

10 

304.2109 

7364.4386 

2 

283.2669 

6385.3089 

7 

294.0007 

6878.3917 

11 

304.4727 

7377.1195 

3 

283.5287 

6597.1171 

8 

294.2625 

6890.6472 

97 0 

304.7345 

7389.8113 

4 

283.7905 

6108.9363 

9 

294.5243 

6902.9135 

1 

304.9963 

7402.5140 

5 

284.0523 

6120.7663 

10 

294.7861 

6915.1908 

2 

305.2581 

7415.2277 

6 

284.3141 

6432.6073 

11 

295.0479 

6927.4791 

3 

305.5199 

7427.9522 

7 

284.5759 

6444.4592 

94 0 

295.3097 

6939.7782 

4 

305.7817 

7410.6877 

8 

284.8377 

6156.3220 

1 

295.5715 

6952.0882 

5 

306.0435 

7453.4340 

9 

285.0995 

6168.1957 

2 

295.8333 

6964.4091 

6 

306.3053 

7466.1913 

10 

285.3613 

6180.0803 

3 

296.0951 

6976.7410 

r- 

/ 

306.5671 

7478.9595 

11 

285.6231 

6191.9758 

4 

296.3569 

6989.0837 

8 

306.8289 

7491.7385 

91 0 

285.8849 

6503.8822 

5 

296.6187 

7001.4374 

9 

307.0907 

7504.5285 

1 

286.1467 

6515.7995 

6 

296.8805 

7013.8019 

10 

307.3525 

7517.3294 

2 

286.4085 

6527.7278 

7 

297.1423 

7026.1774 

11 

307.6143 

7530.1412 

3 

236.6703 

6539.6669 

8 

297.4041 

7038.5638 

98 0 

307.8761 

7542.9640 

4 

286.9321 

6551.6169 

9 

297.6659 

7050.9611 

1 

308.1379 

7555.7970 

5 

287.1939 

6563.5779 

10 

297.9277 

7063.3693 

2 

308.3997 

7568.6421 

6 

287.4557 

6575.5498 

11 

298.1895 

7075.7884 

8 

308.6615 

7581.4976 

7 

287.7175 

6587.5325 

95 0 

298.4513 

7088.2184 

4 

308.9233 

7594.3639 

8 

287.9793 

6599.5262 

1 

298.7131 

7100.6593 

5 

309.1851 

7607.2412 

9 

288.2411 

6611.5308 

2 

298.9749 

7113.1112 

6 

309.4469 

7620.1293 

10 

288.5029 

6623.5463 

3 

299.2367 

7125.5739 

7 

309.7087 

7633.0284 

11 

288.7647 

6635.5727 

4 

299.4985 

7138.0476 

8 

309.9705 

7645.9384 

92 0 

289.0265 

6647.6101 

5 

299.7603 

7150.5321 

9 

310.2323 

7658.8593 

1 

289.2883 

6659.6583 

6 

300.0221 

7163.0276 

10 

310.4941 

7671.7911 

2 

289.5501 

6671.7174 

7 

300.2839 

7175.5340 

11 

310.7559 

7684.7338 

3 

289.8119 

6683.7875 

8 

300.5457 

7188.0513 

99 0 

311.0177 

7697.6874 

4 

290.0737 

6695.8684 

9 

300.8075 

7200.5794 

1 

311.2795 

7710.6519 

5 

290.3355 

6707.9603 

10 

301.0693 

7213.1185 

2 

311. 5413 

7723.6274 

6 

290.5973 

6720.0630 

11 

301.3311 

7225.6686 

3 

311.8031 

7736.6137 

7 

290.8591 

6732.1767 

96 0 

301.5929 

7238.2295 

4 

312.0649 

7749.6109 

8 

291.1209 

6744.3013 

1 

301.8547 

7250.8013 

5 

312.3267 

7762.6191 

9 

291.3827 

6756.4368 

2 

302.1165 

7263.3840 

6 

312.5885 

7775.6382 

10 

291.6445 

6768.5832 

3 

302.3783 

7275.9777 

7 

312.8503 

7788.6681 

11 

291.9063 

6780.7405 

4 

302.6401 

7288.5822 

8 

313.1121 

7801.7090 

93 0 

292.1681 

6792.9087 

5 

302.9019 

7301.1977 

9 

313.3739 

7814.7608 

1 

292.4299 

6805.0878 

6 

303.1637 

7313.8240 

10 

313.6357 

7827.8235 

2 

292.6917 

6817.2779 

7 

303.4255 

7326.4613 

11 

313.8975 

7840.8971 

3 

4 

292.9535 

293.2153 

6829.4788 

6841.6907 

8 

303.6873 

7339.1095 

100 0 

314.1593 

7853.9816 


Circumferences in feet, when the diam contains fractions 
of an inch. See similar process, p 133. 


niam, 

Circumf, 

Diam, 

inch. 

Circumf, 

Diam, 

Circumf, 

Diam, 

Circumf, 

Diam, 

Circumf, 

inch. 

foot 

foot 

inch 

foot. 

inch. 

foot. 

inch. 

foot. 

1 64 

.004091 

7-32 

.057269 

27-64 

.110447 

5-8 

.163625 

53-64 

.216803 

1-32 

.008181 

15-64 

.061359 

7-16 

.114537 

41-64 

.167715 

27-32 

.220893 

3-64 

.012272 

X A 

.065450 

29-64 

.118628 

21-32 

.171806 

55-64 

.224984 

1-16 

.016362 

17-64 

.069540 

15-32 

.122718 

43-64 

.175896 

7-8 

.229074 

5-64 

.020153 

9-32 

.073631 

31-64 

.126809 

11-16 

.179987 

57-64 

.233165 

3-32 

.024544 

19-64 

.077722 


.130900 

45-64 

.184078 

29-32 

.237256 

7-64 

.028634 

5-16 

.081812 

33-64 

.134990 

23-32 

.188168 

59-64 

.241346 

& 

.032725 

21-64 

.085903 

17412 

.139081 

47-64 

.192259 

15-16 

.245437 

.036816 

11-32 

.089994 

35-64 

.143172 


.196350 

61-64 

.249528 

5-32 

.040906 

23-64 

.094084 

9-16 

.147262 

49-64 

.200440 

31-32 

.253618 

11-61 

.044997 

% 

.098175 

37-64 

.151353 

25-32 

.204531 

63-64 

.257709 

3-16 

13-64 

.049087 

.053178 

25-64 

13-32 

.102265 

.106356 

19-32 

39-64 

.155443 

.159534 

51-64 

13-16 

.208621 

.212712 

1 

.261799 












































CIRCULAR ARCS. 


141 


CIRCULAR ARCS. (1 





Fig.l 

Rules for Fig. 1 apply to gll arcs equal 
“ “ Fig. 2 “ “ “ 



or less than, a semi-circle, 
or greater than, a semi-circle. 


Chord, a b, of whole arc, a d b f 

= 2 X \/ radius 2 — (radius — rise) 2 . Fig. 1. 

= 2 X v/radius 2 — (rise — radius) 2 . Fig. 2. 

= 2 X V/rise X (2 X radius — tise). Figs. 1 and 2. 

= 2 X radius X sine of % a c b. Figs. 1 and 2. 

rise 


= 2 X 


tangent of ab d* 


Figs. 1 and 2. 


= 2 X db$ X cosiue of abd.* Figs. 1 and 2. 


= 2 X %/d b 2 — rise 2 . Figs. 1 and 2.g 

= approximately 8 X db% — 3 X Length of arc adb^. Fig. 1. 

For table of chords to radius 1, see p. 105. 


Length, a db f 

_ .. arc a d b in degrees 

= 2 n radius X -—--- . Figs. 1 and 2; 

«= .01745 X radius X arc a d b in degrees. Figs. 1 and 2. 

If the arc contains fractions of a degree, see p. 57. 

= circumference of circle — length of small arc subtending angle acb. Fig. 2. 


approximately 


8 X dbg — chord a b.** 


Fig. 1. 


Continued on p. 141a. See also tables, pp. 143,144 and 145. 


* a b d is = 34 of the angle acb, subtended by the arc. In Fig. 2 the latter angle 
exceeds 180°. 


i db — chord of dib, or of half a db — \/ rise 2 + (^ a 5) 2 . Figs. 1 and 2. 


f If rise 


multiply the result by 

I If rise == 

multiply the result by 

.5 chord, 

1.036 

.25 chord. 

1.0044 

.4 

U 

1.0193 

.2 “ 

1.0021 

.333 

a 

1.0114 

.125 “ 

1.00036 

.3 

u 

1.0083 1 

.1 “ 

1.00015 

* * If rise 

= 

multiply the result by 1 

If rise — 

multiply the result by 

.5 

chord 

1.012 

.25 chord 

1.0015 

.4 

u 

1.0065 

.2 “ 

1.0007 

.333 

tc 

1.0038 

.125 “ 

1.00012 

.3 


1.0028 

.1 “ 

1.00005 































141a 


CIRCULAR ARCS. 


cl 


cl 



Fig.l 



Continued from p. 141. 


Rules for Fig. I apply to all arcs equal to or less than a semi-circle. 

“ “ Fig. 2 “ “ “ or greater than a semi-circle. 


Radius, c a, c d, c i or c b, 

_ 04 a W + ri8e2 , Figs. 1 mid 2. 

2 X rise 


_/4 , Figs. 1 and 2. 

sine of %bcd || 

rise & e _ ^ Fig. 2. 

1 -f cosine of 

For tables of radii to chords of 100 feet, and of 20 metres, see pp. 727 and 728. 

Rise, or middle ordinate, d e f 

— radius — \/ radius 2 — 04 ab ) 2 , Fig. 1. 

= radius + \/ radius 2 — (% a b) 2 , Fig. 2. 

= radius X (1 — cosine of bcd||), Fig. 1. 

= radius X (1 + cosine of bed ||),f Fig. 2. 

__ % , Figs. 1 and 2. 

2 X radius 

*= 34 a b X tangent of a b d* Figs. 1 and 2. 

*= approximately - ^ a ^ — j Fig. 1. 

2 X radius 

When radius = chord a b , the result is 6.7 parts in K>0 too short. 

= ^ X chord a b, the result is 0.7 parts in 100 too short. 

For tables of middle ordinates, seo pp. 726 to 730. 

Side ordinate, as n i, 

= \/radius 2 — e n 2 -f rise — radius, Figs. 1 and 2. 

. , an X iib r- , 

= approximately ---- » rig. 1. 

2 X radius 

For tables of side ordinates, see p. 730. 

*dbd\9 — 34 of the angle acb, subtended by the arc. 

tSri'ctly, this should read 1 minus cosine; but the cosines of angles between 90° 
and 270 must then be regarded as minus or negative. Our rule, therefore, amounts 
to the same thing. 

g d 6 = chord of dib, or of half a d 6, = v/rise 2 + {V^ a b) 2 . Figs. 1 and 2. 
exceeds 180°^ a " Sle 8ubtended b F the arc ' In Fig. 2, the latter angle 


} Figs. 1 and 2 

2 X rise 
Y^ab 


sine of %acb 
rise d e 


1 — cosine of %acb 


, Figs. 1 and 2. 

, Fig. 1. 


























CIRCULAR ARCS. 


14U 


> 


Angle, a c b, subtended toy arc, a db. 

An angle and its supplement (as bee and bed, Fig. 2) have the same sine, the 
same cosine and the same tangent. 

Caution, Ihe following sines, etc., are those of only half acb. 

Sine of J4 a c b = b , Figs. 1 and 2. 
radius 


Cosine of y 2 a c b = —?-" 8 rise > Fig T . = rise — radius , pi 2 


radius 


Tangent of % a c b =- ^ ab 

radius — rise 


, Fig. 1; = - 


radius 
X A a b 


rise — radius 


, Fig. 2. 


rise 


Versed sine of y 2 a c b = - , Figs. 1 and 2. 

radius 


For areas of segments, adbe, see p. 146. 

For areas of sectors, adbe, see p. 146. 

For centers of gravity of arcs, segments and sectors, see pp. 351 a, c and d. 


To describe tire arc of a circle too large for the dividers. 

1st Method. Lot a c ho the chord, and o b the height, of the required arc, as 



laid down on the drawing. On a separate strip of paper, semn, draw a c. o b. and a b 
also b e, parallel to the chord a c. It is well to make 5s. and b e, each a little longer 
than a b. Then cut off the paper carefully aloug the lines s b and b e, so as to leave 
remaining only the strip s a b e m n. Now, if the straight sides s b and b e be applied 
to the drawing, so that any parts of them shall touch at the same time the points a 
and b, or b and c, the point 5 on the strip will be in the circumference of the arc, 
and may be pricked off. Thus, any number of points in the arc may be found, and 
afterward united to form the curve. 

&d Metliod. Draw the span a b ; the rise r c; and a c, b c. From c with radius 



c r describe a circle. Make each of the arcs o t and i l equal to ro or r i; and draw 
c t. cl. Divide ct, cl, cr , each into half as many equal parts as the curve is to be divided 
into. Draw the lines 51, 5 2, 5 3; and a 4, a 5, a 6, extended to meet the first ones at 
e, 8. h. Then e, s, h, are points in one half the curve. Then for the other half, draw 
similar lines from a to 7, 8, 9; and others from 5 to meet them, as before. Trace 
the curve by hand. 




















142 


CIRCULAR ARCS. 


Remark. -It may frequently be of use to remember, that iu any arc bos, not 


c 




exceeding 20° or in other words, whose chord bs is at hast sixteen times its rise, the 
middle ordinate a o, will be one-hall of a c, quite near enough lor many pur¬ 
poses; b c and s c being tangents to the arc.f And vice versa if in such an arc we 
make o c equal a o. then will c be, very nearly, the poiutat which tangen’s from the 
ends of the arc will meet. Also the middle ordinate n, of the half arc o b, or 
o will be approximately % of a o, the middle ordinate oi the whole aic. Indeed, 
this last observation will apply near enough for many approximate uses even if tiie 
arc be as great as 45°; for if in that case we take 34 of oa for the ordinate n . n will 
then be but 1 part in lu3 too small; and therefore the principle may often be used 
in drawings, for finding points iu a curve of too great radius to be drawn by the 
dividers; for in the same manner, % of n will be the middle ordinate for the arc n b 
or n o; and so on to any extent. Below will be found a table by which the 
rise or middle ordinate of a half arc can be obtained with greater 
accuracy when required for more exact drawings. 


CIRCULAR ARCS IN FREQUENT USE, 


The fifth column is of use for findiug points for drawing arcs too largo for the 
beaui-compass, on the principle given above. In even the largest office drawings it 
will not be necessary to use more than the first three decimals uf the fifth column; 
and af er the arc is subdivided into parts smaller than about 35° each, the first two 
decimals .25 will generally suffice. Original. 


Rise 

in 

parts 

of 

chord. 

Degrees 
in whole 
arc. 

For rad 
mult rise 
by 

For 

length of 
arc mult 
chord 
by 

For rise 
of half 
arc 

mult rise 
by 

Rise 

in 

parts 

of 

chord. 

Degrees 
in whole 
arc. 

For rad 
mult rise 
by 

For 

length of 
arc mult 
chord 
by 

For 
rise of 
halfare 
mult 
riseby 

1-50 

o 

9 

r 

9.75 

313. 

1.00107 

.2501 

y 

o 

56 

t 

8.70 

8.5 

1.04116 

.2538 

1-45 

10 

10.75 

253.625 

1.00132 

.2501 

1-7 

63 

46.90 

6.625 

1.05356 

.2549 

1-40 

11 

26.98 

200.5 

1.00167 

.2502 

.155 

68 

53.63 

5.70291 

1.06288 

.2557 

1-35 

13 

4.92 

153.625 

1.00219 

.2502 

1-6 

73 

44 39 

5. 

1.07250 

.2566 

1-30 

15 

15.38 

113. 

1.00296 

.2503 

.18 

79 

11.73 

4.35S03 

1.08128 

.2576 

1-25 

18 

17.74 

78.625 

1.00426 

.2504 

1-5 

87 

12.34 

3.625 

1.10317 

.2593 

1-20 

22 

50.54 

50.5 

1.00665 

.2506 

.207107 

90 


3.41422 

1.11072 

.2599 

1-19 

24 

2.16 

45.625 

1.00737 

. .2507 

.225 

96 

54 67 

2.96913 

1.12997 

.2615 

1-18 

25 

21.65 

41. 

1.00821 

.2508 

% 

106 

15 61 

2.5 

1.15912 

.2639 

1-17 

26 

50.36 

36.625 

1.00920 

.2509 

.275 

115 

14.59 

2.15289 

1.19082 

.2665 

1-1 r> 

28 

30.00 

32.5 

1.01038 

.2510 

.3 

123 

51 30 

1.88889 

1.22495 

.2692 

1-15 

30 

22.71 

28.625 

1.01181 

.2511 


134 

45.62 

1.625 

1.27401 

.2729 

1-14 32 

31.22 

25. 

1.01355 

.2513 

.365 

144 

30.98 

1.43827 

1.32413 

.2766 

1-13 

34 

59.08 

21.625 

1.01571 

.2515 

.4 

154 

38 35 

1.28125 

1.38322 

.2808 

1-12 

37 

50.96 

18.5 

1.01842 

.2517 

.425 

161 

27.52 

1.19204 

1.42764 

.2838 

1-11 

41 

13.16 

15.625 

1.02189 

.2520 

.45 

167 

56.93 

1.11728 

1.47377 

.2868 

1-10 

45 

14.38 

13. 

1.02646 

.2525 

.475 

174 

7.49 

1.05402 

1 52152 

.2899 

1-9 

50 

6.91 

10.625 

1.03260 

.2530 

.5 

180 


1 . 

1.57080 

.2929 


f At 29° o c thus found will be but about 3 parts too short in 100. 

































MENSURATION, 


143 


Lengths of circular arcs. If arc exceeds a semicircle. 

Knowing its chord and height, divide the height by the chord. Find in the column of 
number equal to this quotient. Take out the corresponding number from the column 
Multiply this last number by the length of the given chord. 




T ABLE OF € IRC III 

m A R 

ARCS 


H'ghts. 

Lengths. 

H’ghts. 

Lengths. 

H ghts 

Lengths. 

H’ghts 

Lengths. 

H'ghts. 

.001 

1.00002 

.076 

1.01533 

.151 

1.05973 

•226 

1.13108 

.301 

.002 

1.00002 

.077 

1.01573 

.152 

1.06051 

•227 

1.13219 

. 307 . 

.003 

1.00003 

.078 

1.01614 

.153 

1.06130 

•228 

1.13331 

.303 

.004 

1.00004 

.079 

1.01656 

.154 

1.06209 

•229 

1.13444 

.301 

.005 

1.00007 

.080 

1.01698 

.155 

1.06288 

• 230 

1.13557 

.305 

.006 

1.00010 

.081 

1.01741 

.156 

1.06368 

•231 

1.13671 

.306 

.007 

1.00013 

.082 

1.01784 

.157 

1.06449 

•232 

1.13785 

.307 

.008 

1.00017 

.083 

1.01828 

.158 

1.06530 

• 233 

1.13900 

.303 

.009 

1.00022 

.084 

1.01872 

.159 

1.06611 

• 234 

1.14015 

.309 

.010 

1.00027 

.085 

1.01916 

.160 

1.06693 

•235 

1.14131 

.310 

.011 

1.00032 

.086 

1.01961 

.161 

1.06775 

•236 

1.14247 

.311 

.012 

1.00038 

.087 

1.02006 

.162 

1.06858 

•237 

1.14363 

.317 

.013 

1.00045 

.088 

1.02052 

.163 

1.06941 

•238 

1.14480 

.313 

.014 

1.00053 

.089 

1.02098 

.164 

1.07025 

•239 

1.14597 

.31 4 

.015 

1.00061 

.090 

1.02146 

.165 

1.07109 

•240 

1.14714 

.315 

.016 

1.00069 

.091 

1.02192 

.166 

1.07194 

•241 

1.14832 

•316 

017 

1.00078 

.092 

1.02240 

.167 

1.07279 

.242 

1.14951 

.317 

.018 

1.00087 

.093 

1.02289 

.168 

1.07365 

• 243 

1.15070 

.3'8 

.019 

1.00097 

.094 

1.02339 

.169 

1.07451 

•244 

1.15189 

• 319 

.020 

1.00107 

.095 

1.02389 

.170 

1.07537 

.245 

1.15308 

.320 

.021 

1.00117 

.096 

1.02440 

.171 

1.07624 

.246 

1.15428 

.321 

.022 

1.00128 

097 

1.02491 

.172 

1.07711 

.247 

1.15549 

.322 

.023 

1.00140 

.098 

1.02542 

.173 

1.07799 

.248 

1.15670 

.323 

.024 

1.00153 

.099 

1.02593 

.174 

1.07888 

.249 

1.15791 

.324 

.025 

1 00167 

.100 

1.02646 

.175 

1.07977 

.250 

1.15912 

.325 

.026 

1.C0182 

.101 

1.02698 

.176 

1.08066 

.251 

1.16034 

.326 

.027 

1.00196 

.102 

1.02752 

.177 

1.08156 

.252 

1.16156 

.327 

.028 

1.00210 

.103 

1.02806 

.178 

1.08246 

.253 

1.16279 

.328 

.029 

L00225 

.104 

1.02860 

.179 

1.08337 

.254 

1.16402 

.329 

.030 

1.00240 

.105 

1.02914 

.180 

1.08428 

.255 

1.16526 

.330 

.031 

1.00256 

.106 

1.02970 

.181 

1.08519 

.256 

l. 1 (>650 

.331 

.032 

1.00272 

.107 

1.03026 

.182 

1.08611 

.257 

1.16774 

.332 

.033 

1.00289 

.108 

1.03082 

.183 

1.08704 

.258 

1.16899 

.333 

.034 

1.00307 

.109 

1.03139 

.184 

1.08797 

.259 

1.17024 

.334 

.035 

1.00327 

.110 

1.03196 

.185 

1.08890 

.260 

1.17150 

.335 

.036 

1.00345 

.111 

1.03254 

.186 

1.08984 

.261 

1.17276 

.336 

.037 

1.00364 

.112 

1.03312 

.187 

1.09079 

.262 

1.17403 

.337 

.038 

I.00384 

.113 

1.03371 

.188 

1.09174 

.263 

1.17530 

.338 

.039 

1.00405 

.114 

1.03430 

.189 

1.09269 

.264 

1.17657 

.339 

.040 

1.00426 

.115 

1.03490 

.190 

1.09365 

.265 

1.17784 

.340 

.041 

1.00447 

.116 

1.03551 

.191 

1.09461 

.266 

1.17912 

.341 

.042 

1.00469 

.117 

1.03611 

.192 

1.09557 

.267 

1.18040 

.342 

.043 

1.00492 

.118 

1.03672 

.193 

1.09654 

.268 

1.18169 

.343 

.044 

1.00515 

.119 

1.03734 

.194 

1.09752 

.269 

1.18299 

.344 

.045 

1.00539 

.120 

1.03797 

.195 

1.09850 

.270 

1.18429 

.345 

.016 

1.00563 

.121 

1.03860 

.196 

1.09949 

.271 

1.18559 

.346 

.047 

1.00587 

.122 

1.03923 

.197 

1.10048 

.272 

1.18689 

.347 

.048 

1.00612 

.123 

1.03987 

.198 

1.10147 

.273 

1.18820 

.318 

.049 

1 00638 

.124 

1.04051 

.199 

1.10247 

.274 

1.18951 

.349 

.050 

1.00665 

.125 

1 04116 

.200 

1.10347 

_ .275 

1.19082 

.350 

.051 

1.00692 

.126 

1.04181 

.201 

1.10447 

.276 

X.19214 

.351 

.052 

1 00720 

.127 

1.04247 

.202 

1.10548 

.277 

1.19346 

.352 

053 

1 00748 

.128 

1.04313 

.203 

110650 

.278 

1.19479 

.353 

.054 

1.00776 

.129 

1.04380 

.204 

1.10752 

.279 

1.19612 

.354 

.055 

1.00805 

.130 

1 04447 

.205 

1.10855 

.280 

1.19746 

.355 

.056 

1.00834 

.131 

1 04515 

.206 

1.10958 

.281 

1.19880 

.356 

.057 

1 00864 

.132 

1.04584 

.207 

1.11062 

.282 

1.20014 

.357 

.058 

1.00895 

.133 

1.04652 

.208 

1.11165 

.283 

1.20149 

.358 

.059 

1.00926 

134 

1.04722 

.209 

1.11269 

.284 

1.20284 

.359 

.060 

1.00957 

.135 

1.04792 

.210 

1.11374 

.285 

1.20419 

.360 

.061 

1.00989 

.136 

1.04862 

.211 

1.11479 

.286 

1.20555 

.361 

.062 

1.01021 

.137 

1.04932 

.212 

1.11584 

.287 

1.20691 

.362 

.063 

1.01054 

.138 

1.05003 

.213 

1.11690 

.288 

1.20827 

.363 

.06 4 

1.01088 

.139 

1.05075 

.214 

1.11796 

.289 

1.20964 

.364 

.065 

1.01123 

.140 

1.05147 

.‘215 

1.11904 

.290 

1.21102 

.365 

.066 

1.01158 

.141 

1.05220 

.216 

1.12011 

.291 

1.21239 

.366 

.067 

1.01193 

.142 

1.05293 

.217 

1.12118 

.292 

1.21377 

.367 

.068 

1.01228 

.143 

1.05367 

.218 

1.12225 

.293 

1.21515 

.368 

.009 

1 01264 

.144 

1.05441 

.219 

1.12334 

.294 

1.21654 

.369 

.070 

1.01302 

.145 

1 05516 

.220 

1.12444 

.295 

1.21794 

.370 

.071 

1.01338 

.146 

1.05591 

.221 

1.12554 

.296 

1.21933 

.371 

.072 

1.01376 

.147 

1.05667 

.222 

1.12664 

.297 

1.22073 

.372 

.073 

1.01414 

.148 

1.05743 

.223 

1.12774 

.298 

1.22213 

373 

.074 

1.01453 

.149 

1.05819 

.224 

1.12885 

.299 

1.22354 

.374 

.075 

1.01493 

.150 

1.05896 

•225 i 

1 12997 

.300 

1.22495 

.375 


10 


see p 144. 

heights the 
of lengths. 


No error*. 


Lengths. 


1.22636 
1.2*2778 
1.2*29*20 
1.280(13 
1.23206 
1.23349 
1.23492 
1.23636 
1.23*81 
1.23926 
1.24070 
1.24216 
1.24361 
1.24507 

I. 24654 

J. 24801 
1.24948 
1.25095 
1.25243 
1.25391 
1.25510 
1.25689 
1.25838 
1.25988 
1.26138 
1.26288 
1.26437 
1.26588 
1.26740 
1.26892 
1.27044 
1.27196 
1.27349 
1.27502 
1.27656 
1.27810 
1.27964 
1.28118 
1.28273 
1.28428 
1.28583 
1.28739 
1.28895 
1.294)52 
1.29209 
1.29366 
1.29523 
1.29681 
1.29839 
1.29997 
1.30156 
1.30315 
1.30474 
1.30634 
1.30794 
1.30954 
1.31115 
1.31276 
1.31437 
1.31599 
1.31761 
1.31923 

l .32086 
1 32249 
1.32413 
1.32577 
1.32741 
1.32905 
1.33009 
1.33234 
1.33399 
1.33564 
1.33730 
1.33896 
1.34063 










































144 


MENSURATION. 


TABLE OF CIllC'ULAR ARCS —(Coxtixpitd.) 




H’ghts. 

Lengths. 

H'ghts. 

Lengths. 

H’ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

.376 

1.34229 

.401 

1.38496 

.4 26 

1.42945 

.451 

1.47565 

.476 

1.52346 

Jin 

1.31393 

.402 

1.33671 

.427 

1.43127 

.452 

1.47753 

.477 

1.52541 

.378 

1.34563 

.403 

1.38346 

.428 

1.43309 

.453 

1.47C42 

.478 

1.52736 

.379 

1.3473! 

.404 

1.39021 

.429 

1.43491 

.454 

1.48131 

.479 

1.52(31 

.380 

1.34899 

.405 

1.391(6 

.430 

1.43673 

.455 

1.483.0 

.480 

1.53126 

.341 

1.35068 

.406 

1.39372 

.431 

1.43856 

.456 

1.48509 

.481 


•3S2 

1.35237 

.407 

1.39548 

.432 

1.44039 

.457 

1.48699 

.482 

1.53518 

.383 

1.35406 

.408 

1.39724 

.433 

1.44222 

.458 

1.48819 

.483 

1.53714 

.384 

1.35 >75 

.409 

1.39900 

.434 

1.44405 

.459 

1.49079 

.484 

1.53910 

.38 > 

1.35744 

.410 

1.40077 

.435 

1.44589 

.460 

1.49269 

.486 

1.54106 

.386 

1.35914 

.411 

1.40254 

.436 

1.44773 

.461 

1.49460 

.486 

1.54202 

.357 

1.36084 

.412 

1.40432 

.437 

1.44957 

.462 

1.4S651 

.487 

1.54499 

.388 

1.36254 

.413 

1.40610 

.438 

1.45142 

.463 

1.49842 

.488 

1.54696 

.359 

1.36425 

.414 

1.40738 

.439 

1.45327 

.464 

1.50033 

.48*9 

1.54893 

.390 

1.36596 

.415 

1.40966 

.410 

1.45.512 

.465 

1.50224 

.490 

1.55091 

.391 

1 .<16767 

.416 

1.41145 

.441 

1.45697 

.466 

1.50416 

.491 

1.55289 

.392 

1.36939 

.417 

1.41324 

.442 

1.45813 

.467 

1.50C08 

.492 

1.55487 

.393 

1.37111 

.418 

1.41503 

.443 

1.46069 

.468 

1.50800 

.493 

1.55685 

.394 

1.37283 

.419 

1.41682 

.444 

1.46255 

.469 

1.50992 

.494 

1.55884 

.395 

1.37455 

.420 

1.41861 

.445 

1.46441 

.470 

1.51185 

.495 

1.56083 

.396 

1.37628 

.421 

1.42041 

.446 

1.46628 

.471 

1.51378 

.496 

1.56282 

.397 

1.37801 

.422 

1.42221 

.447 

1.46315 

.472 

1.51571 

.497 

1.56481 

.398 

1.379.4 

.423 

1.42402 

.448 

1.47002 

.473 

1.51764 

.498 

1.56681 

.399 

1.38148 

.421 

1.425S3 

.419 

1.47189 

.474 

1.51958 

.499 

1.56881 

.400 

1.33322 

.425 

1.42764 

.450 

1.47377 

.475 

1.52152 

.500 

1.57080 


~ «* sfiimime, men, as mreciea at top ol 

P - 1 » 0 , f , lhe circle. Then fiud its circumf. From diam take lit of are. The* rem 

Subtract it rrom ch-comf*"* ° f ^ C ' rC ‘ e ' By rU ' e at l ° P ° f P143 fiud the leD e th ° f this smaller “ns. 

The length of 1 degree of a circular arc is equal to .017453 292 520 X its radius 

* “ “ 1 minute “ “ •* •• «• .000290 888 209 X “ “ 

“ “ 1 second “ “ “ “ .000004 818 137 X “ “ 

A ? ?. r< ; of J° ° f , the ear<l, * s Srre»l circle is but 4.6356 feet, lonror than its 

chord. Its length 1869.16 land or statute miles. Earth’s equatorial rad ~3962.5705 miles. Polar 3949.67. 

An are or 1 f rad 1 mile, is 92.1534 feet; a minute is 1.5359 feet: a second is 0256 of a fo#t- 
or Ttry nearly 5-sixteenths of an inch. Arc of 1“, rad 100 ft = 1 74533 fcST ' 



{ 








































MENSURATION. 


145 


To fiml the length of a circular arc by the following table. 


Knowing the rad of the circle, and the measure of the are in deg, min, Ac. 

K he .i enSt ^ s J n the table found respectively opposite to the deg, min, &c, of 
the arc. Mult the sum by the rad of the circle. - . ’ 

Ex. In a circle of 12.43 feet rad, is an arc of 13 deg, 27 min, 8 sec. How long is the arc? 

Here, opposite 13 deg in the table, we And, .2268928 
“ 27 min “ “ “ .0078540 

“ 8 sec “ “ “ .0000388 

, . . Rum =: .2347856 

And .23478o6 X 12.43 or rad = 2.918385 feet, the reqd length of arc. 

LENGTHS OF CIRCFEAR ARCS TO RAO I. 

No errors. 


Deg. 

Length. 

Deg. 

Length. 

Deg. 

1 

.0174533 

61 

1.0646508 

121 

2 

.0349066 

62 

1.0821041 

122 

3 

.0523599 

63 

1.0995574 

123 

4 

.0698132 

64 

1.1170107 

124 

5 

.0872665 

65 

1.1344640 

125 

6 

.1047198 

66 

1.1519173 

126 

7 

.1221730 

67 

1.1093706 

127 

8 

.1396263 

68 

1.7868239 

128 

9 

.1570796 

69 

1.2042772 

129 

10 

.1745329 

70 

1.2217305 

130 

11 

.1919862 

71 

1.2391838 

131 

12 

.2094395 

72 

1.2566371 

132 

13 

.2268928 

73 

1.2740904 

133 

14 

.2443461 

74 

1.2915436 

134 

15 

.2617994 

75 

1.3089969 

135 

16 

.2792527 

76 

1.3264502 

136 

17 

.2967060 

77 

1.3439035 

137 

18 

.3141593 

78 

1.3613568 

138 

19 

.3316126 

79 

1.3788101 

139 

20 

.3490659 

80 

1.3962634 

140 

21 

.3665191 

81 

1.4137167 

141 

22 

.3839724 

82 

1.4311700 

142 

23 

.4014257 

83 

1.4486233 

143 

24 

.4188790 

84 

1.4660766 

144 

25 

.4363323 

85 

1.4835299 

145 

26 

.4537856 

86 

1.5009832 

146 

27 

.4712389 

87 

1.5184364 

147 

28 

.4886922 

88 

1.5358897 

148 

29 

.5061455 

89 

1.5533430 

149 

30 

.5235988 

90 

1.5707963 

150 

31 

.5410521 

91 

1.5882496 

151 

32 

.5585054 

92 

1.6057029 

15*2 

33 

.5759587 

93 

1.6231562 

153 

34 

.5934119 

94 

1.6406095 

154 

35 

.6108652 

95 

1.6580628 

155 

36 

.6283185 

96 

1.6755161 

156 

37 

.6457718 

97 

1.6929694 

157 

38 

.6632251 

98 

1.7104227 

158 

39 

.6806784 

99 

1.7278760 

159 

40 

.6981317 

100 

1.7453293 

160 

41 

.7i55850 

101 

1.7627825. 

161 

42 

.7330383 

102 

1.7802358 

162 

43 

.7504916 

103 

1.7976891 

163 

44 

.7679449 

104 

1.8151424 

164 

45 

.7853982 

105 

1.8325957 

165 

46 

.8028515 

193 

1.8500490 

166 

47 

.8203047 

107 

1.8675023 

167 

48 

.8377580 

108 

1.8849556 

168 

49 

.8552113 

109 

1.9024089 

169 

50 

.8726646 

110 

1.9198622 

170 

51 

.8901179 

111 

1.9.373155 

171 

52 

.9075712 

112 

1.9547688 

172 

53 

.9250245 

113 

1.9722221 

173 

54 

.9424778 

114 

1.9896753 

174 

55 

.9599311 

115 

2.0071286 

175 

56 

.9773844 

116 

2.0245819 

176 

57 

.9948377 

117 

2.0420352 

177 

58 

1.0122910 

118 

2.0594885 

178 

59 

1.0297443 

119 

2.0769418 

179 

60 

1.0471976 

120 

2.0943951 

180 


Length. 

Min. 

Length. 

Sec. 

Length. 

2.1118484 

1 

.0002909 

1 

.0000048 

2.1293017 

2 

.0005818 

2 

.0000097 

2.1467550 

3 

.0008727 

3 

.0000145 

2.1642083 

4 

.0011636 

4 

.0000194 

2.1816616 

5 

.0014544 

5 

.0000242 

2.1991149 

6 

.0017453 

6 

.0000291 

2.2165682 

7 

.0020362 

7 

.0000339 

2.2340214 

8 

.0023271 

8 

.0000388 

2.2514747 

9 

.0026180 

9 

.0000436 

2.2689280 

10 

.0029089 

10 

.0000485 

2.2863813 

11 

.0031998 

11 

.0000533 

2.3038346 

12 

.0034907 

12 

.0000582 

2.3212879 

13 

.0037815 

13 

.0000630 

2.3387412 

14 

.0040724 

14 

.0000679 

2.3561945 

15 

.0043633 

15 

.0000727 

2.3736478 

16 

.0046542 

16 

.0000776 

2.3911011 

17 

.0049451 

17 

.0C00S24 

2.4085544 

18 

.0052360 

18 

.0000873 

2.4260077 

19 

.0055269 

19 

.0000921 

2.4434610 

20 

.0058178 

20 

.0000970 

2.4609142 

21 

.0061087 

21 

.0001018 

2.4783675 

22 

.0063995 

22 

.0001067 

2.4958208 

23 

.0066904 

23 

.0001115 

2.5132741 

24 

.0069813 

24 

.0001164 

2.5307274 

25 

.0072722 

25 

.0001212 

2.5481807 

26 

.0075631 

26 

.0001261 

2.5656340 

27 

.0078540 

27 

.0001309 

2.5830873 

28 

.0081449 

28 

.0001357 

2.6005406 

29 

.0084358 

29 

.0001406 

2.6179939 

30 

.0087266 

30 

.0001454 

2.6354472 

31 

.0090175 

31 

.0001503 

2.6529005 

32 

.0093084 

32 . 

.0001551 

2.6703538 

33 

.0095993 

33 

.0001600 

2.6878070 

34 

.0098902 

34 

.0001648 

2.7052603 

35 

.0101811 

35 

.0001697 

2.7227136 

36 

.0104720 

36 

.0001745 

2.7401669 

37 

.0107629 

.37 

.0001794 

2.7576202 

38 

.0110538 

38 

.0001842 

2.7750735 

39 

.0113446 

39 

.0001891 

2.7925268 

40 

.0116355 

40 

, .0001939 

2.8099801 

41 

.0118264 

41 

.0001988 

2.8274334 

42 

.0122173 

42 

: .0002036 

2.8448867 

43 

.0125082 

43 

.0002085 

2.8623400 

44 

.0127991 

44 

.0002133 

2.8797933 

45 

.0130900 

45 

.0002182 

2.8972466 

46 

.0133809 

46 

.0002230 

2.9146999 

47 

.0136717 

47 

.0002279 

2.9321531 

48 

.0139626 

48 

.0002327 

2.9496061 

49 

.0142535 

49 

.0002376 

2.9670597 

50 

.0145444 

50 

.0002424 

2.9845130 

51 

.0148353 

51 

.0002473 

3.0019663 

52 

.0151262 

52 

.0002521 

3.0194196 

53 

.0154171 

53 

.0002570 

3.0368729 

54 

.0157080 

54 

.0002618 

3.0543262 

55 

.0159989 

55 

.0002666 

3.0717795 

56 

.0162897 

56 

.0002715 

3.0892328 

57 

.0165806 

57 

.0002763 

3.1066861 

58 

.0168715 

58 

.0002812 

3.1241394 

59 

.0171624 

59 

.0002860 

3.1415927 

60 

.0174533 

60 1 

.0002909 



































146 


MENSURATION. 


CIRCULAR SECTORS, RINGS, SEGMENTS, ETC. 

Area of a circular sector, a d b c. Fig. A, 

arc adb 


a 



h 


— - — X radius c a. (For length of arc seo p. 141.) 

M 


— area of entire circle X 


arc a d b in degrees. 


360 


(For minutes and seconds in decimals of a degree, see p. 57.) 



Area of a circular ring, Fig. B, 

= area of larger circle, c d, — area of smaller one, a b. 

^ = .7854 X (sum of diams. cd -f a b) X (diff. of diams. cd — a b.) 
= 1.5708 X thickness c a X sum of diameters c d and a b. 


To find tlie radius of a circle which shall have the same area 
as a given circular ring c a d a b, Fig. B, 


Draw any radius nr of the outer circle ; and from where said radius cuts the 
inner circle at t , draw t s at right angles to it. Then will t 8 be the required radius. 


Breadth, c a b d, of a circular ring, Fig. B, 

= \Z 2 difference of diameters c d and a b. 

= % (diameter c d — n/ 1.2732 area of circle a 6.) 

Area of a circular zone abed, 

— area of circle m n —areas of segments a mb and end, 

(for areas of segments, seo below.) 



A circular lime is a crescent-shaped 
figure, comprised between two arcs ab c 
and a o c of circles of different radii, a d 

Y'--. 

'■ 1 



and a u. 

Area of a circular lnne ab c o 


area of segment abc — area of segment aoc, 
(for areas of segments see below.) 




„ tl ?" area of a circular segment, abed, Figs. C, D, see table 

pp 147, 148. Also, 

Area of Segment a d b n, Fig. A (at top of page) 

== Area of Sector adb c — Area of Triangle abc. 

= ]/ 2 (Arc adb X radius ac — cn X chord a b). For arc adb, see p. 141. 

Having the area of a segment required to he cut off from a 
given circle, to find its chord and rise. 

r Divide the area by the square of the diameter of the circle; look for the quotient 
in the column of areas in the table of areas, pp. 147, 148; take out from the table 
the corresponding number in the column of rises. Multiply this number bv the 
diameter. The product will bo the required rise. Then J 

chord = 2 X “v (diameter 


rise) X rise. 




















MENSURATION, 


147 


TABLE OF AREAS OF CIRCULAR SEGMENTS, Figs C, D. 

^ PXCPPlIs «l S©Hlicil*d©, its area is = area of circle— area 

, n segment whose rise is — (diaru of circle — rise of given segment). l>iam of circle — (square 
ot half chotd — rise) -f- rise, w'hether the segment exceeds a semicircle or not. 


Rise 
div by 
diam of 
circle. 

Area = 

(square 
of diam) 
mult by 

Rise 
div by 
diam of 
circle. 

Area = 

(square 
of diam) 
mult by 

Rise 
div by 
diam o] 
circle. 

1 Area = 

(square 
of diam) 
mult by 

Rise 
div by 
diam ol 
circle. 

Art; a 

(square 
of diam) 
mult by 

Rise 
div by 
diam ol 
circle. 

Area = 

(square 
of diam 
mult by 

.001 

.000042 

.064 

.021168 

.127 

.057991 

.190 

.103900 

.253 

.156149 

.002 

.000119 

.065 

.021660 

.128 

.058658 

.191 

.104686 

.254 

.157019 

.003 

.000219 

.066 

.022155 

.129 

.059328 

.192 

.105472 

.255 

.157891 

.004 

.000337 

.067 

.022653 

.130 

.059999 

.193 

.106261 

.256 

.158763 

.005 

.000471 

.068 

.023155 

.131 

.060673 

.194 

.107051 

.257 

.159636 

.006 

.000619 

.069 

.023660 

.132 

.061349 

.195 

.107843 

.258 

.160511 

.007 

.000779 

.070 

.024168 

.133 

.062027 

.196 

.108636 

.259 

.161386 

.008 

.000952 

.071 

.024680 

.134 

.062707 

.197 

.109431 

.260 

.162263 

.009 

.001135 

.072 

.025196 

.135 

.063389 

.198 

.110227 

.261 

.163141 

.010 

.001329 

.073 

.025714 

.136 

.064074 

.199 

.111025 

.262 

.164020 

.011 

.001533 

.074 

.026236 

.137 

.064761 

.200 

.111824 

.263 

.164900 

.012 

.001746 

.075 

.026761 

.138 

.065449 

.201 

.112625 

.264 

.165781 

.013 

.001969 

.076 

.027290 

.139 

.066140 

.202 

.113427 

.265 

.166663 

.014 

.002199 

.077 

.027821 

.140 

.066S33 

.203 

.114231 

.266 

.167546 

.015 

.002438 

.078 

.028356 

.141 

.067528 

.204 

.115036 

.267 

.168431 

.016 

.002685 

.079 

.02S894 

.142 

.068225 

.205 

.115842 

.268 

.169316 

.017 

.002940 

.080 

.029435 

.143 

.068924 

.206 

.116651 

. .269 

.170202 

.018 

.003202 

.0S1 

.029979 

.144 

.069626 

.207 

.117460 

.270 

.171090 

.019 

.003472 

.082 

.030526 

.145 

.070329 

.208 

.118271 

.271 

.171978 

.020 

.003749 

.083 

.031077 

.146 

.071634 

.209 

.119084 

.272 

.172868 

.021 

.004032 

.084 

.031630 

.147 

.071741 

.210 

.119898 

.273 

.173758 

.022 

.004322 

.085 

.032186 

.148 

.072450 

.211 

.120713 

.274 

.174650 

.023 

.004619 

.086 

.032746 

.149 

.073162 

.212 

.121530 

.275 

.175542 

.024 

.004922 

.087 

.033308 

.150 

.073875 

.213 

.122348 

.276 

.176436 

.025 

.005231 

.088 

.033873 

.151 

.074590 

.214 

.123167 

.277 

.177330 

.026 

.005546 

.089 

.034441 

.152 

.075307 

.215 

.123988 

.278 

.178226 

.027 

.005867 

.090 

.035012 

.153 

.076026 

.216 

.124811 

.279 

.179122 

.028 

.006194 

.091 

.0355S6 

.154 

.076747 

.217 

.125634 

.280 

.180020 

.029 

.006527 

.092 

.036162 

.155 

.077470 

.218 

.126459 

.281 

.180918 

.030 

.006866 

.093 

.036742 

.156 

.078194 

.219 

.127286 

.282 

.181818 

.031 

.007209 

.094 

.037324 

.157 

.078921 

.220 

.128114 

.283 

.182718 

.032 

.007559 

.095 

.037909 

.158 

.079650 

.221 

.128943 

.284 

.183619 

.033 

.007913 

.096 

.038497 

.159 

.080380 

.222 

.129773 

.285 

.184522 

.034 

.008273 

.097 

.039087 

.160 

.081112 

.223 

.130605 

.286 

.185425 

.035 

.008638 

.098 

.039681 

.161 

.081847 

.224 

.131438 

.287 

.186329 

.036 

.009008 

.099 

.040277 

.162 

•0825S2 

.225 

.132273 

.288 

.187235 

.037 

.009383 

.100 

.040875 

.163 

.083320 

.226 

.133109 

.289 

.188141 

.038 

.009764 

.101 

.041477 

.164 

.084060 

.227 

.133946 

.290 

.189048 

.039 

.010148 

.102 

.042081 

.165 

.084801 

.228 

.134784 

.291 

.189956 

.040 

.010538 

.103 

.042687 

.166 

.085545 

.229 

.135624 

.292 

.190865 

041 

.010932 

.104 

.043296 

.167 

.086290 

.230 

.136465 

.293 

.191774 

.042 

.011331 

.105 

.043908 

.168 

.087037 

.231 

.137307 

.294 

.192685 

.043 

.011734 

.106 

.044523 

.169 

.087785 

.232 

.138151 

.295 

.193597 

.044 

.012142 

.107 

.045140 

.170 

.088536 

.233 

.138996 

.296 

.194509 

.045 

.012555 

.108 

.045759 

.171 

.089288 

.234 

.139842 

.297 

.195423 

.046 

.012971 

.109 

.046381 

.172 

.090042 

.235 

.140689 

.298 

.196337 

.047 

.013393 

.110 

.047006 

.173 

.090797 

.236 

.141538 

.299 

.197252 

.048 

.013818 

.111 

.047633 

.174 

.091555 

.237 

.142388 

.300 

.198168 

.049 

.014248 

.112 

.048262 

.175 

.092314 

.238 

.143239 

.301 

.199085 

.050 

.014681 

.113 

.048894 

.176 

.093074 

.239 

.144091 

.302 

.200003 

.051 

.015119 

.114 

.049529 

.177 

.093837 

.240 

.144945 

.303 

.200922 

.052 

.015561 

.115 

.050165 

.178 

.094601 

.241 

.145800 

.304 

.201841 

.053 

.016008 

.116 

.050805 

.179 

.095367 

.242 

.146656 

.305 

.202762 

.054 

.016458 

.117 

.051446 

.180 

.096135 

.243 

.147513 

.306 

.203683 

.055 

.016912 

.118 

.052090 

.181 

.096904 

.244 

.148371 

.307 

.204605 

.056 

.017369 

.119 

.052737 

.182 

.097675 

.245 

.149231 

.308 

.205528 

.057 

.017831 

.120 

.053385 

.183 

.098447 

.246 

.150091 

.309 

.206452 

.058 

.018297 

.121 

.05401 7 

.184 

.099221 

.247 

.150953 

.310 

.207376 

.059 

.018766 

.122 

.054690 

.185 

.099997 

.248 

.151816 

.311 

.208302 

.060 

.019239 

.123 

.055346 

.186 

.100774 

.249 

.152681 

.312 

.209228 

.061 

.019716 

.124 

.056004 

.187 

.101553 

.250 

.153546 

.313 

.210155 

.062 

.020197 

.125 

.056664 

.188 

.102334 

.251 

.154413 

.314 

.211083 

.063 

.020681 

.126 

.057327 

.189 

.103116 

.252 

.155281 

.816 

.212011 







































148 


MENSURATION, 


TABLE OF AREAS OF CIRCULAR SEGMENTS— (Continued.) 


Rise 
div by 
diam o 
circle. 

Area = 

(square 
of diam) 
mult by 

Rise 
div by 
diam o 
circle. 

Area = 

(square 
of diam) 
mult by 

Rise 
div by 
diam ol 
circle. 

Area = 

(square 
of diam) 
mult bv 

Rise 
div by 
iiam o 
circle. 

Area = 

(square 
of diam) 
mult by 

Rise 
div by 
diam o 
circle. 

.464 

.465 

.466 

.467 

.468 

.469 

.470 

.471 

.472 

.473 

.474 

.475 

.476 

.477 

.478 

.479 

.480 

.481 

.482 

.483 

.484 

.485 

.486 

.487 

.488 

.489 

.490 

.491 

.492 

.493 

.494 

.495 

.496 

.497 

.498 

.499 

.500 

I Area = 

(square 
of diam 
mult by 

.316 

.317 

.318 

.319 

.320 

.321 

.322 

.323 

.324 

.325 

.326 

.327 

.328 

.329 

.330 

.331 

.332 

.333 

.334 

.335 

.336 

.337 

.338 

.339 

.340 

.341 

.342 

.343 

.344 

.345 

.346 

.347 

.348 

.349 

.350 

.351 

.352 

.212941 
.213871 
.214802 
.215734 
.216666 
.217600 
.218534 
.219469 
.220404 
.221341 
.222278 
.223216 
.224154 
.225094 
.226034 
.226974 
.227916 
.228858 
.229801 
.230745 
.231689 
.232634 
.2335S0 
.234526 
.235473 
.236421 
.237369 
.238319 
.239268 
.240219 
.241170 
.242122 
.243074 
.244027 
.244980 
.245935 
.246890 

.353 
.354 
.355 
.356 
.357 
.358 
.359 
.360 
.361 
.362 
.363 
.364 
.365 
.366 
.367 
.368 
.369 
.370 
• .371 
.372 
.373 
.374 
.375 
.376 
.377 
.378 
.379 
.380 
.381 1 
.382 
.383 
.384 
.385 
.386 
.387 
.388 
.389 

.247845 
.248801 
.249758 
.250715 
.251673 
.25263- 
.253591 
.254551 
.255511 
.256472 
.257433 
.258395 
.259358 
.260321 
.261285 
.262249 
.263214 
.264179 
.265145 
.266111 
.267078 
.26' 045 
.269014 
.269982 
.270951 
.271921 
.272891 
.273861 
.274832 
.275804 
.276776 
..277748 
.278721 
.279695 
.280669 
.281643 
.282618 

•390 

•391 

•392 

•393 

•394 

•395 

•396 

•397 

•398 

•399 

•400 

•401 

•402 

•403 

•404 

•405 

•406 

•407 

•408 

•409 

•4)0 

•411 

•412 

•413 

•414 

•415 

•416 

•417 

•418 

•419 

•420 

.421 

.422 

.423 

.424 

.425 

.426 

.283593 

.284569 

.285545 

.286521 

.287499 

.288476 

.289454 

.290432 

.291411 

.292390 

.293370 

.294350 

.295330 

.296311 

.297292 

.298274 

.299256 

.300238 

.301221 

.302204 

.303187 

•304171 

.305156 

.306140 

.307125 

.308110 

.309096 

.310082 

.311068 

.312055 

.313042 

.314029 

.315017 

.316005 

.316993 

.317981 

.318970 

.427 

.428 

.429 

.430 

.431 

.432 

.433 

.434 

.435 

.436 

.437 

.438 

.439 

.•140 

441 

.442 

.443 

.444 

.445 

.446 

.447 

.448 

.449 

.450 

.451 

.452 

.453 

.454 

.455 

456 

.457 

.458 

.459 

.460 

.461 

.462 

.463 

.319959 

.320949 

.321938 

.322928 

.323919 

.324909 

.325900 

.326891 

.327883 

.328874 

.32986f 

.330858 

.331851 

.332843 

.333836 

.334829 

.335823 

.336816 

.337810 

.338804 

.339799 

.340793 

.341788 

.342783 

.343778 

•344773 

.345768 

.346764 

.347760 

.348756 

.349752 

.350749 

.351745 

.352742 

.353739 

.354736 

.355733 

.356730 

.357728 

.358725 

.359723 

.360721 

.361719 

.362717 

.363715 

.364714 

.365712 

.366711 

.367710 

.368708 

.369707 

.370706 

.371705 

.372704 

.373704 

.374703 

.375702 

.376702 

.377701 

.378701 

.379701 

.380700 

.381700 

.382700 

.383700 

384699 

.385699 

.386699 

.387699 

.3S8699 

.389699 

.390699 

.391699 

.392699 











































MENSURATION, 


149 


THE ELLIPSE. 




Fig. 1. 




Fig. 2. 



a 

Fig. 3. 


An ellipse i< a curve, e eee, Pig 1, formed by an oblique section of either a cone or a cylinder, pass¬ 
ing through its curved surface, without cutting the base, its nature is such that if two lines, as 
| n/ and n g, Fig. 2, be drawn from any point n in its periphery or circumf, to two certain points/ 
and g, in its long diarn c w, (and called the foci of the ellipse.) their sum will be equal to that of any 
other two lines, as b /, and b g drawn from any other point, as b, in the circumf, to the foci /and g; 
also the sum of any two such lines will be equal to the long diamc tv. The line c w dividing the ellipse 
into two equal parts lengthwise, is called its transverse, or major axis, or long diam ; and a b , which 
divides it equally at righ^jangles to c w, is called the conjugate, or minor axis, or short diam. To 
find the positiou of the foci of an ellipse, from either end, as 6, of the short diam, measure off the 
j dists b f and b g, Fig 2, each equal to o c, or one half the long diam. 

The parameter of an ellipse is a certain length obtained thus; as the long diam: short diam : : 

> short diam : parameter. Any line r v, or s of, Fig 3, drawn from the circumf, to, and at right angles 
to, either diam, is called an ordinate; and the parts c v and v w, b s and a a, of that diam, between 
the ord and the circumf, are called abscisses, or abscisses. 

To find the length of any ordinate, r v or s d, drawn to either 

diam, C tv or b (l. Knowing the absciss, ctiorss, and the two diarns, c w, b a; 


c w 2 : b o 2 :: c v X i> w • r v 3 . 


b efi : c tv 2 ::b s X # a • s d 2 . 


To find the circumf of an ellipse. 

. Mathematicians have furnished practical men with no simple working rule for this purpose. The 
go-called approximate rules do not deserve the name. They are as follows, D being the long diam: 
and d the short one. 


Rule 1. Circumf = 3.1416 .ILiiL • Rule 3. 3.1416 

2 > 


Rule 3. 2.2215DZ-H2; 


this is the same as Rule 2, but in a diff shape. Rulk4.2X|/ D 2 -|-1.4674 cP. Now, in an ellipse 


whose long and short diams are 10 and 2, the circumf is actually 21, very approximately; but rule 1 
; gives it = 18.85; rule 2, or 3, = 22.65; and rule 4, - 20.51. Again, if the diams be 10 and 6. the cir- 
j cumf actually = 25.59; but rule 4 gives 24.72. These examples show that none of the rules usually 
given are reliable. The following one bv the writer, is sufficiently exact for ordinary purposes; not 
being in error probably more than 1 part in 1000. When D is not more than 5 times as long as d , 


Circumf = 3.1416 ^/D 2 (D — of) 2 
2 8.8 


If D exceeds 5 times tl, then in¬ 
stead of dividing (D — d ) 2 by 8.8, div it by 
the number in this table. 

1 '. The following rule originated with Mr. M. 
Arnold Pears, of New South Wales, Australia, 
and was by him kindly communicated to the 
rate than our own, it is much neater. 


Jb O CO»fi__h-CSCIO-t*©r~cC»rt 

' «fit-OOOJ©!N^«000©iftOO©0©©0 

HHHH«->fi9C4W , t»OtCh'QOO 

author. Although not more accu- 


Circumf = 3.1416 d + 2(D — d) — 


_rf(D — d) _ 

\/(D -f- d) X (D + 2d) 


The following 1 table of semi*elliptie arcs was prepared by our rule. 



To use this table, div the height or rise of the arc, by its span or chord. The quot 
will be the height of an arc whose span is 1. Find this quot in the column of 
heights ; and takeout the corresponding number from the col. of lengths. Mult this 
number by the actual span. The prod will be the reqd length. 

When the height becomes .500 of the chord (as at the end of the table) the ellipse 
becomes a circle. When the height exceeds .500 of the chord, as in a 6 c, then take 
a o, or half the chord, as the rise; and div this rise by the long diam b d, for tha 
quot to be looked tor in the col of heights; and to be mult by long diam. We thu» 
get the arc bad, which is evidently equal to a be. 































150 


MENSURATION, 




TABIjE OF LENGTHS OF SEMI-ELLIPTIO ARCS. (Original. 


Height 

Length = 

Height 

Length = 

Height 

Length = 

1 Height 

Length = 

-r span. 

spanxby 

+spun. 

span x by 

■i span. 

span X by 

1 -5- span. 

span X by 

.005 

1.000 

.130 

1.079 

.255 

1.219 

.380 

1.390 

.01 

1.001 

.135 

1.084 

.260 

1.226 

.385 

1.397 

.015 

1.002 

.140 

1.089 

.265 

1.233 

.390 

1.404 

.02 

1.00.3 

.145 

1.094 

.270 

1.239 

.395 

1.412 

.025 

1.004 

.150 

1.099 

.275 

1.245 

.400 

1 419 

.03 

1.006 

.155 

1.104 

.280 

1.252 

.405 

1.426 

4)35 

1.008 

.160 

1.109 

.285 

1.259 

.410 

1.434 

.04 

1.011 

.165 

1.115 

.290 

1.265 

.415 

1.441 

.045 

1.014 

.170 

1.120 

.295 

1.272 

.420 

1.449 

.05 

1.017 

.175 

1.125 

.300 

1.279 

.425 

1.456 

.055 

1.020 

.ISO 

1.131 

.305 

1.285 

.430 

1.464 

.06 

1.023 

.185 

1.137 

.310 

1.292 

.435 

1.471 

4)65 

1.026 

.190 

1.142 

.315 

1.298 

.440 

1.479 

.07 

1.029 

.195 

1147 

.320 

1.305 

.445 

1.486 

.075 

1.032 

.200 

1.153 

.325 

1.312 

.450 

1.494 

.08 

1.036 

.205 

1.159 

.330 

1.319 * 

.455 

1.501 

.0.85 

1.039 

.210 

1.165 

.335 

1.325 

.460 

1.509 

.09 

1.043 

.215 

1.171 

.340 

1.332 

.465 

1.517 

.095 

1.046 

.220 

1.177 

•345 

1.339 

.470 

1.524 

.100 

1.051 

.225 

1.183 

.350 

1.346 

.475 

1.532 

.105 

1.055 

.230 

1.189 

.355 

1.353 

,480 

1.540 

.110 

1.059 

.235 

1.196 

.300 

1.361 

.4S5 

1.547 

1.555 

.115 

1.064 

.240 

1.202 

.365 

1.368 

.490 

.120 

1.069 

.245 

1.207 

.370 

1 375 

.495 

1.563 

.125 

1.074 

.250 

1.213 

.375 

1.382 

.500 

1.571 


__ Area of ail ellipse =: prod of diams x .7854. Ex. D = 10; d = 6. Then 10 X 6 X .7854 
— 47.124 area. The area of an ellipse is a mean proportional between the areas of two circles, de¬ 
scribed on its two diams ; therefore it may be found by mult together the areas of those two circles ; 
and taking the sq rt of the prod. The area of an ellipse is therefore always greater than that of the 
circular section of the cylinder from which it may be supposed to be derived. 

Diam of circofsameareaasagriven ellipse = x .hurt diam ' . 

To find the area of an elliptic segment whose base is paral¬ 
lel to eitlier <liain. Div the height of the segment, bv that diam of which said height 
is a part. From the table of circular segments take out the tabular area opposite the quot. Mult 
together this area, the long diam, and the short diam. 


To <lraw an ellipse. Having its long and short diams a b and c d, Fig. 4. 



Hulk 1. From either end of the short 
diam, as c, lay off the dists c /, c/' , each equal 
to e a, or to one-half of the long diam. The 
points /, /’ are the foci of the ellipse. Pre¬ 
pare a string, /’ »/, or /’ g /, with a loop at 
each end; the total length of string from end 
to end of loop, being equal to the long diam. 

Place pins at/and /'; and placing the loops 
ever them, trace the curve by a pencil, which 
in every position, as atn, or g, keeps the string 
/’ n /, or/’ g f, equally stretched all the time. 

Note. Owing to the difficulty of keeping 
the string equally stretched, this method is 
not as satisfactory as the following. 

Rule 2. On the edge of a strip of paper 
tv s , mark w l equal to half the short diam ; 
and ws equal half the long diam. Then in 
whatever position this strip be placed, keep¬ 
ing l on the long diam, and s on the short 
diam. w will mark a point in the circumf of 
the ellipse. We may thus obtain as many such points as we please; and then draw the curve through 
them bv hand. 6 


Fig. 4. 


them by hand. 

Rule 3. From the two foci / and /’, Fig 4, with a rad equal to any part whatever of the long diam 
describe 4 short arcs, o o o o; also with a rad equal to the remaining part of the long diam describe 
4 other arcs, i i i i. The intersections of these four pairs of arcs, will give four points in the circumf 
In this manner any number of such points may be found, and the curve he drawn by hand. 


To draw a tangent 11, at 

»»/', to the foci; bisect the angle /»/' by 


point W of ail ellipse. Drawn/and 

the line xp ; draw t n t at right angles to x p. 


lo draw a joint wp,of an elliptic urcli, from any point n f in 

* s**’®* 1 - Proceed as in the foregoing rule for a tangent, only omitting ( t ; np will be the 

reqa joint. 



































MENSURATION, 


151 



To draw an oval, or false ellipse. 

When only the long diam a b is given, the following 
will give agreeable curves, of which the span a b will 
not exceed about three times the rise c o. On a b de¬ 
scribe two intersecting circles of any rad; through 
their intersections s, v, draw eg; make s g and v e 
each equal to the diam of one of the circles. Through 
the centers of the circles, draw e y, e h, g d, g t. From 
e describe h i y ; and from g describe d o t. 


• 

l 

Z ^t 

b 

\ 

/ \ 

(c 

W / \ 


3 /X J 


7 


j n 


When the span, m n, and the 
rise, s t, are both given. 

Make any t to and m r, equal to each other, 
but each less than t s. Drawrw; and through 
its center o draw the perp to y, Draw y y z. 
Make n x equal m r, and draw yxb. From x and 
t describe H c and m a; and from y describe 
ate. By making s d equal to s y, we obtain 
the center for the other side of the oval. 

The beauty of the curve will depend upon 
what portion offs is taken for m r and tw. 
When an oval is very flat, more than three cen¬ 
ters are required for drawing a graceful curve; 
but the finding of these centers is quite as trou¬ 
blesome as to draw the correct ellipse. 


y 


On the given line, a s, to draw a 
cyma recta, a c s. 

Find the center c, of us. From a, c, ands, with one-half 
of a s as rad, draw the four small arcs at 0. o. The inter¬ 
sections o,o, are the centers for drawing the cyma, with 
the same rad. By reversing the position of the arcs, we 
obtain the cyma reversa, or ogee, d e /. 





















152 


MENSURATION, 



The common or conic parabola, 

o ft e, Fig 1, is a curve formed by cuttiug a cone in a direction ft a, parallel to its side. 


Tl 


curved line o b c itself is called the perimeter of the parabola; the line o c is called its base; b a i 
height or axi *; b its apex or vertex; any line e s, or o a, Fig 2, drawn from the curve, to, and at rigl: 
angles to, the axis, is an ordinate ; and the part a 6, or a ft, of the axis, between the ordinate and tl 
apex 6, is an abscissa. The focus of a parabola is that point in the axis, where the abscissa b s, 
equal to one-half of the ord e s. The dist from the apex to the focus, is called the focal dist. Tl 
focus may be entirely beyond or outside of the curve itself. Its dist from the apex is fouud thm 
square any ord, as oa ; div this square by the abscissa b a of that ord; div the quot by 4. Tl 
nature of the parabola is such that its abscissas, as b s, b a. &c, are to each other as, or in proportic 
to. the squares of their respective ords e s. o a, &c; that is, as b s : ba : : es 5 : o a 2 ; or 6 s : e :: 6 <; 
o a'l . If the square of any ord be divided by its abscissa, the quot will be a constant quantity ; th: 
is, it will be equal to the square of any other ord divided by its abscissa. This quot or constant quai 
tity is also equal to a certain quantity called the parameter of the parabola. Therefore the paramett 
may be fouud by squaring e *, or o a, ( one-half of the base.) and dividiugsaid square by the beigl 
ft s. or b a, as the case may be. If the square of any ord be divided by the parameter, the quot wi 
be the abscissa of that ord. 


To find the length of a parabolic curve. 

The approximate rule given by various pocket-books, is as follows : 

Length = 2 X V( x /z base) 2 + 1% times the (Height 2 ) 


Fig. 4. 



Where the heightdoes not exceed l-10th of the base, this rule may, for practic: 
purposes, be called exact. With ht — )4 base, it gives about ^ per cent t( 
much; ht = )4 base, about 34 percent; ht= base, about 8)4 per cent; ht; 
twice the base, about 12)4 percent; ht= 10 X base, or more, about 15)4 per cen 

The following by the writer Is eorre* 

within perhaps 1 part in 800, in ail cases ; and wi 
therefore answer for many purposes. 

Let a d b, Fig 3, or w u'd. Fig 4, be the paraboh 
in which are given the base a b or n d ; and ti 
height c d or c a. Imagine the complete fig ad b 
or n a d b, to be drawn : and in either case, assure 
its long diam a ft to be the chord or base; and om 
half the short diam, or c d, to be the height, of 
circular arc. Fiud the length of this circular an 
by means of the rule and table given for that pu 
pose. Then div the chord or base a 6, or n d c 
the parabola, by its height c d or c a. Look f< 
the quot in the column of bases in the followin 
table, and take from the table the correspondin 
multiplier. Mult the length of the circular arc b 
this; the prod will be the length of arc a db, < 
n a d, as the case may be. For bases of parabol: 
less than .05 of the height, or greater than lOtim 
the height, the multiplier is 1, and is very appro 
lmate; or in other words, the parabola will 
of almost exactly the same length as the circul; 
arc. 



Fig. 5. 


To find flic area of a parabola m a n b. 

Mult its base mn, Fig 5, by its height a b; and take %ds of the pro> 
The area of any segment, as u ft v, whose base u v is parallel to m «, 
fouud in the same way, using u v and s 6, instead of m » and a ft. 

To find file area of a parabolic zone, or fru* 

tiun,asmni(v, 

Rui.k 1. First find by the preceding rule the area of the whole parabo 
mbn; then that of the segment ubv; and subtract the last from tl 
first. 

Rulk 2. From the cube of m n . take the cube ofuti; call the diff 
From the square of m n. take the square of u v; call the diff*. Div c l| 
s. Mult the quot by %ds of the height a *. 














MENSURATION. 


153 


Table for lengths of Parabolic Curves. See opp page. (Original.) 


Base. 

Mult. 

Base. 

Mult. 

Base. 

Mult. 

Base. 

-— -1 

Mult. 


.05 

1.000 

1.10 

.999 

215 

.949 

3.20 

.983 


.10 

1.001 

1.15 

.997 

2.20 

.951 

3.30 

.984 


.15 

1.002 

1.20 

.995 

2.25 

.954 

3.40 

.985 


.*20 

1.004 

1.25 

.993 

2.30 

.956 

3.50 

.986 


.25 

1.006 

1.30 

.990 

2.35 

.958 

3.60 

.987 


.30 

1.007 

1.35 

.987 

2.40 

.960 

3.70 

.988 


.35 

1.007 

1.40 

.9S4 

2.45 

.962 

3.80 

.989 


.40 

1.008 

1.45 

.980 

2.50 

.963 

3.90 

.990 

tu 

.45 

1.009 

1.50 

.977 

2.55 

.965 

4.00 

.991 

tl 

.50 

1.010 

1.55 

.974 

2.60 

.967 

4.25 

.992 

ill 

.55 

1.010 

1.60 

.970 

2.65 

.969 

4.50 

.993 


.60 

1.010 

1.65 

.966 

2.70 

.970 

4.75 

.994 

]( 

.65 

1.011 

1.70 

.963 

2.75 

.972 

5.00 

.995 

$: 

.70 

1.011 

1.75 

.960 

2.80 

.973 

5.25 

.996 

tl( 

.75 

1.010 

1.80 

.957 

2.85 

.975 

5.50 

.997 

Ii 

.80 

1.009 

1.85 

.953 

2.90 

.976 

5.75 

.998 

aii 

.85 

1.008 

1.90 

.950 

2.95 

.978 

6.00 

.998 

Q 

.90 

1.006 

1.95 

.946 

3.00 

.979 

7.00 

.999 

l 1 

.95 

1.004 • 

’ 2.00 

.942 

305 

.980 

8.00 

1 .000 

1, 

1.00 

1.002 

2.05 

.944 

3.10 

.981 

10.00 

1.000 


1.05 

1.001 

2.10 

.916 

3.15 

.982 




To draw a parabola, having base c s and height e o. 

cos, Fig 6. Make o t equal to the height eo. Draw ct and 
e t ; and divide eacii of them into any number of equal parts ; 
numbering them as in the Fig. join 1, 1; 2, 2; 3, 3, &c; 
then draw the curve by hand. It will be observed that the 
intersections of the lines 1,1 ; 2, 2, &c, do not give points in 
the curve; but a portion of each of those lines forms a tan¬ 
gent to the curve. By increasing the number of divisions 
on c t and s (, an almost perfect curve is formed, scarcely 
requiring to be touched up by hand. In practice it is best 
Qrst to draw only the center portions of the two lines which 
cross each other just above o ; and from them to work down¬ 
ward; actually drawing only that small portion of each 
successive lower line, which is necessary to indicate the 
jurve. 

Or the parabola may be drawn 
thus: 

Let b c, Fig 7, be the base ; and a d the height. Draw the 
“ectangle b nm c ; div each half of the base into any num- 
oer of equal parts, and number them from the center each 
way. Div n b, and m c into the same number of equal farts ; 
ind number them from the top, downward. From the points 
m 6 c draw vert lines ; and from those at the sides draw lines 
,o d. Then the intersections of lines 1,1; 2, 2, &c, 
will form points in the parabola. As in the pre- 
J ;eding case, it is not necessary to draw the entire 
' ines; but merely portions of them, as shown be¬ 
tween d and c. 

Or a parabola may be drawn by first div the 
Height a b, Fig 5, into any number of parts, either 
;qual or unequal; and then calculating the ordi- 
tates ms, &c; thus, as the height a b : square of 
naif base am:: any absciss b s : square of its 
>rd u *. Take the sq rt for u s. 

Rkm.—W hen the height of a parabola is not 
greater than l-10th part its base, the curve coin¬ 
cides so very closely with that of a circular arc, 
that in the preparation of drawings for suspen¬ 
sion bridges. &c., the circular arc may be em¬ 
ployed; or if no great accuracy is reqd, the circle 
agay be used even when the height is as great as 
pne-eighth of the base. i 

To draw a tangent «> v, Fig. 5, to a parabola, from any point v. 

Draw v s perp to axis a b ; prolong u b until b w equals s b. Join w v. 






































154 


MENSURATION. 


Tlie Cycloid, 

a cb, is the curve described by a point a in the circumference of a circle, 
a 7i, during one complete revolution of the circle, rolled along a straight line 

' ^ t __U ! .1. '_ a . 1 - «-1 tho /vf t Ink 

d 



ab ; which is cubed the base of the 
b cycloid. 

The vertex of the cycloid is at c. 

Base, u 6, = circumference of gene rat¬ 
ing circle an 

= diameter, cd , of generat¬ 
ing circle = 3.141 tied. 

Axis, or height, cd — an. 
Length, acb, = 4cd. 


Area, acbd = 3X area of generating circle, a n 
= 3 =cd?X £*■ = c X 2.3562. 


Center of gravity at g. eg — §cd. 

draw a tangent, eo, from any point e in a cycloid; draw e s at right 
« to the axis c d ; on c d describe the generating circle d c t ; join t c ; from 


To 

angles , _ 

e draw e o parallel to t c. The cycloid is the cn rve of quickest descent; 
so that a body would fall from b to c along the curve bmc , in less time than 
along the inclined plane b i c , or any other line 


SOLIDS. 

THE REGULAR BODIES. 


A regular body, or regular polyhedron, is one which has all its 

sides, and its solid angles, respectively similar and equal to each other. There 
are but five such bodies, as follows: 


Name. 

Bounded by 

Surface 

(— sum of surfaces 
of all the faces). 

Multiply the square 
of the length of 
one edge by 

Volume. 

Multiply the 
cube of the 
length of one 
edge by 

Tetrahedron. 

4 equilateral triangles. 

1.7320 

.1178 

Hexahedron or cube 

6 squares. 

6. 

1 . 

Octahedron. 

8 equilateral triangles. 

3.4641 

.4714 

Dodecahedron. 

12 “ pentagons. 

20.6458 

7.6631 

Icosahedron. 

20 “ triangles. 

8.6602 

2.1817 


Goldinas’ Theorem. To find 


Fig. A. 



the volume of any body fas the 

irregular inassu b cm. Fig A, or the rinj: 
ab cm. Fig B), generated by a complete 
or partial revolution of any figure (as 
abca) around one of its sides (as ac, 
Fig A), or around any other axis (as 
xy, Fig B). 

Volume = surface abca X length 
of arc described by its center of grav¬ 
ity G. 

If the revolution is complete, the arc 
described is = circumference = radius 
o G* X 2it = radius oG*X 6.283136; and 

Volume =surface abca X radius 
o G * X 6.283186. 


If the revolution is incomplete, 

complete . incomplete 
revolution * revolution 


circumference 
found as above 


arc 

described 


* Measured perpendicularly to the axis of revolution. 































MENSURATION. 


155 


PARALLELOPIPED8. 



a 


Fig. 1. 


Fig. 2. 



A parallelopiped is any solid contained within six sides, all of which are 
parallelograms; and those of each opposite pair, parallel to each other. We 
show but four of them ; corresponding to the four parallelograms; namely, the 
cube , Fig 1, which has all its sides equal squares, and all its angles right angles; 
the right rectangular jtrism , Fig 2, has all its angles right angles, each pair of 
opposite faces equal, but not all of its faces equal; the Rhoinbokedron, Fig 3, 
which has all its sides equal rhombuses, and which, like the rhombus, p 119, is 
sometimes called “ rhomb ” ; the Rhombic prism, Fig 4; its faces, rhombuses, or 
rhomboids, each pair of opposite faces equal, but not all its faces equal. All 
parallelopipeds are prisms. 

Volume of any _ area of any face, w perpendicular distance, p, 

ikoil oc /i ^ tn + lia Annncifa fqno 


parallelopiped 
Volume of a cube 


as a, '' to the opposite face. 

=• cube of length of oue edge, 

= 1.90985 X volume of inscribed sphere, 

= 1.27324 X “ “ cylinder, 

= 3.81972 X “ “ cone. 

Diagonal of a cube = diameter of circumscribing sphere, 

= 1.7320508 X length of one edge of cube. 

The diagonal of a rhomb, or of a rhombic prism, cannot be calculated by 
means of its sides and their angles. 


PRISMS. 


Nl 

I 

I 


at \P 


V 


p 


1 

6 


O 



A prism is any solid whose 
two ends are paral lei, simi lar, 
and equal; and whose .sides 
are parallelograms , as Figs 5 
to 10. Consequently the fore¬ 
going parallelopipeds are 
prisms. A right prism is one 
whose sides are perpendic¬ 
ular to its ends, as 5, 6, 7 ; 
when not so, the prism is 
oblique , as 8, 9,10. IVhen all 
the sides of the fi gures wh ich 


form the ends are equal, and the angles included between those sides are also 
equal, the prism is said to be regular: otherwise, irregular. 

Volume of any prism (whether regular or irregular, right or oblique) 
= area of one end X perpendicular distance, p, to the other end, 

= area of cross section perpendicular to the sides X actual length, a b, Figs 
5 to 10, 

= 3 X volume of pyramid whose base and height are — those of the prism. 


To fiml the volume of any frustum* 
of any prism. 

Whose cross section, perpendicular to its sides, 
is either any triangle; any parallelogram; a 
square, (as in Fig 10'^)or a regular polygon of 
any number of sides; no matter how the two 
ends of the fru-tum may be inclined with regard 
to each other; or whether one, or neither of 
them, is parallel to the base of the original 
prism. 


a 

7! 

7 

1 


\ J 

\ 

v 

1 

A 


y 

s -- 

y 


f 


\ 

A 



Volume 
of frustum 


sum of lengths of parallel edges, 
n+12+33+41 

number of such edges 
(4 in Fig 10%) 


Figs. 10 %. 

area of cross section 
X perpendicular 
to such edges. 


* Often misspelt “ frustrum.” 



























































156 


MENSURATION. 


a 



Tliis rule may be used for ascertaining beforehand, the quantity of earth tc 
be removed from a “borrow pit.” The irregular surface of the ground is first 
staked out in squares; (the tape-line being stretched horizontally, when meas¬ 
uring off their sides). These squares should be of sucl 
a size that without material error each of them may bt 
considered to be a plane surface, either horizontal or in¬ 
clined. The depth of the horizontal bottom of the pit 
being determined on, and the levels being taken at every 
corner of the squares, we are thereby furnished with the 
lengths of the four parallel vertical edges of each of the; 
resulting frustums of earth. In b igs 1U 1 4 y may be sup¬ 
posed to represent one of these frustums. 

If the frustum is that of an irregular 4-sided, or polyg¬ 
onal prism, first divide its cross section perpendicular to its sides, into tri¬ 
angles, by lines drawn from any one of its angles, as a, Fig 10)4. Calculate the 
ar> a of each of these triangles separately ; then consider the entire frustum tc 
be made up of so many triangular ones; calculate the volume 
of each of these by ihe preceding rule for triangular frustums 
and add them together, for the volume of the entire frustum. 

"Volume of any frustum of any prism. 

Or of a cylinder. Consider either end to be the base; and fi rid its 
area. Also find the center of gravity c of the other end, and the 
n v — perpendicular distance n c, from the base to said center of gravity 

Fig. 10%. Then Volume of frustum = area of base X»c, Fig 10% 

The slant end, c, is an ellipse. Its area is greater than that of the circular end 
Surface of any prism. Figs 5 to 10, whether right or oblique, regular 
or irregular 

/ circumference measured w . sum of the areas 

= (perpendicular to the sides X actual len g th > a b ) + of the two ends. 

CYLINDERS. 

A cylinder is any solid whose ends are 
parallel, similar, and equal curved figures ; 
and whose sections parallel to the ends 
are everywhere the same as the ends. 
Hence there are circular cylind- rs, ellip¬ 
tic cylinders (or cylmdroids) and many 
others; but when nototherwise expressed, 
the circular one is understood. A right 
cylinder is one wh<se ends are perpen¬ 
dicular to its sides, as Fig. 11; when other¬ 
wise, it is oblique, as Fig 12. If the ends 
of a right circular cylinder be cut so as to 





p 


Fig. 11. 



make it oblique, it becomes an elliptic one; because then both its ends, and all 
sections parallel to them, are ellipses. An oblique circular cylinder seldom 
occurs; it may be conceived of by imagining the two ends of Fig 12 to be circles, 
united by straight lines forming its curved sides 
A cylinder is a prism having an infinite number of sides. 

Volume of any cylinder (whether circular or elliptic, Ac, right or oblique) 

= area of one end X perpendicular distance, p , to the other end, 

- { measured^X —»> ^ • »• 11 — * 

— 3 X volume of a cone whose base and height are those of the cylinder. 
Surface of any cylinder (whether circular or elliptic, Ac, right or oblioue) 

/circumference \ f n , 

= ( measured perpendicularly X actual length, a bj + f th * ^ 

\to the sides, as at c o, r lg 12, / 

Right circular cylinder whose height = diameter. 

Volume = 1^ X volume of inscribed sphere. 

Curved surface = surface of inscribed sphere. 

Area of one end = \ surface <>f inscribed sphere = \ curved surface. 

Entire surface = 1£ X surface of inscribed sphere = 1£ X curved surface. 



















CONTENTS OF CYLINDERS, OR PIPES. 157 


o Contents for one foot in length, in Cub Ft, and in U. S. Gallons of 

( 231 cub ins, or 7.4805 Galls to a Cub Ft. A cub ft of water weighs about 62% lbs ; and a gallon 
about 8% lbs. Diams 3, 8, or 10 times as great, give 4, 9, or 100 times the content. 

[ for the weight of water in pipes, see Table 2 page 246. 

No errors. 


e 


Diam. 

in 

Ins. 

Diam. 
in deci¬ 
mals of 
a foot. 

For 1ft. in 
length. 

Diam. 

in 

Ins. 

Diam. 
in deci¬ 
mals of 
a foot. 

For 1 ft in 
length. 

Diam. 

in 

Ins. 

Diam. 
in deci¬ 
mals of 
a foot. 

For 1 ft. in 
length. 

Cub. Feet. 
Also area in 
sq. ft. 

Gallons of 
231 Cub.Ins. 

Cub. Feet. 

Also area in 

sq. ft. 

Gallons of 

231 Cub.Ins. 

Cub. Feet. 
Also area ii 
sq. ft. 

Gallons of 

231 Cub.Ins. 

M 

.0208 

.0003 

.0025 

% 

.5625 

.2485 

1.859 

19. 

1.5S3 

1.969 

14.73 

5-16 

.0260 

.0005 

.0040 

7. 

.5833 

.2673 

1.999 


1.625 

2.074 

15.51 

% 

.0313 

.0008 

.0057 

A 

.6042 

.2867 

2.145 

20 . 

1.667 

2.182 

16.32 

7-16 

.0365 

.0010 

.0078 

A 

.6250 

.3068 

2.295 

K 

1.708 

2.292 

17.15 

. 34 

0417 

.0014 

.0102 


.6458 

.3276 

2.450 

2 !. 

1.750 

2405 

17.99 

9 16 

.0469 

.0017 

.0129 

8 . 

.6667 

.3491 

2.611 


1.792 

2.521 

18.86 

% 

.0521 

.0021 

.0159 

A 

.6875 

.3712 

2.777 

22 . 

1.833 

2.640 

19 75 

11-16 

.0573 

.0026 

.0193 


.7083 

.3941 

2.948 


1.875 

2.761 

20.66 

% 

.0625 

.0031 

.0230 

/4 

.7292 

.4176 

3.125 

23. 

1.917 

2.885 

21.58 

13-16 

.0677 

.0036 

.0269 

9. 

.7500 

.4418 

3.305 

K 

1.958 

3.012 

22.53 

% 

.0729 

.0042 

.0312 

A 

.7708 

.4667 

3491 

24. 

2.000 

3.142 

23.50 

15-16 

.0781 

.0048 

.0359 

A 

.7917 

.4922 

3.682 

25. 

2.083 

3.409 

25.50 

1 . 

.0833 

.0055 

.0408 

74 

.8125 

.5185 

3.879 

26. 

2.167 

3.687 

27.58 

A 

.1012 

.0085 

.0638 

10 . 

.8333 

.5454 

4.081 

27. 

2.250 

3.976 

29.74 

V, 

.1250 

.0123 

.0918 


.8542 

.5730 

4.286 

28. 

2.333 

4.276 

31.99 

YA 

.1458 

.0167 

.1249 

Kl 

.8750 

.6013 

4.498 

29. 

2.417 

4.587 

3431 

2. 

.1667 

.0218 

.1632 

74 

.8958 

.6303 

4.715 

30. 

2.500 

4.909 

36.72 

A 

.1875 

.0276 

.2066 

u. 

.9167 

.6600 

4.937 

31. 

2.583 

5.241 

39.21 

\/ 

.2083 

.0341 

.2550 

A 

.9375 

.6903 

5.164 

32. 

2.667 

5585 

41.78 

/A 

.2292 

.0412 

.3085 

% 

.9583 

.7213 

5.S96 

33. 

2.750 

5.940 

44.43 

3. 

.2500 

.0491 

.3672 

% 

.9792 

.7530 

5.633 

34. 

2.833 

6.305 

47.13 

A 

.2708 

.0576 

.4309 

12 . 

1 Foot. 

.7854 

5.875 

35. 

2.917 

6.681 

49.98 

i| 

.2917 

.0668 

.4998 


1.042 

.8522 

6.375 

36. 

3 000 

7.069 

52.88 

Va 

.3125 

.0767 

.5738 

13. 

1.083 

.9218 

6.895 

37. 

3.083 

7.467 

55.86 

4. 

.3333 

.0873 

.6528 

Vo 1.125 

.9940 

7.436 

38. 

3.167 

7.876 

58.92 

A 

.3542 

.0985 

.7369 

14. 

1.167 

1.069 

7.997 

39. 

3.250 

8.296 

62.06 

A 

.3750 

.1104 

.8263 

V\ 1.208 

1.147 

8.578 

40. 

3.333 

8.727 

65.28 

k 

.3958 

.1231 

.9206 

15. 

1.250 

1.227 

9.180 

41. 

3.417 

9.168 

68.58 

5. 

.4167 

.1364 

1.020 

V> 1.292 

1.310 

9.801 

42. 

3.500 

9.621 

71.97 

A 

.4375 

.1503 

1.125 

16.' ' 

1.3.33 

1.396 

10.44 

43. 

3.583 

10.085 

75.44 


.4583 

.1650 

1.234 

V> 1.375 

1.485 

11.11 

44. 

3.667 

10.559 

78.99 


.4792 

.1803 

1.349 

17. 

1.417 

1.576 

11.79 

45. 

3.750 

11.045 

82.62 

6 

.5000 

.1963 

1.469 

Vv 1.458 

1.670 

12.49 

46. 

3.833 

11.541 

86.33 


.5208 

.2131 

1.594 

18.' 

1.500 

1.767 

13.22 

47. 

3.917 

12.048 

90.13 

^4 

.5417 

.2304 

1.724 

Kl-542 

1.867 

13.96 

48. 

4.000 

12.566 

94.00 


Table continued, but with the diams in feet. 


Diam. 

Feet. 

Cub. 

Feet. 

u. s. 

Galls. 

Diam. 

Feet. 

Cub. 

Feet. 

U. S. 
Galls. 

Dia. 

Feet. 

Cub. 

Feet. 

U. S. 
Galls. 

Dia. 

Feet. 

Cub. 

Feet. 

U. S. 
Galls. 

1 

12.57 

94.0 

7 

38.49 

287.9 

12 

113.1 

846.1 

24 

452.4 

3384 

A 

14.19 

106.1 

K 

41.28 

308.8 

13 

132.7 

992.8 

25 

490.9 

3672 

y> 

15.90 

119.0 


44.18 

330.5 

14 

153.9 

1152. 

26 

530.9 

3971 

% 

17.72 

132.5 

/'A 

47.17 

352.9 

15 

176.7 

1322. 

27 

572.6 

4283 

5 

19.64 

146.9 

8 

50.27 

376.0 

16 

201.1 

1504. 

28 

615.8 

4606 

A 

21.65 

161.9 


56.75 

424.5 

17 

227.0 

1698. 

29 

660.5 

4941 

Vo 

23.76 

177.7 

9 

63.62 

475.9 

18 

254.5 

1904. 

30 

706.9 

5288 

% 

25.97 

194.3 


70.88 

530.2 

19 

283.5 

2121. 

31 

754.8 

5646 

5 

28.27 

211.5 

10 

78.54 

587.6 

20 

314.2 

2350. 

32 

804.3 

6017 


30.68 

229.5 


86.59 

647.7 

21 

346.4 

2591. 

33 

855.3 

6398 

k 

33.18 

248.2 

11 

95.03 

710.9 

22 

380.1 

2844. 

34 

907.9 

6792 

% 

35.79 

267.7 

A 

103.90 

777.0 

23 

415.5 

3108. 

35 

962.1 

7197 




























































158 


CONTENTS AND LININGS OF WELLS, 


CONTEXTS AND EININGS OF WELLS. 

For diams twice as great as those in the table, for the cub yds of digging, take out those opposi 
one half of the greater diam; and mult them by 4. Thus, for the cub yds in each foot of depth of 
well 31 feet in diam, first take out from the table those opposite the diam of 15>* feet; namely 6.981 
Then 6.989 X 4 — ‘27.956 cub yds reqd for the 31 ft diam. But for the stone lining or walling, brie 
or plastering, mult the tabular quantity opposite half the greater diam, by ‘2. Thus, the perches i 
stoue walling for each foot of depth of a well of 31 ft diam, will be *2.073 X 2 rr 4.146. If the wall 
more or less than one foot thick, within usual moderate limits, it will generally be near enough f< 
practice to assume that the number of perches, or of bricks, will increase or decrease in the same Dr 
portion. r 

The size of the bricks is taken at 8% X 4 X 2 inches: and to be laid dry, or without mortar. ] 
practice an addition of about 5 per cent should be made for waste. The brick lining is supposed 
be 1 brick thick, or 8% ius. ** 

CAUTION. — Be careful to observe that the diams to be used for the digcini 

are greater than those for the walling, bricks, or plastering. No errors. 



For each foot of depth. 


For each foot of depth. 

Diam. 

For this 
col use the 
Diameter 

For these three cols use the 
diam in clear of the lining. 

Diam. 

For this 
col use the 
Diameter 

For these three cols use th< 
diam in clear of the lining 

in 

Feet. 

of the 
Digging. 

Stone 
Liuing 
l ft thick. 
Perches of 
25 Cub Ft. 

No. of 
Bricks in 
a Lining 
1 Brick 
thick. 

Square 

in 

Feet. 

of the 
Digging. 

Stone 

Lining 

1 ft thick. 
Perches of 
25 Cub Ft. 

No. of 
Bricks in 
a Lining 
1 Brick 
thick. 

Squa 

Yard 

ofPla 

terinj 


Cub Yds. 
of 

Digging. 

Yards of 
Plaster¬ 
ing. 


Cub Yds. 
of 

Digging. 

i. 

.0291 

.2513 

57 

.3491 

X 

5.107 

1.791 

750 

4.625 

X 

.0155 

.2827 

71 

.4364 

X 

5.301 

1.822 

764 

4.713 

l A 

.0654 

.3142 

85 

.5236 

X 

5.500 

1.854 

778 

4.800 

X 

.0801 

.3456 

99 

.6109 

14. 

5.701 

1.885 

792 

4.887 

2. 

.1164 

.3770 

114 

.6982 

X 

5.907 

1.916 

806 

4.974 

X 

.1473 

.4084 

128 

.7855 

X 

6.116 

1.948 

820 

5.062 

A 

.1818 

.4398 

142 

.8727 

X 

6.329 

1.979 

834 

5.149 

X 

.2200 

.4712 

156 

.9600 

15. 

6 545 

2.011 

849 

5.236 

3. 

.2618 

.5027 

170 

1.047 

X 

6.765 

2.042 

863 

5.323 

X 

3073 

.5341 

184 

1.135 

X 

6.989 

2.073 

877 

5.411 

X 

.3563 

.5655 

198 

1.222 

X 

7.216 

2.105 

891 

5.498 

X 

.4091 

.5969 

212 

1.309 

16. 

7.447 

2.136 

905 

5.585 

4. 

.4654 

.6283 

227 

1.396 

X 

7.681 

2.168 

919 

5 673 

X 

.5254 

.6597 

241 

1.484 

X 

7.919 

2.199 

933 

5.760 

X 

.5890 

.6912 

255 

1.571 

X 

8-161 

2.231 

948 

5.847 

X 

.6563 

.7226 

269 

1.658 

17. 

8.407 

' 2.262 

962 

5.934 

5. 

.7272 

.7540 

283 

1.745 

X 

8.656 

2.293 

976 

6.022 

X 

.8018 

.7&54 

297 

1.833 

X 

8.908 

2.325 

990 

6.109 

X 

.8799 

.8168 

311 

1.920 

X 

9.165 

2.356 

1004 

6.196 

X 

.9617 

.8482 

326 

2.007 

18. 

9.425 

2.388 

1018 

6.283 

6. 

1.047 

.8796 

340 

2.095 

X 

9.688 

2.419 

1032 

6.371 

X 

1.136 

.9111 

354 

2.182 

X 

9.956 

2.450 

1046 

6.458 

A 

1 229 

.9425 

368 

2.269 

X 

10.23 

2.482 

1061 

6.545 

X 

1.325 

.9739 

382 

2.356 

19. 

10.50 

2.513 

1075 

6.633 

7. 

1.425 

1.005 

396 

2.444 

X 

10.78 

2.545 

1089 

6.720 

X 

1.529 

1.037 

410 

2.531 

X 

11.06 

2.576 

1103 

6.807 

X 

1.636 

1.068 

425 

2.618 

X 

11.35 

2.608 

1117 

6.894 

X 

1.747 

1.100 

439 

2.705 

20. 

11.64 

2.639 

1131 

6.982 

8. 

1.862 

1.131 

453 

2.793 

X 

11.93 

2.670 

1145 

7.069 

Y\ 

1.980 

1.162 

467 

2.880 

X 

12.22 

2.702 

1160 

7.156 

X 

2.102 

1.194 

481 

2.967 

X 

12.52 

2.733 

1174 

7.243 

X 

2.227 

1.225 

495 

3.054 

21. 

12.83 

2.765 

1188 

7.331 

9. 

2.356 

1.257 

509 

3.142 

X 

13.14 

2.796 

1202 

7.41i 

X 

2.489 

1.288 

523 

3.229 

X 

13.45 

2.827 

1216 

7.505 

% 

2.625 

1.319 

538 

3.316 

X 

13.76 

2.859 

12.30 

7 5931 

in H 

2.765 

1.351 

552 

3.404 

22. 

14.08 

2.890 

1244 

7.680j 

10. 

2.909 

1.382 

566 

3.491 

X 

14.40 

2.922 

1259 

7.767 1 

X 

3.056 

1.414 

580 

3.578 

X 

14.73 

2.953 

1273 

7.854 

Y 

3.207 

1.445 

594 

3.665 

X 

15.06 

2.985 

1287 

7.942 

X 

3.362 

1.477 

608 

3.753 

23. 

15.39 

3 016 

1301 

8.029 

11. 

3.520 

1.508 

622 

3 840 

X 

15.72 

3.047 

1315 

8.116 

Y 

3.682 

• I 539 

637 

3.927 

X 

16.06 

3.079 

1329 

8.203 

Y 

3.847 

1.571 

651 

4.014 

X 

16.41 

3.110 

1343 

8.291 

X 

4.016 

1.602 

665 

4.102 

24. 

16.76 

3.142 

1357 

8.378i 

12. 

4.189 

1.634 

679 

4.189 

X 

M 

17.11 

3.173 

1372 

8.465 

X 

4.365 

1.665 

693 

4.276 

17.40 

3 204 

1386 

8.552 ’ 

Yi 

4.545 

1.696 

707 

4.364 

X 

17.82 

3.236 

1400 

MU0 
8.727 j 

X 

4.729 

1.728 

721 

4.451 

25. 

18.18 

3.267 

1414 

13. 

4.916 

1.759 

736 

4.538 

A c 

nb y«l 

= 202 I 

r . S. ga 

Is. & 


If porches are named in a contract, it is necessary, in order to prevent fraud 

to specify the number of cub feet contained in the perch ; for stone-quarriers have one perch stone 
masons another, Ac. Kngineers, on this account, contract by the cubic yard. The perch should b 
done away with entirely ; Perches of 26 cub ft X .926 = cub yds ; and cub yds-j- .926- pers of 25 cub fl 








































CYLINDllIC UNGULAS, ETC. 


159 


CIRCULAR CYIiINl>ItIC UJJGULAS. 

I. Wlien tlie cutting: plane does not cut the base. Figs 13,14. 

m 

I 

I 
I 
« 

! 

I 
I 

! 


c 

Fig. 13. Fig. 14. 

Aolume'l — area of base o g X 34 SUIU °f greatest <fc least perp heights, on, cm, 
ot f = / area of cross sec measd v y 2 sum of greatest and least lengths, 
ungula ) ( perp to sides, as x, A gm,ol, measd along the sides. 

f circtimf measd perp half sum of greatest and least lengths, 
| to sides, as at x, A gm, o t, measd along the sides. 

Add areas of ends if required. 

For areas of sections perpendicular to the sides, see Circles, pp 123, &c. 
For areas of sections oblique to the sides, see The Ellipse, pp 149, &c. 


Area of 1 

curved, > = 

surface j 



II. When the cutting- plane touches the base. Figs A to D. 



m n 


(whether 
right or 
oblique) 


Curved 

surface 


Fig A = ( %ab 3 — ac X area admb* of base) 

Fig B = %cb 2 Xmn. 

Fig C = (% a 6 3 + a c X area admb* of base) 

Fig I> -= 34 area of circle ymf X mn 
= y, volume of cylinder xy mn. 

ac X length of arc dmb\) 


m n 
meas'd perp. 
to base 


a m 


Fig A = (ab X my 
Fig B — my X mn. 

Fig C = (abXmy + acX length of arc dmb%) 

Fig 1> = 34 circumference of base f my X mn 
= 34 curved surface of cylinder xyinn. 

i'or area of sloping plane, see p 150. 


(right 
ungula 
only) 


mn 

am' 

mn 

am' 


* For area of base (segment of circle), see pp 146 to 148. 
f For circles, see pp 123, etc. 

X For length of circular arc, see pp 141, etc. 

11 


































































IGO 


PYRAMIDS AND CONES. 


PYRAMIDS AND CONES. 




A pyramid, Figs. 1, 2, 3, is any solid which has, foT its base, a plane figure 
of any number of sides, aud, for its sides, plane triangles all terminating at one 
point d , called its apex , or top. When the base is a regular figure, the pyramid 
is regular; otherwise irregular. For regular figures, see Polygons, p. 110. 

A cone, Figs. 4 and 5, is a solid, of which the base is a curved figure; ami 
which mav be considered as made or generated by a line, of which one end if 
stationary at a certain point d, called the apex or top, while the line is being 
carried around the circumference of the base, which may be a circle, ellipse 
or other curve. A cone may also be regarded as a pyramid with an infinite 

number of sides. , . , , . . 

The axis of a pyramid or cone, is a straight line d o in Figs. 1, 2, 4; and a i ll 
Figs. 3 and 5, from the apex d, to the center of gravity of the base. VV hen the 
axis is perpendicular to the base, as in Figs. 1, 2, 4, the solid is said to be a righ 
one; when otherwise, as Figs 3, 5, an oblique one. W hen the word cone is use< 
alone, the right circular cone, Fig. 4, is understood. If such a cone be cut, as a 
11 obliquely to its base, the new base 11 will be an ellipse; and the cone d 1 
becomes an oblique elliptic one. Fig. 5 will represent either an oblique circula 
cone, or an oblique elliptic one, according as its base is a circle or an ellipse. 

Volume of pyramid or cone, regular or irregular, right or oblique. 

Volume = % area of base X perpendicular height d o, Figs. 1 to 5. 

= % volume of prism or cylinder having same area of base an< 
same perpendicular height. 

% volume of hemisphere of same base and same height. 

Or, a cone, hemisphere and cylinder, of the same base and same height, hav 
volumes as 1, 2 and 3. 

Area of surface of sides of right regular pyramid or right circular cone. 

Area = % circumference of base X slant height.* - 

In the cone , this becomes 


Area = X slant height, 

radius of base 


Add area of base 
if required. 



Area of surface of olrlique elliptic cone, d t 

Fig. 5j, cut from a right circular cone, dss. From ihe poiij 
c where the axis do of the right circular cone cuts the ollipt 
base It, measure a perpendicular, r, iu any direction, to tl) 
curved surface of the cone. Call the area of the elliptic bas 
tt, “a”; and the height du measured perpendicularly to sai 
base “ h.” Then 

ah 3 X volume of cone 
Curved Surface = - = --- * 


Add area of base if required. 

^ No measurement has been devised for the surface of a 

olrlique circular cone. 


* in the pyramid, this slant height must be measured along the middle of oi 
of the sides, aud not aloDg one of the edges. 




















PYRAMIDS AND CONES. 


160a 


To find the surface of an irregular pyramid. 

\V hethei right or oblique, each side must be calculated as a separate triancle (see 
p. 110); and the several areas added together. Add the area of base if required. 


FRUSTUMS OF PYRAMIDS AND CONES. 



Fig-. 6. 


Fig. 7. 



Frustum of pyramid (Fig. 6) 

parallel. 


or of cone (Fig. 7) with base and trp 


Volume (regular or irregular, right or oblique) 


= i/ y- perpendicular v / area / area > / area area \ 

/3 /'S height oo A ^of top • of base ' V of top * of base/ 

= 1/ V Perpendicular .. / area , area . 4 ar ea of a section \ 

/& A height oo * l of top of base T* Parallel to, and midway I 

x between, base and top / 

= (for frustum of right circular cone, Fig. 7, only) 


VsX 


perpendicular 
height o o 


X 3.1416 X (o<2 + QS 2 + 0 t 


os 


Surface of frustum of right regulir pyramid or cone, with top and base parallel. 
Figs. 6 and 7. 

= V f c i rcnm ference i circumference \ s’ant * 

2 \ of top *" of base / ^ height s t 

Add areas of top and base if required. v 


In the frustum of a right circular cone, this becomes 

Surface = ^ /radius , radius\ v slant* 

\of top * of base/ height st 

(^r == 3.1416) . Add areas of top and base if required. 


Frustum of lrregu’ar or oblique pyramid or cone. Surface =* 
um of surfaces of sid s, each of which must be treated as a trapezoid. See p. 120. 


* In the frustum of the/?yrawicZ (Fig 6), this slant height must be measured along 
he middle of one of the sides (as at t s), and not along one of the edges. 















1606 


PllISMOIDS. 


PRISMOIDS. 




A prismoid is sometimes defined as a solid having for its ends two parallel 
plane figures, connected by other plane figures on which, aud through every point 
of which, a straight line may be drawn from one of the two parallel ends to the 
other. These connecting planes may be parallelograms or not, aud parallel to each 

^ Tills definition would include the cube and all other parallelepipeds; 
the prism; the cylinder (considered as a prism having an infinite number of sides); 
the pyramid and cone (in which one of the two parallel ends, i e the one forming the 
apex,' is considered to be infinitely small), and their frustums with top and base 

parallel; and the wedge. „ ... 

But the use of the term prismoid is frequently restricted to six-sided solids 
in which the two parallel ends are unequal quadrangles; and the connecting planes 
trapezoids; as iu Figs. 1 and 2; aud, by some writers, to cases where the parallel 
quadrangular ends are rectangles. 

The following “prismoidal formula” applies to all the foregoing solids 
and to others, as noted below. 


Let A = the area of one of the two parallel ends 

a “ “ the other of the two parallel ends. 

M = “ “a cross section midway between, and parallel to, the tw< 
parallel ends. 

L = the perpendicular distance between the two parallel ends. 

Then 

T w A + a + 4M 

Volume = L X - „ - 

& 

= L X mean area of cross section. 

The following six figures represent a few of the irregular solids which fall under th 
above broad definition of “prismoid,” and to which the prismoidal formula apylief 
They maybe regarded as one-chain lengths of railroad cuttings; a o being the length 
or perpendicular (horizontal) distance between the two parallel (vertical) ends. 
















WEDGES. 


161 


The prtgmoldal formula applies also to the 

other spherical segments; also to any sections such as a 6 


sphere, hemisphere, and 
cd, and onidbc, of the 



cone, in which the sides a d, a c, or od, i c, are straight; as they are only when the 
cutting plane adc passes through the apex or top a. Also to the cylinder 
when a plaue parallel to the sides passes through both ends; but not if the plane 
102 is oblique, as in Ike fig., though never erring more than 1 in 142. In this last 
case we must imagine the plane to be extended until it cuts the side of the cylinder 
likewise extended ; and then by page 159 find the solidity of the ungula thus formed. 
Then find the solidity of the small ungula above w y also thus foimed, and subtract 
it from the large one. 

This very extended applicability of the prismoidal formula was first discovered 
and made known, by Ellwood Morris, C. E., of Philadelphia, in 1840. 


WEDGES. 



A wedge 

Is usually defined to he a solid, Figs. 8 and 9, generated by a plane triangle, anc, 
moving parallel to itself, in a straight line. This definition requires that the two 
triangular ends of the wedge should be parallel; but a wedge may be shaped as in 
Fig. 10 or 11. We would therefore propose the following definition, which embraces 
all the figs.; besides various modifications of them. A solid of five plane faces ; one 
of which is a parallelogram abed, two opposite sides of which, as ac and b d, are 
united by means of two triangular faces acn, and bdm, to an edge or Hdo n m, 
y^rallel to the other opposite sides ab and cd. The parallelogram abed may bo 
either rectangular, or not; the two triangular faces may bo similar, or not; and the 
same with regard to the other two faces. The following rule applies equally to all: 


Volume 

of wedge 


-Ye X 


Sum of lengths 
of the 3 edges 
ab -{■ c d nrn 


X 


perp htp from 
edge to back 


width of 

X hack (abed) 
meas’d perp to a b. 


























162 


MENSURATION. 


SPHERES OR GLOBES. 


A Sphere 

Is a solid generated by the revolution of a semicircle around its diameter. 

.1 r . ..X* _ _ 1.-. :r. M/.iii.li.dont frnm o nortflill Tlftlllt Pfl.ll fill I Ilf 


Is a solid generaieu uy mo rnuiuiwu w «, —--- Every 

point in the surface of a sphere is equidistant from a certain point called the center. 
Anv line passing entirely through a sphere, and through its center, is called its axis. 
or diameter. Any circle described on tl e surface of a sphere, from the center of 
the sphere as the center of the circle, is called a great circle of that sphere ; in other 
words any entire circumference of a sphere is a great circle. A sphere has a greater 
content or solidity than any other solid with the same amount of surface; so that if 
the shape of a sphere be any way changed, its couteut will be reduced. The inter¬ 
section of a sphere with any plane is a circle. 


Volume of sphere* 


= * tc radius 3 


Y & 7r diameter 3 
, circumference 3 


4.1888 radius 3 
0.5236 diameter 3 


% 


0.01 GS9 circumference 3 


7T 


= Y, diameter X area of surface 
= % diameter X area of great circle 
■= % volume of circumscribing cylinder 
= 0.5236 volume of circumscribing cube. 


Area of surface of sphere* 

= 47 r radius 2 = 12.5664 radius 2 

= 7 r diameter 2 — 3.1416 diameter 1 

circumference 2 


0.3183 circumference 2 


7T 


= diameter X circumference 
= 1 X area of great circle 

= area of circle whose diameter is equal to twice diameter of sphere. 
= curved surface of circumscribing cylinder 
6 X volume 


diameter. 


Radius of sphere 
3 


3 


volume 


— V T 


7T 


= 0.62035 3 y/ volume 


= ; 


Area of surface 


7T 


= v/ . 07958 x area of surface 


Circumference of sphere (see also rules for circles, p. 123.) 

= S \/ fi 7T 2 volume — 3 y/ 59.2176 volume 

= \/tt area of surface = y / 3.1416 area of surface 

area of surface 
diameter. 


♦For tallies of surfaces and solidities (volumes) of spheres, sei 

pp. 163 to 165. If the diameter is measured in inches, divide the surfaces in the tabl* 
by 144. if it is required to reduce them to squaro feet; and divido the solidities bj 
1728, if required in cubic feet. 




















MENSURATION. 


163 


SPHERES. (Original.) 

Some errors of 1 in the last figure only. 


Diam. 

j Surface. 

j Solidity. 

Diam. 

Surface. 

— - 

j Solidity. 

1 Diam. 

Surface. 

1 

' Solidity. 

Diam. 

Surface. 

J Solidity. 

« 

1 64 

.00077 


13-32 

18.190 

7.2949 

X 

170.87 

210.03 

X 

921.33 

2629.6 

1 32 

.00307 

.00002 

7-16 

18.666 

7.5829 

X 

176.71 

220.89 

X 

934.83 

2687.6 

3-64 

.00690 

.00005 

15-32 

19.147 

7.8783 

% 

182.66 

232.13 

X 

948. fit 

2746.5 

I 16 

.01227 

.00013 

X 

19.635 

8.1813 

X 

188.69 

243.73 

X 

962.12 

2806.2 

3-32 

.02761 

.00043 

17-32 

20.129 

8.4919 

Va 

194.83 

255.72 

Ya 

975.91 

2866.8 

X 

.04909 

.00102 

9-16 

20.629 

8 8103 

8. 

201.06 

268.08 

X 

989.80 

2928.2 

5-32 

.07670 

.00200 

19-32 

21.135 

9.1366 

X 

207.39 

280.85 

Ya 

1003.8 

2990.5 

3 16 

.11045 

.00345 

% 

21.648 

9.4708 

X 

213.82 

294.01 

18. 

1017.9 

3053.6 

7-32 

.15033 

.00548 

21-32 

22.166 

9.8131 

% 

220.36 

307.58 

X 

1032.1 

3117.7 

X 

.19635 

.00818 

11-16 

22.691 

10.164 

X 

226.98 

321.56 

X 

1046.4 

3182.6 

9-32 

.24851 

.01165 

23-32 

23.222 

10.522 

% 

233.71 

335.95 

X 

1060.8 

3248.5 

5-16 

.30680 

.01598 

3 4 

23.758 

10.889 

X 

240.53 

350.77 

X 

1075.2 

3315.3 

11-32 

.37123 

.02127 

25-32 

24.302 

11.265 

Va 

247.45 

366.02 

Ya 

1089.8 

3382.9 

% 

.44179 

.02761 

13-16 

24.850 

11.649 

9. 

254.47 

381.70 

X 

1104.5 

3451.5 

13-32 

.51848 

.03511 

27-32 

25.405 

12.041 

X 

261.59 

397.83 

Ya 

1119.3 

3521.0 

7-16 

.60132 

.04385 

Va 

25.967 

12.443 

X 

268.81 

414.41 

19. 

1134.1 

3591.4 

15-32 

.69028 

.05393 

29-32 

26.535 

12.853 

% 

276.12 

431.44 

X 

1149.1 

3662.8 

X 

.78540 

'06545 

15-16 

27.109 

13.272 

X 

283.53 

448.92 

X 

1164.2 

3735.0 

17-32 

.88664 

.07850 

31-32 

27.688 

13.700 

Ya 

291.04 

466.87 

X 

1179.3 

3808.2 

9-16 

.99403 

.09319 

3. 

28.274 

14.137 

X 

298.65 

485.31 

X 

1194.6 

3882.5 

19-32 

1.1075 

.10960 

1-16 

29.465 

15.039 

Va 

306.36 

504.21 

Ya 

1210.0 

3957.6 

% 

1.2272 

.12783 

X 

30.680 

15.979 

10. 

314.16 

523.60 

X 

1225.4 

4033.7 

21-32 

1.3530 

.14798 

3-16 

31.919 

16.957 

X 

322.06 

543.48 

Ya 

1241.0 

4110.8 

11-16 

1.4849 

.17014 

X 

33.183 

17.974 

X 

330.06 

563.86 

20. 

1256.7 

4188.8 

23-32 

1.6230 

.19442 

5-16 

34.472 

19.031 

% 

338.16 

584.74 

X 

1272.4 

4267.8 

X 

1.7671 

.22089 

Va 

35.784 

20.129 

X 

346.36 

606.13 

X 

1288.3 

4347.8 

25-32 

1.9175 

.24967 

7-16 

37.122 

21.268 

% 

354.66 

628.04 

X 

1304.2 

4428.8 

13-16 

2.0739 

.28084 

X 

38.484 

22.449 

X 

363.05 

650.46 

X 

1320.3 

4510.9 

27-32 

2.2365 

.31451 

9 16 

39.872 

23.674 

Va 

371.54 

673.42 

Ya 

1336.4 

4593.9 

% 

2.4053 

.35077 

% 

41.283 

24.942 

11. 

380.13 

696.91 

X 

1352.7 

4677 9 

29-32 

2.5802 

.38971 

11-16 

42.719 

26.254 

X 

388.83 

720.95 

Va 

1369.0 

4763.0 

15-16 

2.7611 

.43143 

X 

44.179 

27.611 

X 

397.61 

745.51 

21. 

1385.5 

4849.1 

31-32 

2.9483 

.47603 

13-16 

45.664 

29.016 

% 

406.49 

770.64 

X 

1402.0 

4936 2 

I. 

3.1416 

.52360 

% 

47.173 

30.466 

X 

415.48 

796.33 

X 

1418.6 

5024.3 

1-32 

3.3410 

.57424 

15-16 

48.708 

31.965 

% 

424.56 

822.58 

X 

1435.4 

5113.5 

1-16 

3.5466 

.62804 

4. 

50.265 

33.510 

X 

433.73 

849.40 

X 

1452.2 

5203.7 

3-32 

3.7583 

.68511 

1-16 

51.848 

35.106 

y» 

443.01 

876.79 

Ya 

1469.2 

5295.1 

X 

3.9761 

.74551 

X 

53.456 

36.751 

12. 

452.39 

904.78 

X 

1486.2 

5387.4 

5-32 

4.2000 

.80939 

3 16 

55.089 

38.448 

X 

461.87 

933.34 

Ya 

1503.3 

5480.8 

3-16 

4.4301 

.87681 

X 

56.745 

40.195 

X 

471.44 

962.52 

22. 

1520.5 

5575.3 

7-32 

4.6664 

.94786 

5-16 

58.427 

41.994 

% 

481.11 

992.28 

X 

1537 9 

5670.8 

X 

4.9088 

1.0227 

y» 

60.133 

43.847 

X 

490.87 

1022.7 

X 

1555.3 

5767.6 

9-32 

5.1573 

1.1013 

7-16 

61.863 

45.752 

% 

500.73 

1053.6 

X 

1572.8 

5865.2 

5-16 

5.4119 

1.1839 

X 

63.617 

47.713 

X 

510.71 

1085.3 

X 

1590.4 

5964.1 

11-32 

5.6728 

1.2704 

9-16 

65.397 

49.729 

Va 

520.77 

1117.5 

Ya 

1608.2 

6064.1 

Va 

5.9396 

1.3611 

% 

67.201 

51.801 

13. 

530.93 

1150.3 

X 

1626.0 

6165.2 

13-32 

6.2126 

1.4561 

11-16 

69.030 

53.929 

X 

541.19 

1183.8 

Ya 

1643.9 

6267.3 

7-16 

6.4919 

1.5553 

% 

70.883 

56.116 

X 

551.55 

1218.0 

23. 

1661.9 

6370.6 

15-32 

6.7771 

1.6590 

13-16 

72.759 

58.359 

% 

562 00 

1252.7 

X 

1680.0 

6475.0 

X 

7.0686 

1.7671 

Va 

74.663 

60.663 

X 

572.55 

1288.3 

X 

1698.2 

6580.6 

17-32 

7.3663 

1.8799 

15-16 

76.589 

63.026 

Va 

583.20 

1324.4 

% 

1716.5 

6687.3 

9-16 

7.6699 

1.9974 

5. 

78.540 

65.450 

X 

593.95 

1361.2 

X 

1735.0 

6795.2 

19-32 

7.9798 

2.1196 

1-16 

80.516 

67.935 

Va 

604.80 

1398.6 

Ya 

1753.5 

6904.2 

b A 

8.2957 

2.2468 

X 

82.516 

70.482 

14. 

615.75 

1436.8 

X 

1772.1 

7014.3 

21-32 

8.6180 

2.3789 

3 16 

84.541 

73.092 

X 

626.80 

1475.6 

Ya 

1790.8 

7125.6 

11-16 

8.9461 

2.5161 

X 

86.591 

75 767 

X 

637.95 

1515.1 

24. 

1809.6 

7238.2 

23-32 

9.2805 

2.6586 

5-16 

88.664 

78.505 

% 

649.17 

1545.3 

H 

1828.5 

7351.9 

X 

9.6211 

2.8062 

Vs 

90.763 

81.308 

X 

660 52 

1596.3 

X 

1847.5 

7466.7 

25-32 

9.9678 

2.9592 

7-16 

92.887 

84.178 

Ya 

671.95 

1637.9 

X 

1866.6 

7583.0 

13-16 

10.321 

3.1177 

X 

95.033 

87.113 

X 

683.49 

1680.3 

£2 

1885.8 

7700.1 

27-32 

10.680 

3.2818 

9-16 

97.205 

90.118 

Va 

695.13 

1723.3 

% 

1905.1 

7818.6 

% 

11.044 

3.4514 

% 

99.401 

93.189 

15. 

706.85 

1767.2 

X 

1924.4 

7938.3 

29-32 

11.416 

3.6270 

11-16 

101.62 

96.331 

X 

718.69 

1811.7 

Ya 

1943.9 

8059.2 

15 16 

11.793 

3.8083 

% 

103.87 

99.541 

X 

730.63 

1857.0 

25. 

1963.5 

8181.3 

31-32 

12.177 

3.9956 

13-16 

106.14 

102.82 

% 

742.65 

1903.0 

X 

1983.2 

&304.T 

2. 

12.566 

4.1888 

Va 

108.44 

106.18 

X 

754.77 

1949.8 

X 

2002.9 

8429.2 

1-32 

12.962 

4.3882 

15-16 

110.75 

109.60 

% 

767.00 

1997.4 

Ya 

2022.9 

8554.9 

1-16 

13.364 

4.5939 

6. 

113.10 

113.10 

X 

779.32 

2045.7 

X 

2042.8 

8682.0 

3 32 

13.772 

4.8060 

X 

117.87 

120.31 

Va 

791.73 

2094.8 

Ya 

2062.9 

8810.3 

u 

14.186 

5.0243 

X 

122.72 

127.83 

16. 

804.25 

2144.7 

X 

2083.0 

8939.9 

/b 

5-32 

14.607 

5.2493 

% 

127 68 

135.66 


816.85 

2195.3 

Ya 

2103.4 

9070.6 

a. 16 

15.033 

5.4809 

X 

132.73 

143.79 

X 

829.57 

2246.8 

26. 

2123.7 

9202.8 

7-32 

15.466 

5.7190 

% 

137.89 

152.25 

% 

842.40 

2299.1 

X 

2144.2 

9336.2 

V* 

15 904 

5.9641 

X 

143.14 

161.03 

X 

855.29 

2352.1 

X 

2164.7 

9470.8 

9-32 

16.349 

6.2161 

Va 

148.49 

170.14 

Ya 

868.31 

2406.0 

% 

2185.5 

9606.7 

5-16 

16.800 

6.4751 

7. 

153.94 

179.59 

X 

881.42 

2160.6 

X 

2206.2 

9744.0 

1 1 -3‘2 

17.258 

6 7412 

X 

159 49 

189.39 

Va 

894.63 

2516.1 

% 

2227.1 

9882.5 


17.721 

7.0144 

X 

165.13 

199.53 

17. 

907.93 

2572.4 

X 

2248.0 

10022 































1G4 


MENSURATION, 


SPHERES — (Continued.) 


a 

_CJ 

5 

Surface. 

Solidity. 

Diam. 

Surfaoe. 

Solidity. 

Diam. 

Surfaoe. 

Solidity. 

Diam. 

Surface. 

Solidity. 

X 

2269.1 

10164 

% 

4214.1 

25724 

fa 

6756.5 

52222 

X 

98G6.0 

92574 

J7. 

2290.2 

10306 

X 

4243.0 

25988 

X 

6792.9 

52645 

X 

9£40.2 

83194 

X 

2311.5 

10450 

X 

4271.8 

26254 

X 

6829.5 

53071 

X 

99)84.4 

93M i 

yi 

2332.8 

10595 

37. 

4300.9 

20522 

X 

6866.1 

53499 

X 

1C029 

9443; 

% 

235-1.3 

10741 

X 

4330.0 

26792 

X 

6902.9 

53929 

X 

10073 

9506f 

y 

23T5.8 

10889 

X 

4359.2 

27063 

47. 

6C39.9 

54362 

X 

10118 

9569: 

% 

2397.5 

11038 


4388.5 

27337 

X 

6976.8 

547S7 

'/a 

10163 

£6334 

X 

2419.2 

11189 

X 

4417.9 

27612 

X 

7013.9 

55234 

57. 

1C 207 

£696" 

y» 

2441.1 

11341 

X 

4447.5 

27889 

fa 

7050.9 

55674 

X 

10252 

974)04 

as. 

2463.0 

11494 

X 

4477.1 

28168 

X 

7088.3 

56115 

X 

10297 

9824; 

M 

2485.1 

11649 

x 

4506.8 

28449 

X 

7125.6 

56559 

fa 

10342 

988£! 

y* 

2507.2 

11605 

38. 

4536.5 

28731 

X 

7163.1 

57006 

X 

1C 387 

9954) 

fa 

2529.5 

11 £62 

X 

4566.5 

28016 

X 

7200.7 

57455 

X 

10432 

10019 

y 

2551.8 

12121 

X 

4o96.4 

29302 

48. 

7238.3 

57yC6 

X 

10478 

10084! 

y* 

2574.3 

12281 

fa 

4626.5 

295S0 

X 

7276.0 

583C0 

X 

10523 

10150 

x 

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12770 

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4717.3 

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7389.9 

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10660 

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2665.0 

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2687.8 

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10751 

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4809.0 

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7504.5 

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x 

2734.0 

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X 

4839.9 

31661 

49. 

7543.1 

61601 

X 

10844 

10617! 

y» 

2757.3 

13614 

fa 

4870.8 

31964 

X 

7581.6 

62074 

'/a 

108 £0 

10685- 

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X 

4£01.7 

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7620.1 

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59. 

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4S32.7 

32577 

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63026 

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10983 

10822 

SO. 

2827.4 

14137 

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4964.0 

32886 

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7697.7 

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11029 

10890! 

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2851.1 

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33197 

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7736.7 

63989 

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11076 

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2874.8 

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40. 

5026.5 

33510 

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7775.7 

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11029 

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2898.7 

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33826 

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7814.8 

64£61 

X 

11169 

110991 

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2922.5 

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5089.6 

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50. 

7854.0 

65450 

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11216 

11169'* 

% 

2916.6 

15039 

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5121.3 

31462 

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7893.3 

65941 

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11263 

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2970.6 

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5153.1 

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7£32.8 

66436 

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11310 

11309! 

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5184.9 

341C6 

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7972.2 

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11357 

113801 

31. 

3019.1 

155£9 

X 

5216.8 

35431 

X 

6011.8 

67433 

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3043.6 

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5248.9 

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X 

8051.6 

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fa 

11452 

11523’ 

X 

3068.0 

15979 

41. 

5281.1 

36087 

X 

8091.4 

68439 

X 

11499 

11594: 

% 

3092.7 

16172 

X 

5313.3 

36418 

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8131.3 

68946 

X 

11547 

11666! 

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3117.3 

16366 

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5345.6 

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51. 

8171.2 

69496 

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11595 

11739! 

% 

3142.1 

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5378.1 

37086 

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8211.4 

69967 

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3166.9 

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3192.0 

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11957! 

32. 

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38104 

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8332.3 

71519 

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11786 

12031! 

X 

3242.2 

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5508.9 

38448 

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8372.8 

72040 

fa 

11834 

12105! 

X 

3267.4 

17563 

42. 

5541.9 

88792 

X 

8413.4 

72565 

X 

11882 

12179- 

fa 

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17768 

X 

5574.9 

39140 

X 

8454.1 

73092 

X 

11931 

122531 

X 

3318.3 

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X 

5608.0 

39490 

52. 

8494.8 

73622 

X 

11980 

123284 

X 

3343.9 

181S2 

X 

5611.3 

39841 

X 

8535.8 

74154 

X 

12028 

124034 

X 

3369.6 

18392 

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5674.5 

40194 

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62. 

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12478! 

X 

3395.4 

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5708.0 

40551 

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33. 

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X 

5741.5 

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12706) 

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12322 

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3525.7 

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5876.5 

42360 

53. 

8824.8 

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12371 

12937 

% 

3552.1 

19907 

fa 

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X 

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79060 

63. 

124 69 

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X 

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12519 

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34. 

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43846 

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£034.1 

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X 

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81876 

X 

12718 

134864 

X 

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21501 

X 

6151.5 

4 ’-367 

54. 

9160.8 

82418 

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12768 

13565) 

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X 

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86521 

X 

13121 

141324 

X 

3959.2 

23425 

X 

6432.7 

48513 

55. 

9503.2 

87114 

X 

13172 

142149 

% 

3987.2 

23674 

% 

6168.3 

48916 

X 

9546.5 

87709 

X 

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142964 

X 

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X 

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65. 

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X 

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X 

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X 

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X 

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X 

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% 

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X 

9720.6 

90117 

fa 

13427 

146297 

X 

4128.3 

24942 

46. 

6617.6 

50965 

X 

9764.4 

90726 

X 

13478 

14713F 

% 

4156.9 

25201 

X 

6683.7 

51382 

X 

9808.1 

91338 

X 

13530 

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X 

4185.5 

23461 

X 

6720.0 

51801 

56. 

9852.0 

91953 

X 

13562 

148828 




























MENSURATION, 


165 


s 

cJ 


66 . 


x 

x 

X 

% 

x 

x 

x 


67. 


X 

X 

Vs 

X 

X 


68 . 


69 


70. 


X 

X 

% 

X 

X 

X 

x 

x, 

X 

X 

X 

X 

X 

X 

X 

X 

X 


71. 


T2. 


x 

x 

x 

% 

x 

x 

X 

x 

X 

X 

X 

34 

X 


i\ X 


73. 


71. 


x 

x 

x 

X 

X 

X 

% 

x 

x 

x 

x 


SPH ER£S —(Continued.) 


Surface. 

1 

Solidity. 

Diam. 

13633 

149680 

1 X 

13685 

150533 

X 

13737 

151390 

X 

13789 

152251 

X 

13841 

153114 

75. 

13893 

153980 

X 

13946 

154850 

X 

13998 

155724 

% 

14050 

156600 

x 

14103 

157480 

X 

14156 

158363 

X 

14208 

159250 

X 

14261 

160139 

76. 

14314 

161032 

X 

14367 

161927 

X 

14420 

162827 

X 

14474 

163731 

X 

14527 

164637 

X 

14580 

165547 

X 

14634 

166460 

X 

14688 

167376 

77. 

14741 

168295 

X 

14795 

169218 

X, 

14849 

170145 

% 

14903 

171074 

x 

14957 

172007 

X 

15012 

172944 

X 

15066 

173883 

X 

15120 

174828 

78. 

15175 

175774 

X 

15230 

176723 

X 

15284 

177677 

% 

15339 

178635 

X 

15394 

179595 

X 

15449 

180559 

X 

15504 

181525 | 

X 

15560 

182497 

79. 

15615 

183471 

X 

15670 

181449 | 

X 

15726 

185430 

X 

15782 

186414 

X 

15837 

187402 | 

X 

15893 

188:394 

X 

15949 

189389 

X 

16005 

190387 

80. 

16061 

191389 

X 

16117 

192395 

X 

16174 

193104 

% 

16230 

194417 1 

X 

16286 

195433 | 

X 

16343 

196453 

X 

16400 

197476 

X 

16456 

198502 

81. 

16513 

199532 

X 

16570 

200566 I 

X 

16628 

201604 I 

X 

16685 

202645 I 

X 

16742 

203689 

X 

16799 

204737 

X 

16857 

205789 

X 

16914 

206844 

32. 

16972 

207903 

X 

17030 

208966 

X 

170,88 

210032 

X 

17146 

211102 | 

X 

17204 

212175 

X 

17262 

213252 

X 

17320 

214333 

X 

17379 

215417 

<3. 


3 

03 


17437 

17496 

17554 

17613 

17672 

17731 

17790 

17849 

17908 

17968 

18027 

18087 

18116 

18206 

18266 

18326 

18386 

18446 

18506 

18566 

18626 

18687 

18748 

18809 

18869 

18930 

18992 

19053 

19111 

19175 

19237 

19298 

19360 

19422 

19483 

19545 

19607 

19669 

19732 

19794 

19856 

19919 

19981 

20044 

20106 

20170 

20232 

20296 

20358 

20422 


20485 

20549 

20612 

20676 

20740 

20804 

20867 

20932 

20996 

21060 

21124 

21189 

21253 

21318 


21382 

21448 

21512 

21578 

21642 


1 

Solidity. 

| Diam. 

216505 

X 

217597 

X 

218693 

X 

219792 

x 

220894 

% 

222001 

X 

223111 

x 

224224 

84. 

225341 

X 

226463 

X 

227588 

X 

228716 

X 

229848 

X 

230984 

x 

232124 

X 

233267 

85. 

234414 

X 

235566 

X 

236719 

X 

237879 

X 

239011 

X 

210206 

x 

241376 

x 

242551 

86. 

243728 

: x 

244908 

X 

246093 

X 

247283 

I X 

248475 

X 

249672 

X 

250873 

X 

252077 

87. 

253284 

X 

254496 

X 

255713 

% 

256932 

X 

258155 

% 

259383 

X 

260613 

X 

261848 1 

88. 

263088 | 

X 

264330 

X 

265577 | 

X 

266829 I 

X 

268083 

X 

269342 

X 

270604 

X 

271871 

CO 

273141 

X 

271416 

X 

275694 

X 

276977 

X 

278263 

X 

279553 

X 

280847 

X 

282145 

90. 

2834-17 

X 

284754 

X 

286061 

X 

287378 

X 

288696 

X 

290019 

X 

291345 

X 

292674 

n. 

294010 

X 

295347 

X 

296691 

X 

298036 

X 

299388 

X 


Surface. 

1 l 

Solidity. 

Diam. 

Surface. 

Solidity. 

21708 

300743 

h 

26446 

404406 

21773 

302100 

■ y 

26518 

406060 

21839 

303463 

92. 

26590 

407721 

21904 

304831 

h 

26663 

409384 

21970 

306201 

h 

26735 

411054 

22036 

307576 

h 

26808 

412726 

22102 

308957 

X 

26880 

414405 

22167 

310340 

X 

26953 

416086 

22234 

311728 

X 

27026 

417774 

22300 

313118 

X 

27099 

419164 

22366 

314514 

93. 

27172 

421161 

22432 

315915 

X 

27245 

422662 

22499 

317318 

X 

27318 

424567 

22565 

318726 

X 

27391 

426277 

22632 

320140 

X 

27464 

427991 

22698 

321556 

X 

27538 

429710 

22765 

322977 

X 

27612 

431433 

22832 

324402 

X 

27686 

433160 

22899 

325831 

94. 

27759 

434894 

22966 

327264 

X 

27833 

436639 

23034 

328702 

X 

27907 

438373 

23101 

330142 

X 

27981 

440118 

23168 

331588 

X 

28055 

441871 

23235 

333039 

X 

28130 

44362,7 

23303 

334492 

x 

28204 

445387 

23371 

335951 

x 

28278 

447151 

23439 

337414 

95. 

28353 

448920 

23506 

338882 

X 

28428 

450695 

23575 

340352 

X 

28503 

452475 

23643 

341829 

x 

28577 

454259 

23711 

343307 

X 

28652 

456047 

23779 

344792 

% 

28727 

457839 

23847 

346281 

X 

28802 

459638 

23916 

347772 

X 

28878 

461439 

23984 

349269 

96. 

28953 

463248 

24053 

350771 

X 

29028 

465059 

24122 

352277 

X 

29104 

466875 

24191 

353785 

X 

29180 

468697 

24260 

355301 | 

X 

29255 

470524 

24328 

356819 

X 

29331 

472354 

24398 

358342 

X 

29407 

474189 

24467 

359869 | 

X 

29483 

476029 

24536 

361400 

97. 

29559 

477874 

24606 

362935 

X 

29636 

479725 

24676 

364476 

X 

29712 

481579 

24745 

366019 

X 

29788 

483438 

24815 

367568 

X 

29865 

485302 

24885 

369122 | 

X 

29942 

487171 

24955 

370678 | 

X 

30018 

489045 

25025 

372240 

X 

30095 

490924 

25095 

373806 | 

98. 

30172 

492808 

25165 

375378 

X 

30249 

494695 

25236 

376954 

X 

30326 

496588 

25306 

378531 

X 

30404 

498186 

25376 

380115 

X 

30481 

500388 

25447 

381704 

X 

30558 

502296 

25518 

383297 

X 

30636 

504208 

25589 

384894 1 

X 

30713 

506125 

25660 

386496 

99. 

30791 

508047 

25730 

388102 

X 

30869 

509975 

25802 

389711 

X 

30947 

511906 

25873 

391327 

X 

31025 

513843 

25944 

392945 

X 

31103 

515785 

26016 

394570 

X 

31181 

517730 

26087 

396197 

X 

31259 

519682 

26159 

397831 

X 

31338 

521638 

26230 

399468 

00. 

31416 

523598 

26302 

401109 




26374 

i 

402756 
























































































166 


SEGMENTS, ETC., OF SPHERES, 


To find the solidity of a spherical segment. 


s 



Rule 1. Square the rad o », of its base; mult this square by 3 : to 
the prod add the square of its height o s ; mult the sum by the height 
o s ; and mult this last prod bv .5236. 

Rule 2. Mult the diam ab of the sphere by 3; from the prod 
take twice the height o s of the segment; mult the rem by the square 
of the height o *; and mult this prod by .5236. . . 

The solidity of a sphere being %ds that of its circumscribing cylin- 
der. if we add to any solidity in the table, its half, we obtain that 
of a cylinder of the same diam as the sphere, and whose height 
equals’its diam. | 


To find the curved surface of a spherical segment. 


Rule 1. Mult the diam a b of the sphere from which the segment is cut, by 3.1416; 
mult the prod by the height o s of the seg. Add area of base if reqd. Rem. Having the diam n r 
of the seg, and its height o s, the diam a b of the sphere may be found thus : Div the square of half 
the diam n r, by its height os; to the quot add the height o s. Rule 2. The curved surf of either 
a segment, last Fig, or of a zone, (next Fig,) bears the same proportion to the surf of the whole 
sphere, that the height of the seg or zone bears to tbe diam of the sphere. Therefore, first find the 
surf of the whole sphere, either by rule or from the preceding table: mult it by the height of the seg 
or zone; div the prod by diam of sphere. Rule 3. Mult the cireumf of the sphere by the height o • 

of the seg. 




To And the solidity of a spherical zone. 

Add together the square of the rad e d, the square of rad o b, 
and t$d of the square of the perp height eo; mult the sum by 
1.5708; and mult this prod by the height eo. 

To find the curved surface of a spher¬ 
ical zone. 

Rule 1. Mult together the diam m n of the sphere ; the height 
e o of the zone, and the number 3.1416. Or see preceding Rule 2 
for surf of segments. Rule 2. Mult the circumf of the sphere, by 
the height of the zone. 

To find the solidity of a hollow spher¬ 
ical shell. 

Take from the foregoing table the solidities of two spheres having 
the diams a b, and c d. Subtract the least from the greatest. Her* 
a c or b d is the thickness of the shell. 


THE EEEIPSOID, OR SPHEROID, 

Is ft solid generated by the revolution of an ellipse around either its long or its short diam. Whe* 

around the long (or transverse) diam, as at a. Fig 1, it is an oblong' or pro- l 
late spheroid; when around the short (or conjugate) one, as at m, in Fig 2, 

it is oblate. 




For the solidity in either case, mult the fixed diam or axis bv the smuti-e 

of the revolving one; and mult the prod by .5236. 3 1 c ! 






















PARABOLOIDS, ETC 


167 


THE PARABOLOID, OR PARABOLIC CONOID, 

next Fig, is a solid generated by the revolution of a parabola a cb, around its axis, c r. 

For its solidity mult the area of its base, by half its height, r c. Or mult 

together the square of the rad a t of the base; the height r c; and the number 1.5708. 


For the solidity of a frustum, 

mb g h, the ends of which are perp to the axis r c; add together the 
squares of the two diams a b and g h; mult the sum by the height r l; 
mult the prod by the decimal .3927. 

To find tlie surface of a paraboloid. 

Mult the rad a r of its base, by 6.2832; div the prod by 12 times the 
square of the height rc; call the quotp. Then add together the square 
of the rad a r, and 4 times the square of the height r c. Cube the 
sum; take the sq rt of this cube; from the sq rt subtract the cube of 
the rad a r. Mult the rem by p. 

Either the solidity, or the surface of a frustum, a b g h, when g h is 
parallel to a b, may be found by calculating for the whole paraboloid, 
and for the upper portion c g h, as two separate paraboloids, and taking their difif. 



THE CIRCULAR SPIN D 

Is a solid ab n y generated by the revolution of a circular segment 
ab n e a, around its chord a n as an axis. 

To find its solidity. 

.Rule 1. First find the area of a b e, or half the generating circular 
segment. Then to the square of a e, add the square of be; div the sum 
by b y; from the quot take b e; mult the rem by the area of a e 6; call 
theprodp. Cube a e; div the cube by 3; from the quot take p. Multthe 
rem by 12.5664. 

Rule 2. When the dist o e is known, from the center of the circle to 
the cen of the spindle, then mult that dist o e. by the area of a b e; call 
the prod p; cube a e: div the cube by 3 ; from the quot take p ; mult the 
rem by 12.5664. 

To find its surface. 

Rule 1. First find the length of the circular arc abn; and mult it by 
the dist o e from the center of the circle to the center of the spindle. Call 
the prod p. Next mult the length a n of the spindle, by the rad o b of the circle. From the prod 
takep; mult the rem by 6.2832. 

Rule 2. First find the length of the arc abn. Square a e; also square b e; add these squares 
together; div their sum by by; call the quot s; and mult it by a n; call the prod p. Next from stake 
b e ; mult the rem by the length of the arc abn. Subtract the prod from p ; mult the rem by 6.2832. 

To find the solidity of a middle zone of a circular spindle, 

As h8kp 

e2 —X <7 — ^oeX area of g h l j X 6.2832. 




LE, 



CIRCULAR RINGS. 


Volume = 
Surface = 


area of cross section of bar v \ sura of inner and outer ^ « 141503 
of which ring is made A diameters, a a and b b ^ ' 

circumference of bar y. J sum of inner and outer y g 141593 
of which ring is made A diameters, a a and bb ^ ' 








168 


LAND SURVEYING, 


LAND SURVEYING. 



d 


In surveying a tract of 
ground, the sides which com¬ 
pose its outline are desig- 
uated by numbers in the 
order in which they occur. 
That eud of each side which 
first presents itself in the 
course of the survey, may be 
called its near end ; and the 
other its far end. The num¬ 
ber of each side is placed at its 
far end. Thus, in Fig 1, the 
survey being supposed to 
commence at the corner 6, 
and to follow the direction 
of the arrows, the first side 
is 6, 1; and it3 uumber ia 
placed at its far eud at 1; 
and so of the rest. Let N S 
be a meridian line, that is, a 
north and south line; and 
K IV an east and west line. 
Then iu any side which runs 
northwardly, whether due 

north, or northeast, as side 2; or northwest, as sides 5 and 1, the dist in a due north direction 
between its uear end aud its far end, is called its northing; thus, a 1 is the uorthing of side 1; 1 b 
the uorthiug of side 2; 4 c of side 5. In like manner, if any side runs iu a southwardly direction, 
whether due south, or southcastwardly, as side 3; or southwestwardly, as sides 4 and 6. the corre¬ 
sponding dist iu a due sou th direction between its near end aud its far end, is called its southing ; thus, 
d 3 is the southing of side 3; 3 eof side 4; / 6of side 6. Both northings and southings are included in the 
general term Difference of latitude of a side; or more commonly, but erroneously, its latitude. The dist 
due east, or due west, between the near and the far end of any side, is in like manner called the easting, 
or westing of that side, as the case may be ; thus, fi a is the westing of side 1; 5/ of side 6; c5 of side 
5; e 4 of side 4 ; and h 2 is the casting of side 2; 2 d of side 3. Both eastings aud westings are included 
in the general term Departure of a side; implying that the side departs so far from a north or south 
direction. We may employ the directions (or courses, or beariugs, as they are usually called) of sides, 
ns verbs; aud say that a side norths, wests, southeasts, Ac. We shall call the uorthiugs, southings, 
Ac, the Ns, Ss, Es. aud Ws ; the latitudes, lats ; aud the departures, deps. The Traverse 

Table consists of the lats, (or Ns aud Ss;) and the deps, (or Es and Ws,) corresponding to difT angles 
or courses, for a side whose Ieugth is 1; therefore, to obtain the actual lat and dep of anv given side, 
those tnkeu from the table must be mult by the Ieugth of the side. Beyond 44°, the lats and deps 
of this table must be read upward from the bottom of the page. The angles in the table are those 
which the course or bearing of any side would make with a meridian line drawn through either end 
of said side; but it is self-evident that what would be N from one eud, would be S from the other; aud 
so of E and W ; in other words, the angle is the same at both ends; but the direction is reversed. 

Perfect accuracy is unattainable in auy operation involving the measurements of angles and dists. 
That work is accurate enough, which cannot be made more so without an expenditure more than com¬ 
mensurate with the object to be gained. The writer conceives that the accuracy essential to constitute 
practically fair surveying is purely a matter of dollars and cents. In the purchase and sale of tracts 
of land, such as farms,&c, an uncertainty of aboutl part in 200 respecting the content, and consequently 
respecting the price, probably never prevents a transfer; and on this principle we assume that a survey 
which proves itself within that limit, may ordinarily be regarded as accurate enough. There is no 
great difficulty in attaining this limit, which, if exceeded, is the result of bad work Many circum- 
stances combine to render tnfiing errors absolutely unavoidable:* thev alwavs become apparent 

t, W0r i° o U , , e ♦ e i n °, t fJ ani1 ? ince the nla P or C |ot ° r the survey, and the calcula- 
tions for ascertaining the content, should be consistent within themselves, we do what is usuallv called 
correcting the errors, but what in fact is simply humoring them in, no matter how scientific the pro- 
cess may appear. I\ e distribute them all around the survey. Two methods are used for this purpose, 
both based upon precisely the same principle; one of them mechanical, by means of drawing; tho 
other more exact, but much more troublesome, by calculation. We shall describe both ; but will state 
now, that by proportioning the scale of the plot in the following manner, the mechanical met>-® 

,d ° f a Ta- eC !, draftSman - 8uf »ciently exact for all ordinary purposes Add all 
the sides in feet together, and div the sum by their number, for the average length. r>iv this average 
by 8; the quot will be the proper scale in feet per inch. In other words, take about 8 ins to represent 
an aveiage side.t We shall take it for granted, that an engineer does not consider it accurate work to 


* A 100 ft measuring chain may vary its length 5 feet per mile, between winter and summer by 
mere change or temperature; and by this alone we shall make a difference of about 1+ acres in a lot 

? IfnTT' wh m a C0Rtai ? s acres. Even this error amounts to 1 acre in 533. Not one farmer 
in a hundred would dream of paying for a scrupulously correct survey and plot of his propertv. 

t It will seldom happen that precisely this quotient will be adopted for a scale. For iustance if the 
quot should be 738 f «et, or 83 feet, per inch, we should adopt the most convenient near number if 
smaller the better, as 700, or 80; or rather more than 8inches to a side. With a scale so proportioned 
and with good drawing instruments, the error in protracting (excluding of course errors of the field 
work) will rarely exceed about part of the periphery of the plot; and the area may be found 

mechanically by dividing the plot into triangles, within ^ 1 part or the truth. This remark applies 

particularly to such plots as may be protracted, and computed within a period so short as nm 
to allow the paper to contract or expaud appreciably by atmospheric changes, a larger" seal 
will insure proportionally greater accuracy. The young assistant should practise nlotti a 
from perfectly accurate data; as, for iustauce, from the example given iu the table, p. 175 or 
















1.AND SURVEYING, 


169 


measure his angles to the nearest quarter of a degree, which is the usual practice among land-survey 
ors. They can, by means of the engineer’s transit, nowin universal use on our public works, be readily 
measured within a minute or two; and being thus much more accurate than the compass courses, 
(which cannot be read off so closely, and which are moreover subject to many sources of error,) they 
serve to correct the latter in the office. The noting of the courses, however, should not be confined to 
the nearest quarters of a degree, but should be read as closely as the observer can guess at the minutes. 
The back courses also should be taken at every corner, a3 an additional check, and for the detection 
of local attraction. It is 
well in taking the com¬ 
pass bearings, to adopt ft o 

as a rule, always to point Z 

the north of the compass- 
' box toward the object 
whose bearing is to be 
taken, and to read off 
from the north end of the 
needle. A person who 
uses indifferently the N 
and the S of the box, and 
of the needle, will be very 
liable to make mistakes. 

It is best to measure the 
least angle (shown by 
dotted arcs, Fig 2,) at the 
corners; whether it be 
exterior, as that at corner 
5; or interior, as all the 
others; because it is al¬ 
ways less than 180°; so 
that there is less danger 
of reading it off incor¬ 
rectly, than if it exceeded 
180°; taking it for grant¬ 
ed that the transit instrument is graduated from the same zero to 180° each way ; if it is graduated 
from zero to 360 3 the precaution is useless. When the small angle is exterior, subtract it from 360° 
for the interior one. 

Supposing the field work to be finished, and that we require a plot from which the contents may 
be obtained mechanically, by dividing it into triaugles, (the bases and heights of which may be 
measured by scale, and their areas calculated one by one,) a protraction of it may be made at once 
from the field notes, either by using the angles, or by first correcting the hearings by means of the 
angles, and then using them. The last is the best, because in the first the protractor must be moved 
to each angle; whereas in the last it will remain stationary while all the bearings are being pricked 
off. Every movement of it increases the liability to errors. The manner of correcting the bearings 
is explained on the next page. 

In either case the protracted plot will certainly not close precisely; not only in consequence of errors in 
the field work, but also in theprotractiug itself. Thus the last side, No 6, Fig 2, instead of closing in at 
corner 6, will end somewhere else, say, for instance, at t ; the dist t 6 being the closing error, which, 
however, as represented in Fig 2, is more than ten times as great, proportionally to the size of the 
survey, as would be allowable in practice. Now to humor-in this error, rule through every corner 
a short line parallel to tfi; and, in all cases, in the direction from t (wherever it may be) to the 
starting point 6. Add all the sides together ; and measure f 6 by the scale of the plot. Then begin¬ 
ning at corner 1, at the far end of side 1, say, as the 

Sum of all . Total closing . . . . Error 

the sides • error t 6 • • clae 1 • for side 1. 



Lay off this error from 1 to a. 

Sum of all . 

the sides • 


Then at corner 2, say, as the 
Total closing , . Sum of 

error 1 6 • • sides 1 and 2 


Error 
for side 2. 


Which error lay off from 2 to h ; and so at each of the corners; always using, as the third term, the 
sum of the sides between the starting point and the given corner. Finally, joiu the points a, b, c, 
d, e, 6; and the plot is finished. 

The correction has evidently changed the length of every side; lengthening some and shortening 
others. It has also changed the angles. The new lengths and angles may with tolerable accuracy 
be found by means of the scale and protractor; and be marked on the plot instead of the old ones. 


from those to be found in books on surveying. This is the only way in which he can learn what is 
meant by accurate work. His semicircular’protractor should be about 9 to 12 ins in diam. and gradu¬ 
ated to 10 min. His straight edge and triangle should be of metal; we prefer German silver, which 
does not rust as steel does; and they should be made with scrupulous accuracy by a skilful instru¬ 
ment-maker. A very fine needle, with a sealing-wax head, should be used for pricking off dists and 
angles; it must be held vertically; and the eye of the draftsman must be directly over it. The lead 
pencil should be hard (Faber’s No. i is good for protracting), and must be kept to a sharp point by 
rubbing on a fine file, after using a knife for removing the wood. The scale should be at least as long 
as the longest side of the plot, and should be made at the edge of a strip of the same paper as the plot 
is drawn on. This will obviate to a considerable extent, errors arising from contraction and expan¬ 
sion. Unfortunately, a sheet of paper does not contract and expand in the same proportion length¬ 
wise and crosswise, thus preventing the paper scale from being a perfect corrective. In plots of com¬ 
mon farm surveys, &c, however, the errors from this source may be neglected. For such plots as may 
)e protracted, divided, and computed within a time too short to admit of appreciable change, theordi- 
lary scales of wood, ivory, or metal may be used; but satisfactory accuracy cannot be obtained with 
:hem on plots requiring several days, if the air be meanwhile alternately moist and dry, or subject to 
considerable variations in temperature. What is called parchment paper is worse in this respect than 
good ordinary drawing-paper. 

With the foregoing precautions we may work from a drawing, with as much accuracy as is usually 
attained in the field work. 






170 


LAND SURVEYING. 


When the plot has many sides, this calculating the error for each oi them becomes tedious; and 
since, in a well-performed survey aud protraction, the entire error will be but a very small quantity, 
jit should not exceed about —part of the periphery,) it may usually be divided among the sides by 
merely placing about >4, 14 , and % of it at corners about J 4 i and % way around the plot; and at 

intermediate corners propor¬ 



tion it by eye. Or calculation 
may be avoided entirely by 
drawing a line a 6 of a length 
equal to the united lengths 
of all the sides; dividing it 


into distances a, 1; 1, 2 ; Ac, equal to the respective sides. Make 6 c equal to the entire closing error; 
join a c ; and draw 1,1’; 2, 2’, &c, which will give the error at each corner. 

When the plot is thus completed, it may be divided by fine pencil lines into triangles, whose 
bases and heights may be measured by the scale, in order to compute the coutents. With care in 
both the survey aud the drawing, the error should not exceed about _part of the true area. At 
least two distinct sets of triangles should be drawn and computed, as a guard against mistakes; and if 
the two sets differ in calculated contents more than about part, they have not been as carefully 
prepared as they should have been. The closing error due to imperfect field-work, may be accurately 
calculated, as we shall show, and laid down on the paper before beginning the plot; thus furnishing 
a perfect test of the accuracy of the protraction work, which, if correctly done, will not close at the 
point of beginning, but at the point which indicates the error. But this’ calculation of the error, by 
a little additional trouble, furnishes data also for dividing it by calculation among the diff sides : 
besides the means of drawing the plot correctly at once, without the use of a protractor ; thus ena¬ 
bling us to make the subsequent measurements and computations of the triangles with more cer¬ 
tainty. 

We shall now describe this process, but would recommend that even when it is employed, and 
especially in complicated surveys, a rough plot should first be made and corrected, bv the first of the 
two mechanical methods already alluded to. It will prove to be of great service in using the method 
by calculation, inasmuch as it furnishes an eye check to vexatious mistakes which are otherw ise apt 
to occur : for, although the principles involved are extremely simple, and easily remembered when 
once understood, yet the continual changes in the directions of the sides will, without great care 
cause us to use Ns instead of Ss; Es instead of Ws, &c. 

We suppose, then, that such a rough plot has been prepared, and that the angles, bearings and 
distances, as taken from the field book, are figured upon it in lend pencil. 

Add together the interior angles formed at all the corners : call their sum a Mult the number of 
sides by 180°; from the prod subtract 360° : if the remainder is equal to the sum o it is a proof that 
the angles have been correctly measured.* This, however, will rarelv if ever occur • there will 
always be some discrepancy ; but if the field work has been performed with moderate care this will 
not exceed about two min for each angle. In this case div it in equal parts among all the angles 
adding or subtracting, as the case may be, unless it amounts to less than a min to each angle w hen 
it may he entirely disregarded in common farm surveys. The corrected angles mav then be marked 
on the plot in ink, and the pencilled figures erased. We will suppose the cwrected ones to be as 

8I10WI1 1U Jr lg o» 


Next, by means of these 
corrected angles, correct the 
bearings also, thus, Fig 3 ; 
Select some side (the longer 
the better) from the two ends 
of which the bearing and the 
reverse bearing agreed ; thus 
showing that that bearing 
was probably not influenced 
by local attraction. Let side 
2 be the one so selected ; as¬ 
sume its bearing, N 75°32' E, 
as taken on the ground, to be 
correct; through either en<j 
of it, as at its far end 2, draw 
the short meridian line; par¬ 
allel to which draw others 
through every corner. Now, 
having the bearing of side 2, 
N 75° 32' E, and requiring 
that of side 3, it is plain that 
the reverse bearing from cor¬ 
ner 2 is S 75° 32' W; and 
that therefore the angle 1, 2, 
771 , is 75°32'. Therefore, if we 
take 75° 32' from the entire 
corrected angle 1,2,3, or 144° 
57', the rem 69° 25' will be 
the angle m 23 ; consequently 
„ .. ,, . , the bearing of side 3 must be 

, *'• * <or finding the bearing of side 4, we now have the angle 23 a of the reverse bearing of 
Buie 3, also equal to t>9‘ 25'; and if we add this to the entire corrected angle 234, or to69° 32', we have 
the angle a .14—69 25' -)- 69° 32' = 138° 57' ; which taken from 180°, leaves the angle 6 34 = 41°3'; 
consequently the bearing of side 4 must be S 41° 3' W. For the bearing of side 5 we now have the 
angle 34 c —41 3', which taken from the corrected angle 345, or 120° 43', leaves the angle c 45 ~79° 
40 ; consequently the bearing of side5 must be N 79° 40' W. At corner 5, for the bearing of side 6 
we have the angle 45 d = 40', w'hioh taken from 133° 10', leaves the angle d 56 = 53°30' ; conse¬ 

quently the bearing of side 6 must be S 53° 30' W. And so with each of the sides, nothing but 



_* Because in every straight-lined figure the sum of all its interior angles is equal to twice as manv 
right angles as the figure has sides, minus 4 right angles, or 360°,. 3 

















. LAND SURVEYING, 


171 


<2j 




careful observation is necessary to see how the several angles are to be employed at each corner. 
Rules sire sometimes given for this purpose, but unless frequently used, they are soon forgotten. 
The plot mechanically prepared obviates the necessity for such rules, inasmuch as the principle of 
proceeding thereby becomes merely a matter of sight, and teuds greatly to prevent error from using 
the wrong bearings; while the protractor will at once detect any serious mistakes as to the angles, 
and thus prevent their being carried farther along. After having obtained all the corrected bearings, 
they may be figured on the plot instead of those taken in the field. Thev will, however, require a 
still further correction after a while, since they will he affected by the adjustment of the closing error. 

A\*e now proceed to calculate the closing error <6 of Fig 2, which is done on the principle that in a 
correct survey the northings will be equal to the southings, and the eastings to the westings. Pre¬ 
pare a table of 7 columns, as below, and in the first 3 cols place the numbers or the sides, and their cor¬ 
rected courses; also the dists or lengths of the sides, as measured on the rough plot, ifsuchaone 
has been prepared ; but if not, then as measured on the ground. Let them be as follows: 


Side. 

Bearing. 

Dist. Ft. 

Latitudes. 

Departures. 

N. 

S. 

E. 

W. 

1 

N 16° 40' W 

1060 

1015.5 



304. 

2 

N 75° 32' E 

1202 

300.3 


1163.9 


3 

S 69° 25' E 

1110 


390.2 

1039.2 


4 

S 41° 3' W 

850 


641. 


558.2 

5 

N 79° 40' W 

802 

143.9 



789. 

6 

S 53° 30' W 

705 


419.3 


566.7 




1459.7 

1450.5 

2203.1 

2217.9 




1450.5 



2203.1 




9.2 

Error in 

Error in 

14.8 





Lat. 

Dep. 



Now find the N, S, E, W, of the several sides, and place them in the corresponding four columns, 
thus: By means of the Traverse Table tiud out the lat and dep for the augle of each course. Mult each 
of them by the length of the side; and place the prod in the corresponding col of N, S, E, W. Thus, 
for side 1, which is 1060 feet long, the latitude from the traverse table for 16° 40' is .9580; and the 
departure is .2868; and .9580 X 1060 = 1015.5 lat; which, since the side norths, we put in the N 
col. Again, .2868 X 1060 = 304 dep; which, since the side wests, we put in the W col. Proceed 
thus with all. Add up the four cols; find the diff between the N and S cols; aud also betw een 
the E and W ones. In this instance we find that the Ns are 9.2 feet greater than the Ss; and that 
the Ws are 14.8 ft greater than the Es; in other words, there is a closing error which would cause a 
correct protraction of our first three cols, to terminate 9.2 feet too far north of the starting point; and 
24.8 feet too far west of it. So that by placing this error upou the paper before beginning to protract, 
we should have a test (%r the accuracy of the protracting work ; but, as before remarked, a little more 
trouble will now'enable us to div the error proportionally among all the Ns, Ss, Es, and Ws, and thereby 
give as data for drawing the plot correctly at once, without using a protractor at all. 

To divide the errors, prepare a table precisely the same as the foregoing, except that the hor spaces 
are farther apart; and that the addings-up of the old N, S, E, W columns are omitted. The additions 
here noticed are made subsequently. 

The new table is on the next page. 


Rkmark. The bearing'and the reverse bearing from the two ends 
of a line will not read precisely the same angle; and the difference varies with the 
latitude and with the length of the line, but not in the same proportion with either. 
It is, however, generally too small to be detected by the needle, being, according to 
Gummere, only three quarters of a minute in a line one mile long in lat 40°. In 
higher lats it is more, and in lower ones less. It is caused by the fact that meridians 
or north and south lines are not truly parallel to each other; but would if extended 
meet at the poles. 

Hence the only bearing that can be run in a straight line, 

with strict accuracy, is a true N and S one; except on the very equator, where alone a due E and W 

one will also be straight. But a true curved E ami W line may be found 

anywhere with sufficient accuracy for the surveyor's purposes thus. Having first by means or the N 
star p 177, or otherwise got a true N and S bearing at the starting point, lay off from it 90°, for a true 
E and W bearing at that point. 'This E aud W bearing will be tangent to the true E and W curve. 
Run this tangent carefully ; aud at intervals (say at the end of each mile) lay off from it (towards 


offsets will mark poiuts in the true E aud W curve. 

Latitude M or S. 

15° 20° 25° 30° 35° 40° 45° 50° 55° 

dMlsets in ft one mile from starting point. 

.179 .243 .311 .385 .467 .559 .667 .795 .952 


5^ 

10° 

.058 

.118 


point. 

These 

60=> 

65° 

1.15 

1.43 


Or, any offset in ft = .6666 X Total Dist in miles 2 X Nat Tang of Lat. 

A rhumb line is any one that crosses a meridian obliquely, that is, is 

neither due N and S, nor E and W. 

































172 


LAND SURVEYING, 


Side. 

Bearing. 

Dist. Ft. 

Latitudes. 

Departures. 



N. 

S. 

E. 

W. 

1 

N 10° 40' W 

1060 

1015.5 

1.7 



304.0 

2.7 




1013.8 



... 301.3 

2 

N 75° 32' E 

1202 

300.3 

1.9 


1163.9 

3.1 


3 

S 69° 25' E 

1110 

298.4 

•-•••••• •••••• 

390.2 

1.8 

... 1167.0 

1039.2 

2.9 


4 

S 410 3 ' w 

S50 


392 ... 

641.0 

1.3 

... 1042.1 

558.2 

2.2 





642.3... 



6 

N 79° 40' W 

802 

143.9 

1.3 



789.0 

2.1 




142.6... 

<•••••••••••«• 

• •••• •••••••• 

... 786.9 

6 

S 53° 30' W 

705 


419.3 

1.1 


566.7 

1.8 





420.4... 


564 9 





! 




5729 
Sum of 
Sides. 

1454.8 

Cor’d Ns. 

1454.7 

Cor’d Ss. 

2209.1 

Cor’d Es. 

2209.1 

Cor’d Ws. 


Now wc have already found by the old table that the Ns and the Ws are too long; consequently 
they must be shortened ; while the Ss, and Es, must be lengthened; all in the following proportions : 
As the 

Sum or all . Any given .. Total err of . Err of lat, or dep, 
the sides • side • • lat or dep • of given side. 

Thus, commencing with the lat of side 1, we have, as 

Sum of all the sides. . Sidel. .. Total lat err. . Lat err of side 1 

5729 • 1060 • • 9.2 • x.7 

Now as the lat of side 1 ia north, it must be shortened ; hence it becomes = 1015.5_1.7 = 1015 8 as 

figured out in the new table. Again we have for the departure of side 1, 

Sum of all the sides. . Sidel. .. Total dep err. . Dep err of side 1 

5729 • 1060 • • 14.8 • 2.7 

Now as the dep of sidel is west, it must be shortened; hence it becomes 304 — 2 7 =: 301 3 as fitmrpd 
out in the new table. ’ 


Proceeding thu3 with each 
side, we obtain all the corrected 
lats and deps as shown in the 
new table ; where they are con¬ 
nected with their respective 
sides by dotted lines; but in 
practice it is better to cross out 
the original ones when the cal¬ 
culation is finished and proved. 
If we now add up the 4 cols of 
corrected N, S, E,W,we find that 
the Ns = the Ss; and the Es = 
theirs; thus proving that the 
work is right. There is, it is 
true, a discrepancy of .1 of a ft 
between the Ns, and the Ss; but 
this is owing to our carrying 
out the corrections to only one 
decimal place ; and is too small 
to be regarded. Discrepancies 
of 3 or 4 tenths of a foot will 
sometimes occur from this 
cause; but may be neglected. 
The corrected lats and deps 
must evidently change the 

side; but without knowing either of these, we can now plot the survey by*menus of Unfcorrected 



Fig. 
















































LAND SURVEYING, 


173 


a ^ one * The principle is self-evident, explaining itself. First draw a meridian line 
NS, r lg 4 ; and upou it tix on a point 1, to represent the extreme, west* corner of the survey. 

1 hen from the point 1, prick off by scale, northward, the dist 1, 2 —the corrected northing 298.4 
of side 2, taken from the last table ; from 2' southward prick off the dist 2', 3', the corrected south¬ 
ing 392 of side 3; from 3' southward prick off 3% 4', — southing 642.3 of side 4; from 4' northward 
prick off 4', 5' — northing 142.6 of side 5; from 5' prick off southward 5', 6'“southing of side 6.f 
Then from the poiuts 2', 3\ 4', 5', 6 f , draw indefinite lines due eastward, or at right angles to the 
meridian line. Make by scale, 2', 2 =: corrected departure of side 2; and join 1, 2. Make 3 r , 3 —dep 
of side 2-f-dep of side 3 ; and join 2, 3; make 4', 4 = 3', 3 — dep of side 4; and join 3, 4; make 5', 5 
4 , 4 dep of side 5; and join 4, 5 ; make 6', 6 — 5', 5 — dep of side 6; and join 5, 6 t Finally join 
6, 1; and the plot is complete. If scrupulous accuracy is not required, the contents may be found by 
> the mechanical method of triangles; the bearings, by the protractor; and the lengths of the sides, 
by the scale ; all with an approximation sufficient for ordinary purposes; and perhaps quite as close 
aa by the method by calculation, when, as is customary, the bearings are taken only to the nearest 

quarter of a degree. We have already said that with a scale of feet per inch = _ verage lengttl of Sldeg » 

1 8 
the error of area need not exceed the 'Jo'ffth part. 

But if it is required to calculate the area of the corrected survey with rigorous exactness, it may 
be done on the following 
principle, (see Fig 5.) If a m 
meridian line N S be sup- N 
posed to be drawn through 
the extreme west corner 1 of 
a survey; and lines (called 
middle distances) drawn (as 
the dotted ones in the Fig) 
at right angles to said me¬ 
ridian, from the c enter of 
each side of the survey; 
then if each of the middle 
dists of such sides as have 
northings, be mult‘by the 
corrected northing of its cor¬ 
responding side; and if each 
of the middle dists of such 
sides as have southings, be 
mult by the corrected south¬ 
ing of its corresponding 
side; if we add all the north 
prods into one sum ; and all 
the south prods into another 
> sum; and subtract the least 

of these sums from the great- w 6 

est, the rem will be the area 



* The extreme east corner would answer as well, with a slight change in the subsequent oper¬ 
ations, as will become evident. 

t Instead of pricking off these northings and southings in succession, from each other, it will be 
wore correct in practice to prepare first a table showing how far each of the poiuts 2',3', Ac, is north 
or south from 1. This being done, the poiuts can be pricked off north or south from 1, without mov¬ 
ing the scale each time; and of course with greater accuracy. Such a table is readily formed. Rule 
it as below; and in the first three columus place the numbers of the sides (starting with side 2 from 
point I;) and their respective corrected northings and southings. The formation of the 4th and 5th 
cols by menus of the 3d and 4th ones, explains itself. Its accuracy is proved by the final result 
being 0. 






Dist X or S from Point 1. 

Side. 

N. lat. 

S. lat. 


N. 

S. 

2 

298.4 



298.4 


3 


392. 



93.6 

4 


642.3 



735.9 

5 

142.6 




593.3 

6 


420.4 



1013.7 

1 

1013.8 



000.0 

000.0 


I A similar table should be prepared beforehand for the dists of the points 2, 3, 4, Ac, east from the 
meridian line. It is done in the same manner, but requires one col less, as all the dists are on the 
same side of the mer line. Thus, starting from point 1, with side 2: 


Side. 

E. dep. 

W. dep. 

2 

1167.0 


3 

1042.1 


4 


556.0 

5 


786.9 

6 


564.9 

1 


301.3 


Dist east from 
meridian line. 


1167.0 

2209.1 

1653.1 
866.2 
301.3 
000.0 


This work likewise proves itself by the final result being 0. 


12 



































174 


LAND SURVEYING. 


of the surrey.* The corrected northings and southings we have already found ; as also the eastings 
and westings. The middle dists are found by means of the latter, by employing their halves; adding 
half eastings, and subtracting half westings. Thus it is evident that the middle dist 2' of side 2, is 
equal to half the easting of side 2. To this add the other half easting of side 2, and a half easting 1 
of side 3; and the suiu is plainly equal to the middle dist 3' of side 3. To this add the other half 
easting of side 3, and subtract a-half westing of side 4, for the middle dist 4' of side 4. From this 
subtract the other half westing of side 4, and a half westing of side 5, for the middle dist 5' of Bid? 

5; and so on. The actual calculation may be made thus: 

Half easting of side 2 = -■ = 583.5 E = mid dist of side 2. 

2 583.5 E 


1042.1 

Half casting of side 3= - 

2 


1167.0 E 
521.0 E 


1688.0 E 
521.0 E 


mid dist of side 3. 


556 2209.0 E 

Hair westing of side 4 rz — = 278.0 W 

2 - 

1931.0 E = mid dist of side 4. 
278.0 W 

786.9 1653.0 E 

Hair westing of side 5 = -= 393.5 W 

2 - 

1259.5 E = mid dist of side 5. 

393.5 W 


564.9 866.0 E 

Half westing of side 6 =-= 282.4 W • 

2 - 

583.6 E r= mid dist of side 6. 

282.4 W 


301.3 301.2 E 

Half westing of side 1 =-= 150.6 W 

2 - 

150.6 E — mid dist of side 1. 





- 

1 


The work always proves itself by the last two resnlts being'equal. 

like ? hu folU>w i n &- in the first 4 cols of which place the numbers or the sides, 
or ,‘ e “or.tbtngs. and southings. Mult each middle dist by its corresponding northing 

or southing, and place the products in their proper col. Add up each col; subtract the least from the 


Side. 


1 

2 

3 

4 

5 

6 


Middle dist. 


150.6 

583.5 
1688 
1931 
1259.5 

583.6 


Northing. 


1013.8 

298.4 


142.6 


Southing. 


392 

642.3 


420.4 


North prod. 


152078 

174116 


179605 


506399 


Sonth prod. 


661696 

1240281 


245345 


2147322 

506399 


43560)1640923(37.67 Acres, 


Pi oof. To illustrate the principle upon which this 
rule is based let ah, be, and ca, Fig 6, represent in 
order the 3 sides of the triangular plot of a survey, with 
a meridian line df drawn through the extreme west cor- 
wer * a. Ijet hues b d and cf be drawn from each corner, 
perp to the meridian line; also from the middle of each 
side draw lines we, mn, so, also perp to meridian ; and 
representing the middle dists of the sides. Then since 
the sides are regarded in the order ab,bc,ca, it is 
plain that ad represents the northing of the side ah: 
fa the northing of ca; and d/ the sonthing of he. 

if 7 e mu l* the northin 9 ad of the side a h, by its 
and dist ew, the prod is the area of the triangle ahd. 
Jn .ike manner the northing fa of the side ca, mult by 
its mid dist s o, gives the area of the triangle a cf. Again 
the southing df of the side h c, mult by its mid dist mn, 
gives the area cf the entire tig d hefd. If from this 
area we subtract the areas of the two triangles ahd 
and acf, the rem is evidently the area of the plot a he. 
so with any other plot, however complicated. 













































LAND SURVEYING, 


175 


greatest. Tlie rera will be the area of the survey in sq ft; which, div by 435G0, (the number of sq ft 
In an acre,) will be the area in acres; in this instance, 37.67 ac. 

it now remains only to calculate the corrected bearings and lengths of the sides of the survey, all 
of which are necessarily changed by the adoption of the corrected lats and deps. To fiud the bearing 
of any side, div its departure (E or W) by its lat (N or S); in the table of nat tang, find the quot; 

301.3 W 

the angle opposite it is the reqd angle of bearing. Thus, for the course of side 1, we have - - 


r= .2972—nat tang; opposite which in the table is the reqd angle, 16° 33'; the bearing, therefore, is 
N 16 3 33' W. 


Again : for the dist or length of any side, from the table of nat cosines take the cos opposite to 
the angle of the corrected bearing ; divide the corrected lat (N or S) of the side by the cos. Thus, 
for the dist of side 1, we fiud ojiposite 16° 33', the cos .9586. And 


Lat. Cos. 

1013.8 -7- .9586 = 1057.6 the reqd dist. 


The following table contains all the corrections of the foregoiug survey ; consequently, if the bear¬ 


Side. 

Bearing. 

Dist. Ft. 

1 

N 16° 33' W 

1057.6 

2 

N 75° 39' E 

1204.0 

3 

S 69° 23' E 

1113.3 

4 

S 40° 53' W 

849.6 

5 

N 7 9° 44' W 

800.1 

6 

S 53° 21' W 

704.3 


ings and dists are correctly plotted, they will close perfectly. The young assistant is advised to 
practise doing this, as well as dividing the plot into triangles, and computing the content, in this 
manner he will soon learn w'hat degree of care is necessary to insure accurate results. 

The following hints may often be of service. 

1st. Avoid taking bearings and 
dists along a circuitous bound¬ 
ary line like a b c, Fig 7; but run 
the straight line a c; and at 
right angles to it, measure off¬ 
sets to the crooked line. 5Jd. 

Wishing to survey a straight 
line from a to c, but being una¬ 
ble to direct the instrument 
precisely toward c, on account 
of intervening woods, or other 
obstacles; first run a trial line, 
as a m, as nearly in the proper 
direction as can be guessed at. 

Measure m c, and say, as a m is to m c, so is 100 ft to ? Lay off a o equal to 100 ft, and o s equal 
to ? ; and run the final line a s c. Or. if m c is quite small, calculate offsets like o s for every 100 ft 
along a m, and thus avoid the necessity for running a second line. 8d. When c is visible from a, but 
the intervening ground difficult to measure along, on account of marshes, &c, extend the side y a 
to good ground at t: then, making the angle y t d equal to y a c, run the line t n to that point d at 
which the angle nd c is found by trial to be equal to the angle at d. It will rarely be necessary to 
make more than one trial for this point d ; for, suppose it to be made at x, see where it strikes a c a t 
f; measure i c, and continue from x, making xd~ic. 4th. In case of a very irregular piece of 
land, or a lake, Fig 8, surround it by straight lines. Survey these, and at right angles to them, 
measure offsets to the crooked boundary. 5th. Surveying a straight line from w toward y, lg 9. 


t X d 

I ------»-p- 




Fig. 9. 

an obstacle. o, is met. To pass it, lay off a right angle w t u ; measure any t u 1 make ftir = 
90°; measure « v ; make u v i = 90°; make v i = t u\ make v i y ~ 9*>°. Then is ti-uv, and 
iy is in the straight liue. Or, with less trouble, at g make t g a = 60°; measure any g a; make 
« a s =60° ; and as = ga: make a s i - 60°. Then is g s = g a or a s ; and i *, continued toward 
«, is in the straight line. 6th. Being between two objects, m and n, and wishing to place myself in 
range with them, I lav a straight rod c b on the ground, and point it to one of the objects ni; then 
going to the end r. 1 find that it does not point to the other object. By successive trials, 1 find the 
position e d, in which it points to both objects, and consequently is in range with them. If no rod 



























176 


LAND SUllVEYING, 


is at hand, two stones will answer, or two chain-pins. A plumb-line (a pebble tied to a piece of 
thread) will add to the accuracy of ranging the rod, or stones, &c. 


THE FOLLOWING TABLE 


gives deductions or udditlons to be made every 100 ft as actually chained along sloping 

ground, in order to reduce the sloping measurements to horizontal ones. Even when it is so nearly 
level that the eye cannot detect the slope, an over-measurement of an inch or two in 100 ft may 
readily occur. It is plain, that, if we measure all the undulations of the ground, we shall get 
greater totals than if we measure hor, as is supposed always to be done ; but since few surveyors 
pretend to measure hor until the slope becomes appareut to the eye, their lines are usually too iong 
by from one to two ins in 100 feet. To counteract this to some extent, chains are frequently mado ' 
from one to two ins longer than 100 feet; and for ordinary purposes the precaution is a good one. 
When greater accuracy is required the chainmen should be attended by a third person, with a rod 
and slope-level, for taking the inclinations of the ground. These deductions being made, the remain¬ 
der will be the actual hor dist. 


Fiff. 10 K- 


For example, in Fig lOJsj, each 100 ft a o measured up or down the 
slope a e plainly corresponds to the shorter horizontal distance a c; the 
difference or deduction being c n. Taking a o as Rad, then a c is the 
cosine, and c u the versed sine of the angle e an of the slope. There¬ 
fore a o multiplied by the nat. cosine of the angle e a n gives the reduced 
hor dist a c ; which taken from a o gives the deduction cn of ourtable. 

But If while chaining along the slope n e we wish to drive 
stakes that shall correspond with hor diftta a n of 100 ft, it is evident 
that we must add c n to each 100 ft a o, as shown at x e; and the stake 

must be driven at e. instead of at o. Observe that xe = cn must 

. be measured horizontally. 

When the ground is very sloping, all this calculation may be avoided where great accuracy is not 
required, by actually bolding the chain horizontal, as nearly as can be judged by eye, aud finding 

by means of a plumb line, where its raised end would strike the ground. A whole chain at a time 

cannot be measured in this way; but shorter distances must be taken as the ground reauircs • at 
times, on very steep ground, not more than 5 or 10 feet. See note, p 113. ’ 



Table of Deductions or Additions to be made per 100 feet, 
in chaining* over sloping- ground. 

IW ORDER TO REDUCE THE INCLINED MEASUREMENTS TO HORIZONTAL ONES 

See pp 354, 723, 724, and 725 for other tables. 


Slope 

in 

Deg. 

Deduct 

Feet. 

Rise in 
100 ft 
hor. 

Slope 

in 

Deg. 

Deduct 

Feet. 

Rise in 
100 ft 
hor. 

Slope 

in 

Deg. 

Deduct 

Feet. 

Rise in 

100 rt 

hor. 

Slope 

in 

Deg. 

Deduct 

Feet. 

Rise in 
100 ft 
hor. 

x 

X 

X 

1 

X 

X • 
X 

2 

X 

X 

X 

s 

X 

X 

X 

4 

X 

X 

X 

5 

.001 

.004 

.009 

.015 

.024 

.034 

.047 

.061 

.077 

.095 

.115 

.137 

.161 

.187 

.214 

.244 

.275 

.308 

.343 

.381 

.436 

.873 

1.309 
1.746 
2.182 
2.619 
3.055 
3.492 
3.929 
4.366 
4.803 
5.241 
5.678 
6.116 
6.554 
6.993 
7.431 
7.870 

8.309 
8.749 

X 

X 

X 

6 

X 

X 

X 

7 

X 

X 

X 

8 

X 

X 

X 

9 

X 

X 

X 

10 

.420 

.460 

.503 

.548 

.594 

.643 

.693 

.745 

.800 

.856 

.913 

.973 

1.035 

1.098 

1.164 

1.231 

1.300 

1.371 

1.444 

1.519 

9.189 

9.6*29 

10.07 

10.51 

10.95 

11.39 

11.84 
12.28 
1*2.72 

13.17 
13.61 
14.05 
14.50 

14.95 

15.39 

15.84 
16.29 
16.73 

17.18 
17.63 

X 

X 

X 

11 

X 

X 

X 

1 *2 

X 

X 

X 

13 

X 

X 

X 

14 

X 

X 

X 

15 

—I- 

1.596 

1.675 
1.755 
1.837 
1.9*21 
2.008 
2.095 

2.185 
2.277 
2.370 
2.466 
2.563 
2.662 
2.763 
2.866 
2.970 
3.077 

3.185 
3.295 
3.407 

18 08 
18.53 
18.99 
19.44 
19.89 
20.35 
20.80 

21.26 
21.71 
2*2.17 
2*2.63 
23.09 
23 55 
24.01 
24.47 
24.93 
25.40 
25.86 
26.33 
26.79 

X 

X 

X 

16 

X 

X 

X 

17 

X 

X 

18 

X 

X 

X 

19 

X 

X 

X 

20 

3.521 

3.637 

3.754 

3.874 

3.995 

4.118 

4.243 

4.370 

4.498 

4.6*28 

4.760 

4.894 

5.030 

5.168 

5.307 

5.448 

5.591 

5.736 

5.882 

6.031 

27.26 
27.73 
28.20 
28.67 
29.15 
29.62 
30.10 
30.57 
31.05 
31.53 
3*2.01 
3*2.49 
32.98 
33.46 
33 95 
34.43 
34.92 
35.41 
35.90 
36.40 


Chain and Pins. 


mering into hard ground, the pins may be of this shape 
and size, 11 or 12 ins long, % inch thick, % wide, head 
2H wide, with a circular hole of 1 % dinni. Each nin 
should have a strip of bright red flannel tied to its top 
Fig*. 11. that it may be readily found among grass, Ac by the 

hind chainniau. The length of the chain shouid b* 

for this purpose. While locating it is well to h ? nv» < \h Ver E f ® W da * VS ’ au<1 t, ? e tar Set-rod may be used 

st “‘ *•«o t «. L^S5f;iiWio» bXJ? c iK szz, r..si*«r.“a,! 00 
































LAND SURVEYING 


177 


Nat Sines of Polar l>ists of Polaris or X. Star. 


Year. 

Siue. 

Year. 

Sine. 

Year. 

Sine. 

Year. 

Sine. 

Year. 

Sine. 

Year. 

Sine. 

1880 

1881 

1882 

.0232 

.0231 

.0230 

1883 

1884 

1885 

.0229 

.0229 

.0228 

1886 

1887 

1888 

.0227 

.0226 

.0225 

1889 

1890 

1891 

.0224 

.0223 

.0222 

1892 

1893 

1894 

.0221 

.0220 

.0219 

1895 

1896 

1897 

.0218 

.0217 

.0216 


Nat Secants of North Latitudes. 


Lat. 

Sec. 

Lat. 

Sec. 

Lat. 

Sec. 


Sec. 

Lat. 

Sec. 


Lat. 

Sec. 

0° 

1.000 

24° 

1.095 

I/O 

1.196 

4° 

1.296 

4° 

1.433 


52° 

1.624 

2 

1.001 

4 

1.099 


1.199 

% 

1.301 

46 

1.440 


4 

1.633 

4 

1.002 

25 

1.103 

k 

1.203 

40 

1.305 

4 

1.446 


4 

1.643 

5 

1.004 

4 

1.108 

34 

1.206 

4 

1.310 

4 

1.453 


4 

1.652 

6 

1.006 

26 

1.113 

4 

1.210 

4 

1.315 

4 

1.460 


53 

1.662 

7 

1.008 

4 

1.117 

\/ 

1.213 

4 

1.320 

47 

1.466 


4 

1.671 

8 

1.010 

27 

1.122 

74: 

1.217 

41 

1.325 

4 

1.473 


4 

1.681 

9 

1.013 

4 

1.127 

35 

1.221 

4 

1.330 

4 

1.480 


4 

1.691 

10 

1.016 

28 

1.133 

l 4 

1.225 

4 

1.335 

74 

1.487 


54 

1.701 

11 

1.019 

4 

1.138 


1.228 

4 

1.340 

48 

1.495 


4 

1.712 

12 

1.022 

29 

1.143 


1.232 

42 

1.346 

4 

1.502 


% 

1.722 

13 

1.026 

4 

1.149 

36 

1.236 

4 

1.351 

4 

1.509 


74 

1.733 

14 

1.031 

30 

1.155 

4 

1.240 

4 

1.356 

74 

1.517 


55 

1.743 

15 

1.035 

4 

1.158 


1.244 

/4 

1.362 

49 

1.524 


4 

1.754 

16 

1.040 

4 

1.161 


1.248 

43 

1.367 

4 

1.532 


4 

1.766 

17 

1.046 

% 

1.164 

37 

1.252 

4 

1.373 

4 

1.540 


4 

1.777 

18 

1.052 

31 

1.167 

4 

1.256 

4 

1.379 

4 

1.548 


56 

1.788 

19 

1.058 


1.170 

17 

1.261 

74 

1.384 

50 

1.556 


4 

1.800 

20 

1.064 

4 

1.173 

74 

1.265 

44 

1.390 

4 

1.564 


4 

1.812 

21 

1.071 

4 

1.176 

38 

1.269 

4 

1.396 

4 

1.572 


74 

1.824 

4 

1.075 

32 

1.179 

4 

1.273 

4 

1.402 

74 

1.581 


57 

1.836 

22 

1.079 

% 

1.182 

4 

1.278 

4 

1.408 

51 

1.589 


4 

1.849 

4 

1.082 

4 

1.186 

4 

1.282 

45 

1.414 

4 

1.598 


4 

1.861 

23 

1.086 

4 

1.189 

39 

1.287 

4 

1.420 

4 

1.606 


74 

1.874 

4 

1.090 

33 

1.192 

4 

1.291 

4 

1.427 

4 

1.615 


58 

1.887 


To find a Meridian Line (a true North and South line) by 
means of the North Star. (Polaris.) 

The north star appears to describe a small circle, n n', &c, Fig 14, around the true north point, or 
north pole, as a center. The rad of this circle is estimated by the angle between the star and the 
pole, as measured from the earth ; and is called the polar diet of the star. This polar dist be¬ 
comes seconds, or very nearly % of a minute less every year. On Jan 1, 1885, it is, approx¬ 

imately, 1° 17 ( 58". On the first of 1890, it will be about 1° 16' 41", &c. When, in its revolution, the 
star is farthest east or west from the pole, as at n' or n", it is said to be at its greatest J2 or W 
elongation. Then its apparent motion for several min is nearly vert, and consequently affords 
the best opportunity for an observation in the simple manner here described. The arrows in Fig 14 
show the direction in which the stars appear to move from east to west when the spectator faces the 
north. 

The latitude of the place must be known approximately. Taking it at the closest one in our fore¬ 
going table, the error in the position of the meridian will not exceed half a min of azimuth in lat 57°, 
or one quarter min in lat 40° ; and still less in lower lats. 

About 3 ft above ground fix firmly, perfectly level, and as nearly east and west as may be, a 
smooth narrow piece of board, about 3 ft long, to serve as a kind of 
tahle. Also prepare another piece a a, about a foot long; and fasten 
to it, at right angles, a compass-sight, or a strip of thin metal, with 
a straight slit, (shown by a black line in the fig.) about 6 ins long 
and -jir inch wide. This piece of board is to be slid along the 
table, as the observer follows the motions of the star toward the 
east or west: looking at it through the vert slit. Plant a stout 
pole, about 20 ft long, firmly in the ground, with its top as nearly 
north as possible from the middle of the table. Its top should 
lean 2or 3 ft toward either the east or the west; and a plumb- 
line must be suspended from its top, with a bob weighing one or two lbs, which may swing in a 
bucket of water placed on the ground. This is to prevent the line from being so easily moved by 
slight currents of air; and for further steadiness, the pole itself should be well braced from within, a 


b 



Fig-. 12. 






















































































178 


LAND SURVEYING 



Fig. 13. 


few feet below its top. The proper diet a o, of the pole p o. from the table t a, may be found 
thus: Make an angle n ms, equal approximately to the lat of the place. Open a pair of dividers 
to equal, by auy convcnieut scale, the height t a of the table; and draw t a. Then take, by the 

same scale, the height, p o, of the pole above ground ; and place 
it upon the sketch, so that the top p shall be by scale a ft 01 
two above m n. Then a o, by the same scale, will be about the 
dist reqd; probably from 3 to 5 yards. A deviation of a ft or so 
from this will not be important. 

The correct clock time at any place, for the elongation, may be 
found within a few min from the following table. 

Instead of a pole and plumb-line, the writer would suggest a 
planed, straight-edged board planted vert and braced; its side 
toward the observer. '| 

The observer should be at his station at least % of an hour ia 
advance of the time. Placing the board a a, upon the table and 
in range with the plumb-line and star, he will watch both of them 
through the vert slit; sliding the board along the table, so as to 
keep the slit in the range as long as the star continues to move 
toward the east or west, as the case may be. Au assistant must 
hold a candle, or lantern, on a pole near the plumb-line, to enable 
the observer to see the latter. As the star approaches its elonga- 
tion, it will appear to move nearly vert for several min, so that it can be seen without moviug the 
slit. When certain by this that the star has reached its elongation, confine the sliding board to the 
table by sticking a few tacks around its edges. Then let a third person, with another candle, go off 
some dist, (a hundred yards or more if convenient.) in a direction toward the star ; and then drive 
a stake as directed by the observer, who will take care that it is exactly in range with the slit and 
plumb-line. Another stake must then be driven exactly under either the slit or the plumb-line. 
Haviug thus placed the two stakes in the range of the elongation, defer the remainder of the operation 
until morning. From the tables given above take out the sine of the polar dist, and also the secant 
of the lat. Mult these together. The prod will be the nat sine of an angle called the azimuth 
of the star. Find the sine in table, p 60, &c. and the angle which corresponds to it. This azimuth 
angle will be between 1° 20' and 2° 30', according to lat. Place an instrument over the S stake, sight 
to the N one, and lay off this angle to the E if the elong was W, or vice versa, and drive a stake to 
mark it. This last direction is true N and S. It might be supposed that after driving the first two 
stakes, a true meridian could be had by merely laying off the polar dist, by means of a compass or 
transit; but this is not so. Place the compass over the south stake, and take sight to the north one. 
if, then, the north end of the needle points east of the line, the variation of the compass is east and 
vice versa. ’ 

Times by a correct clock of Elongations of the Iff. Star. 

oo?^ u< L ed fr ? m U - S - Coast Surve y table * calculated for April 1, 1883, to April 1, 1884. and for lat . 
38 u N ; but will answer within about 5 minutes for auy lat up to 60° N, and until 1890. 

Times of Eastern Elongations. 


Pav of 
Month. 

Apr. 

May. 

June. 

July. 

Au*. 

Sep. 

1 

7 

13 

19 

25 

H. M. 

6 41 A M 

6 18 “ 

5 54 “ 

5 30 “ 

5 7“ 

H. M. 

4 43 A M 

4 20 “ 

3 56 “ 

3 32 “ 

3 9“ 

H. M. 

2 41 A M 

2 18 “ 

1 54 “ 

1 30 “ 

1 7 “ 

H. M. 

12 43 A M 
12 20 “ 

11 52 PM 

11 29 “ 

11 5 “ 

H. M. 

10 37 P M 
10 14 “ 

9 50 “ 

9 27 “ 

9 3“ 

11. M. 

8 36 P M 

8 12 “ 

7 48 “ 

7 25 “ 

7 1 “ 

Times of Western Elongations. 

Pay of 
Mouth. 

Oct. 

Nov. 

Dee. 

Jan. 

Feb. 

Mar. 

1 

7 

13 

19 

25 

H. M. 

6 31 A M 

6 8“ 

5 44 “ 

5 21 “ 

4 57 “ 

H. M. 

4 29 A M 

4 6“ 

3 42 “ 

3 19 “ 

2 55 “ 

H. M. 

2 32 A M 

2 8“ 

1 44 “ 

1 21 “ 

12 57 “ 

H. M. 

12 30 A M 
12 6 “ 

11 39 P M 

11 15 “ 

10 51 “ 

H. M. 

10 24 P M 
10 00 » 

9 36 “ 

9 13 “ 

8 49 “ 

H M. 

8 30 P M t 
8 6“ 1 
7 43 “ 

7 19 “ 

6 55 “ 


1 S lh0 ”’ lte “'*• 11 enough h> .note .he .tae 

north .torn'!, on tii. merlii’," Mo“i e ‘p.l^ o?bSf<.w’l, 

Will nave to watch Alioth above the north star. Watch through the movable slit , A, * ne 

the same vert lino with the north star. Then put in two stakes as before and thev w.n K» h 0 ," 
in a tiue north and south line. But to be more exact, either lay off (to’tho E ipAlioth^sTtove 




















































LAND SURVEYING 


179 


Polaris, or to the W if below) an azimuth Angle of 11 minutes; or else do not drive the stakes when 
the two stars are in a vert line, but note the time, and then wait 24.5 minutes. Then take the 
range to Polaris alone. 1 his last range will be true N and S within 2 or 3 minutes depending on lat. 

until 1890. A transit with illuminated crosS-wires can plainly be used in¬ 
stead of the plumbiiue, &c, but is more troublesome except to an expert. A very correct method 
adapted to both N and S lats, is to take the two ranges to any N or S circumpolar star on the same 

hight; when it is at any two equal altitudes. Half way between them 

ViU be true N and S. 



There can be no 
difficulty in find¬ 
ing Alioth, as it 
is one of the 7 
bright stars in 
the fine constel- 
lation so well 
known as the 
Great Bear, or 
the Wagon and 
Horses. Alioth is 
the horse nearest 
tothe fore-wheels 
of the wagon. 
The two hind- 
wheels t t are 
known to every 
schoolboy as the 
“Pointers,” be¬ 
cause they point 
nearly in the di- 
rection to the 
North Star. The 
relative positions 
of these 7 stars, 
as shown in Fig 
14, are tolerably 
oorreet. 











(PO 

2 

4 

fi 

8 

10 

12 

14 

16 

18 

20 

22 

21 

26 

28 

30 

32 

34 

36 

38 

40 

42 

41 

46 

48 

50 

52 

54 

56 

58 

1°0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

31 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

lOA' 


fc*P0 

58 

56 

54 

52 

50 

*8 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

15 c 0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 


TRAVERSE TABLE, 


Traverse Table for a Distance = 1. 


Dep. 

or 

E. W. 


.0000 

90’0 

.0006 

58 

.0012 

56 

.0017 

54 

.0023 

52 

.0029 

50 

.0035 

48 

.0041 

46 

.0047 

44 

.0052 

42 

.0058 

40 

.0064 

38 

.0070 

36 

.0076 

34 

.0081 

32 

.0087 

30 

.0093 

28 

.0099 

26 

.0105 

24 

.0111 

22 

.0116 

20 

.0122 

18 

.0128 

16 

.0134 

14 

.0140 

12 

.0145 

10 

.0151 

8 

.0157 

6 

.0163 

4 

.0169 

2 

.0175 

89' > 0 

.0180 

58 

.0186 

56 

.0192 

54 

.0198 

52 

.0204 

50 

.0209 

48 

.0215 

46 

.0221 

44 

.0227 

42 

.0233 

40 

.0239 

38 

.0244 

36 

.0250 

34 

.0256 

32 

.0262 

30 

.0268 

28 

.0273 

26 

.0279 

24 

.0285 

22 

.0291 

20 

.0297 

13 

.0302 

16 

.0308 

14 

.0314 

12 

.0320 

10 

.0326 

8 

.0332 

6 

.0337 

4 

.0343 

2 

.0349 

88=0' 


Lat. 

or 

N. S. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 

2°0' 

.9994 

.0349 

2 

.9994 

.0355 

4 

.9993 

.0361 

6 

.9993 

.0366 

8 

.9993 

.0372 

10 

.9993 

.0378 

12 

.9993 

.0384 

14 

.9992 

.0390 

16 

.9992 

.0396 

18 

.9992 

.0401 

20 

.9992 

.0407 

22 

.9991 

.0413 

24 

.9991 

.0419 

26 

.9991 

.0425 

28 

.9991 

.0430 

30 

.9990 

.0436 

32 

.9990 

.0442 

34 

.9990 

.0148 

36 

.9990 

.0454 

38 

.9989 

.0459 

40 

.9989 

.0465 

42 

.9989 

.0471 

44 

.9989 

.0477 

46 

.9988 

.0483 

48 

.9988 

.0488 

50 

.9988 

.0494 

52 

.9987 

.0500 

54 

.9987 

.0506 

56 

.9987 

.0512 

58 

.9987 

.0518 

3°0' 

.9986 

.0523 

2 

.9986 

.0529 

4 

.9986 

.0535 

6 

.9985 

.0541 

8 

• 9985 

.0547 

10 

.9985 

.0552 

12 

.9984 

.0558 

14 

.9984 

.0564 

16 

.9984 

.0570 

18 

.9983 

.0576 

20 

.9983 

.0581 

22 

.9983 

.0587 

24 

.9982 

.0593 

26 

.9982 

.0599 

28 

.9982 

.0605 

30 

.9981 

.0610 

32 

.9981 

.0616 

34 

.9981 

.0622 

36 

.9980 

.0628 

38 

.9980 

.0634 

40 

.9980 

.0640 

42 

.9979 

.0645 

44 

.9979 

.0651 

46 

.9978 

.0657 

48 

.9978 

.0663 

50 

.9978 

.0669 

52 

.9977 

.0674 

54 

.9977 

.0680 

56 

.0976 

.0686 

58 

.9976 

.0692 

4°0' 

.9976 

.0698 


Dep. 

Lat. 


or 

or 


E. W. 

N. S. 


Lat. 

or 

N. S. 


88°0' 

4°0' 

.9976 

58 

2 

.9975 

56 

4 

.9975 

54 

6 

.9974 

52 

8 

.9974 

50 

10 

.9974 

48 

12 

.9973 

46 

14 

.9973 

44 

16 

.9972 

42 

18 

.9972 

40 

20 

.9971 

38 

22 

.9971 

36 

24 

.9971 

34 

26 

.9970 

32 

28 

.9970 

30 

30 

.9969 

28 

32 

.9969 

26 

34 

.9968 

24 

36 

.9968 

22 

38 

.9967 

20 

40 

.9967 

18 

42 

.9966 

16 

44 

.9966 

14 

46 

.9965 

12 

48 

.9965 

10 

50 

.9964 

8 

52 

.9964 

6 

54 

.9963 

4 

56 

.9963 

2 

58 

.9962 

87°0' 

5°0' 

.9962 

58 

2 

.9961 

56 

4 

.9961 

54 

6 

.9960 

52 

8 

.9960 

50 

10 

.9959 

48 

12 

.9959 

46 

14 

.9958 

44 

16 

.9958 

42 

18 

.9957 

40 

20 

.9957 

38 

22 

.9956 

36 

24 

.9956 

34 

26 

.9955 

32 

28 

.9955 

30 

30 

.9954 

28 

32 

.9953 

26 

34 

.9953 

24 

36 

.9952 

22 

38 

.9952 

20 

40 

.9951 

18 

42 

.9951 

16 

44 

.9950 

14 

46 

.9949 

12 

48 

.9949 

10 

50 

.9948 

8 

52 

.9948 

6 

54 

.9947 

4 

56 

.9946 

2 

58 

.9946 

86°0' 

6°0' 

.9945 


T)ep. 

or 

F.. W. 


Dep. 

or 

E. W. 


.0698 

.0703 

.0709 

.0715 

.0721 

.0727 

.0732 

.0738 

.0744 

.0750 

.0756 

.0761 

.0767 

.0773 

.0779 

.0785 

.0790 

.0796 

.0802 

.0808 

.0814 

.0819 

.0825 

.0831 

.0837 

.0843 

.0848 

.0854 

.0860 

0866 

.0872 

.0877 

.0883 

.0889 

.0895 

.0901 

.0906 

.0912 

.0918 

.0924 

.0929 

.0935 

.0941 

.0947 

.0953 

.0958 

.0964 

.0970 

.0976 

.0982 

.oils; 

.0993 

.0999 

.1005 

.1011 

.1016 

.1022 

.1028 

.1034 

.1039 

.1045 








































































6°0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

7°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

on' 


181 

80~0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

79°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

78°0' 


TRAVERSE TABLE 


averse Table for a Distance = 1. (Continci 


Dep. 

or 

E. W. 



Lat. 

or 

N.S. 

Dep. 

or 

E. \V\ 



Lat. 

or 

N.S. 

.1045 

8400' 

8°0' 

.9903 

.1392 

82°0' 

10°0' 

.9848 

.1051 

58 

2 

.9902 

.1397 

58 

2 

.9847 

.1057 

56 

4 

.9901 

.1403 

56 

4 

.9846 

.1063 

54 

6 

.9900 

.1409 

54 

6 

.9845 

.1068 

52 

8 

.9899 

.1415 

52 

8 

.9844 

.1074 

50 

10 

.9899 

.1421 

50 

10 

.9843 

.1080 

48 

12 

.9898 

.1426 

48 

12 

.9842 

.1086 

46 

14 

.9897 

.1432 

46 

14 

.9841 

.1092 

44 

16 

.9896 

.1438 

44 

16 

.9840 

.1097 

42 

18 

.9895 

.1444 

42 

18 

.9839 

.1103 

40 

20 

.9894 

.1449 

40 

20 

.9838 

.1109 

38 

22 

.9894 

.1455 

38 

22 

.9837 

.1115 

36 

24 

.9893 

.1461 

36 

24 

.9836 

.1120 

34 

26 

.9892 

.1467 

34 

26 

.9835 

.1126 

32 

28 

.9891 

.1472 

32 

28 

.9834 

.1132 

30 

30 

.9890 

.1478 

30 

30 

.9833 

.1138 

28 

32 

.9889 

.1484 

28 

32 

.9831 

.1144 

26 

34 

.9888 

.1490 

26 

34 

.9830 

.1149 

24 

36 

.9888 

.1495 

24 

36 

.9829 

.1155 

22 

38 

.9887 

.1501 

22 

38 

.9828 

.1161 

20 

40 

.9886 

.1507 

20 

40 

.9827 

.1167 

18 

42 

.9885 

.1513 

18 

42 

.9826 

.1172 

16 

44 

.9884 

.1518 

16 

44 

.9825 

.1178 

14 

46 

.9883 

.1524 

14 

46 

.9824 

.1184 

12 

48 

.9882 

.1530 

12 

48 

.9823 

.1190 

10 

50 

.9881 

.1536 

10 

50 

.9822 

.1196 

8 

52 

.9880 

.1541 

8 

52 

.9821 

.1201 

6 

54 

.9880 

.1547 

6 

54 

.9820 

.1207 

4 

56 

.9879 

.1553 

4 

56 

.9818 

.1213 

2 

58 

.9878 

.1559 

2 

58 

.9817 

.1219 

8300' 

9°0 

.9877 

.1564 

81°0' 

ll°0' 

.9816 

.1224 

58 

2 

.9876 

.1570 

58 

2 

.9815 

.1230 

56 

4 

.9875 

.1576 

56 

4 

.9814 

.1236 

54 

6 

.9874 

.1582 

54 

6 

.9813 

.1242 

52 

8 

.9873 

.1587 

52 

8 

.9812 

.1248 

50 

10 

.9872 

.1593 

50 

10 

.9811 

.1253 

48 

12 

.9871 

.1599 

48 

12 

.9810 

.1259 

46 

14 

.9870 

.1605 

46 

14 

.9808 

.1265 

44 

16 

.9869 

.1610 

44 

16 

.9607 

.1271 

42 

18 

.9869 

.1616 

42 

18 

.9806 

.1276 

40 

20 

.9868 

.1622 

40 

20 

.9805 

.1282 

38 

22 

.9867 

.1628 

38 

22 

.9804 

.1288 

36 

24 

.9866 

.1633 

36 

■ 24 

.9803 

.1294 

34 

26 

.9865 

.1639 

34 

26 

.9802 

.1299 

32 

28 

.9864 

.1645 

32 

28 

.9800 

.1305 

30 

30 

.9863 

.1650 

30 

30 

.9799 

.1311 

28 

32 

.9862 

.1656 

28 

32 

.9798 

.1317 

26 

34 

.9861 

.1662 

26 

34 

.9797 

.1323 

24 

36 

.9860 

.1668 

24 

36 

.9796 

.1328 

22 

38 

.9859 

.1673 

22 

38 

.9795 

.1334 

20 

40 

.9858 

.1679 

20 

40 

.9793 

.1340 

18 

42 

.9857 

.1685 

18 

42 

.9792 

.1346 

16 

44 

.9856 

.1691 

16 

44 

.9791 

.1351 

14 

46 

.9855 

.1696 

14 

46 

.9790 

.1357 

12 

48 

.9854 

.1702 

12 

48 

.9789 

.1363 

10 

50 

.9853 

.1708 

10 

50 

.9787 

.1369 

8 

52 

.9852 

.1714 

8 

52 

.9786 

.1374 

6 

54 

.9851 

.1719 

6 

54 

.9785 

.1380 

4 

56 

.9850 

.1725 

4 

56 

.9784 

.1386 

2 

58 

.9849 

.1731 

2 

58 

.9783 

.1392 

82°0' 

10°0' 

.9848 

.1736 

80°0' 

12°0' 

.9781 

Lat. 

or 

N.S. 



Dep. 

or 

E. W. 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 




























































2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

41 

46 

48 

50 

52 

54 

56 

58 

5°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

92 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 


58 

56 

{4 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

i2 

10 

8 

6 

4 

2 

i°0’ 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 


TRAVERSE TABLE, 


Traverse Table for a Distance = 1. (Continued.) 


Dep. 

or 

E. W. 


.2079 

.2084 

.2090 

.2096 

.2102 

.2108 

.2113 

.2119 

.2125 

.2130 

.2136 

.2142 

.2147 

.2153 

.2159 

.2164 

.2170 

.2176 

.2181 

.2187 

.2193 

.2198 

.2204 

.2210 

.2215 

.2221 

.2227 

.2232 

.2238 

.2244 

.2250 

.2255 

.2261 

.2267 

.2272 

.2278 

.2284 

.2289 

.2295 

.2300 

.2306 

.2312 

.2317 

.2323 

.2329 

.2334 

.2340 

.2346 

.2351 

.2357 

.2363 

.2368 

.2374 

.2380 

.2385 

.2391 

.2397 

.2402 

.2408 

.2414 

.2419 


Lat. 

or 


N. S. 




Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

78°0’ 

14°0 

.9703 

.2419 

76°0' 

16°0' 

.9613 

58 

2 

.9702 

.2425 

58 

2 

.9611 

56 

4 

.9700 

.2431 

56 

4 

.9609 

54 

6 

.9699 

.2436 

54 

6 

.9608 

52 

8 

.9697 

.2442 

52 

8 

.9606 

50 

10 

.9696 

.2447 

50 

10 

.9605 

48 

12 

.9694 

.2453 

48 

12 

.9603 

46 

14 

.9693 

.2459 

46 

14 

.9601 

44 

16 

.9692 

.2464 

44 

16 

.9600 

42 

18 

.9690 

.2470 

42 

18 

.9598 

40 

20 

.9689 

.2476 

40 

20 

.9596 

38 

22 

.96»7 

.2481 

38 

22 

.9595 

36 

24 

.96,86 

.2487 

36 

24 

.9593 

34 

26 

.9684 

.2493 

34 

26 

.9591 

32 

28 

.9683 

.2498 

32 

28 

.9590 

30 

30 

.9681 

.2504 

30 

30 

.9588 

28 

32 

.9680 

.2509 

28 

32 

.9587 

26 

34 

.9679 

.2515 

26 

34 

.9585 

24 

36 

.9677 

.2521 

24 

36 

.9583 

22 

38 

.9676 

.2526 

22 

38 

.9582 

20 

40 

.9674 

.2532 

20 

40 

.9580 

18 

42 

.9673 

.2538 

18 

42 

.9578 

16 

44 

.9671 

.2543 

16 

44 

.9577 

14 

46 

.9670 

.2549 

14 

46 

.9575 

12 

48 

.9668 

.2554 

12 

48 

.9573 

10 

50 

.9667 

.2560 

10 

50 

.9572 

8 

52 

.9665 

.2566 

8 

52 

.9570 

6 

54 

.9664 

.2571 

6 

54 

.9568 

4 

56 

.9662 

.2577 

4 

56 

.9566 

2 

58 

.9661 

.2583 

2 

58 

.9565 

77°0' 

I5°0- 

.9659 

.2588 

75°0' 

17°0- 

.9563 

58 

2 

.9658 

.2594 

58 

2 

.9561 

56 

4 

.9656 

.2599 

66 

4 

.9560 

54 

6 

.9055 

.2605 

54 

6 

.9558 

52 

8 

.9653 

.2611 

52 

8 


50 

10 

.9652 

.2616 

50 

10 

.9555 

48 

12 

.9650 

.2622 

48 

12 

.9553 

46 

14 

.9649 

.2628 

46 

14 

.9551 

44 

16 

.9617 

.2633 

44 

16 

.9549 

42 

18 

.9646 

.2639 

42 

18 

.9548 

40 

20 

.9614 

.2644 

40 

20 

.9546 

38 

22 

.9642 

.2650 

38 

22 

.9544 

36 

24 

.9641 

.2656 

36 

24 

.9542 

34 

26 

.9639 

.2661 

34 

26 

.9541 

32 

28 

.9638 

.2667 

32 

28 

.9539 

30 

30 

.9636 

.2672 

30 

30 

.9537 

28 

32 

.9635 

.2678 

28 

32 

.9535 

26 

34 

.9633 

.2684 

26 

34 

.9534 

24 

36 

.9632 

.2689 

24 

36 

.9532 

22 

38 

.9630 

.2695 

22 

38 

.9530 

20 

40 

.9628 

.2700 

20 

40 

.9528 

18 

42 

.9627 

.2706 

18 

42 

.9527 

16 

44 

.9625 

.2712 

16 

44 

.9525 

14 

46 

.9624 

.2717 

14 

46 

.9523 

12 

48 

.9622 

.2723 

12 

48 

.9521 

10 

50 

.9621 

.2728 

10 

50 

.9520 

8 

52 

.9619 

.2734 

8 

52 

.9518 

6 

54 

.9617 

.2740 

6 

54 

.9516 

4 

56 

.9616 

.2745 

4 

56 

.9514 

2 

58 

.9614 

.2751 

2 

58 

.9512 

76°0' 

16°0' 

.9613 

.2756 

74°0' 

18°0' 

.9511 



Dep. 

Lat. 



Dep. 



or 

or 



or 



E. W. 

N. S. 



E. W. 


















































2 

4 

6 

8 

10 

12 

14 

16 

18 

SO 

22 

24 

86 

28 

30 

92 

84 

36 

38 

40 

42 

44 

40 

48 

50 

52 

54 

56 

58 

j : >0 

2 

4 

6 

8 

10 

12 

1 t 

16 

is 

80 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

40 

48 

50 

52 

54 

56 

58 

ion 


B°0 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

r°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 


TRAVERSE TABLE 


averse Table for a Distance = 1. (Continue! 





Lat. 

or 

N. S. 

.3090 

72°0' 

20 r 0' 

.9397 

.3096 

58 

2 

.9395 

.3101 

56 

4 

.9393 

.3107 

54 

6 

.9391 

.3112 

52 

8 

.9389 

.3118 

50 

10 

.9387 

.3123 

48 

12 

.9385 

.3129 

46 

14 

.9383 

.3134 

44 

16 

.9381 

^ .3140 

42 

18 

.9379 

.3145 

40 

20 

.9377 

.3151 

38 

22 

.9375 

.3156 

36 

24 

.9373 

.3162 

34 

26 

.9371 

.3168 

32 

28 

.9369 

.3173 

30 

30 

.9367 

.3179 

28 

32 

.9365 

.3184 

26 

34 

.9363 

.3190 

24 

36 

.9361 

.3195 

22 

38 

.9359 

.3201 

20 

40 

.9356 

.3206 

18 

42 

.9354 

.3212 

16 

44 

.9352 

.3217 

14 

46 

.9350 

.3223 

12 

48 

.9348 

.3228 

10 

50 

.9346 

.3234 

8 

52 

.9344 

.3239 

6 

54 

.9342 

.3245 

4 

56 

.9340 

.3250 

2 

58 

.9338 

.3256 

71°0' 

21°0' 

.9336 

.3261 

58 

2 

.9334 

.3267 

56 

4 

.9332 

.3272 

54 

6 

.9330 

.3278 

52 

8 

.9327 

.3283 

50 

10 

.9325 

.3289 

48 

12 

.9323 

.3294 

46 

14 

.9321 

.3300 

44 

16 

.9319 

.3305 

42 

18 

.9317 

.3311 

40 

20 

.9315 

.3316 

38 

22 

.9313 

.3322 

36 

24 

.9311 

.3327 

34 

26 

.9308 

.3333 

.32 

28 

.9306 

.3338 

30 

30 

.9304 

.3344 

28 

32 

.9302 

.3349 

26 

34 

.9300 

.3355 

24 

36 

.9298 

.3360 

22 

38 

.9296 

.3365 

20 

40 

.9293 

.3371 

18 

42 

.9291 

.3376 

16 

44 

.9289 

.3382 

14 

46 

.9287 

.3387 

12 

48 

.9285 

.3393 

10 

50 

.9283 

.3398 

8 

52 

.9281 

.3404 

6 

54 

.9278 

.3109 

4 

56 

.9276 

.3415 

2 

58 

.9274 

.3420 

70°0' 

22°0' 

.9272 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 


Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

.3420 

70°0 

22°0 

.9272 

.3426 

58 

2 

.9270 

.3431 

56 

4 

.9267 

.3437 

54 

6 

.9265 

.3442 

52 

8 

.9263 

.3448 

50 

10 

.9261 

.3453 

48 

12 

.9259 

.3458 

46 

14 

.9257 

.3464 

44 

16 

.9254 

.3469 

42 

18 

.9252 

.3475 

40 

20 

.9250 

.3480 

38 

22 

.9248 

.3486 

36 

24 

.9245 

.3491 

34 

26 

.9243 

.3497 

32 

28 

.9241 

.3502 

30 

30 

.9239 

.3508 

28 

32 

.9237 

.3513 

26 

34 

.9234 

.3518 

24 

36 

.9232 

.3524 

22 

38 

.9230 

.3529 

20 

40 

.9228 

.3535 

18 

42 

.9225 

.3540 

16 

44 

.9223 

.3546 

14 

46 

.9221 

.3551 

12 

48 

.9213 

.3557 

10 

50 

.9216 

.3562 


52 

.9214 

.3567 

6 

54 

.9212 

.3573 

4 

56 

.9210 

.3578 

2 

58 

.9207 

.3584 

69°0 

23°0' 

.9205 

.3589 

58 

2 

.9203 

.3595 

56 

4 

.9200 

.3600 

54 

6 

.9198 

.3605 

52 

8 

.9196 

.3611 

50 

10 

.9194 

.3616 

48 

12 

.9191 

.3622 

46 

14 

.9189 

.3627 

44 

16 

.9187 

.3633 

42 

18 

.9184 

.3638 

40 

20 

.9182 

.3643 

38 

22 

.9180 

.3649 

36 

24 

.9178 

.3654 

34 

26 

.9175 

.3660 

32 

28 

.9173 

.3665 

30 

30 

.9171 

.3670 

28 

32 

.9168 

.3676 

26 

34 

.9166 

.3681 

24 

36 

.9164 

.3687 

22 

38 

.9161 

.3692 

20 

40 

.9159 ’ 

.3697 

18 

42 

.9157 

.3703 

16 

44 

.9154 

.3708 

14 

46 

.9152 

.3714 

12 

48 

.9150 

.3719 

10 

50 

.9147 

.3724 

8 

52 

.9145 

.3730 

6 

54 

.9143 

.3735 

4 

56 

.9140 

.3741 

2 

58 

.9138 

.3746 

68°0' 

24°0' 

.9135 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 










































184 


TRAVERSE TABLE, 


Traverse Table for a Distance = 1 . (Coxtinued.) 



I., at. 
or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. \V\ 


24' ’0’ 
2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

80 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

25°0’ 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

26 0’ 

.9135 

.9133 

.9131 

.9128 

.9126 

.9124 

.9121 

.9119 

.9116 

.9114 

.9112 

.9109 

.9107 

.9(04 

.9102 

.9100 

.9097 

.9095 

.9092 

.9090 

.9088 

.9085 

.9083 

.9080 

.9078 

.9075 

.9073 

.9070 

.9068 

.9066 

.9063 

.9061 

.9058 

.9056 

.9053 

.9051 

.9048 

.9016 

.9043 

.9041 

.9038 

.9036 

.9033 

.9031 

.9028 

.9026 

.9023 

.9021 

.9018 

.9016 

.9013 

•9011 

.9008 

.9006 

.9003 

.9001 

.8998 

.8996 

.8993 

.8990 

.8988 

.4067 

.4073 

.4078 

.4083 

.4089 

.4094 

.4099 

.4105 

.4110 

.4115 

.4120 

.4126 

.4131 

.4136 

.4142 

.4147 

.4152 

.4158 

.4163 

.4168 

.4173 

.4179 

.4184 

.4189 

.4195 

.4200 

.4205 

.4210 

.4216 

.4221 

.4226 

.4231 

.4237 

.4242 

.4247 

.4253 

.4258 

.4263 

.4268 

.4274 

.4279 

.4284 

.4289 

.4295 

.4300 

.4305 

.4310 

.4316 

.4321 

.4326 

.4331 

.4337 

.4342 

.4347 

.4352 

.4358 

.4363 

.4368 

.4373 

.4378 

.4384 ( 

66°0' 

58 

56 

54 

52 

60 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

65°0 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

>4 °0' 2 

26°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

27°0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

8°0' 

.8988 

.8985 

.8983 

.8980 

.8978 

.8975 

.8973 

.8970 

.8967 

.8965 

.8962 

.8960 

.8957 

.8955 

.8952 

.8949 

.8947 

.8944 

.8942 

.8939 

.8336 

.8934 

.8931 

.8928 

.8926 

.8923 

.8921 

.8918 

.8915 

.8913 

.8910 

.8907 

.8905 

.8902 

.8899 

.8897 

.8894 

.8892 

.8889 

.8886 

.8884 

.8881 

.8878 

.8875 

.8873 

.8870 

<8867 

.8865 

.8862 

.8859 

.8857 

.8854 

.8851 

.8849 

.8846 

.8843 

.8840 

.8838 

.8835 

.8832 

.8829 

.4384 

.4389 

.4394 

.4399 

.4405 

.4410 

.4415 

.4420 

•4425 

.4431 

.4436 

.4441 

.4446 

.4452 

.4457 

.4462 

.4467 

•4472 

.4478 

.4483 

.4488 

.4493 

.4498 

.4504 

.4509 

.4514 

.4519 

.4524 

.4530 

.4535 

.4540 

.4545 

.4550 

.4555 

.4561 

.4566 

.4571 

.4576 

.4581 

.4586 

.4592 

.4597 

.4602 

.4607 

.4612 

.4617 

.4623 

.4628 

.4633 

.4638 

.4643 

.4648 

.4654 

.4659 

.4664 

.4669 

.4674 

.4679 

.4684 

.4690 

.4695 

64°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

63°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

62°0’ : 

28°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

29°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

o°o- 

.8829 

.8827 

.8824 

.8821 

.8819 

.8816 

.8813 

.8810 

.8808 

.8805 

.8802 

.8799 

.8796 

.8794 

.8791 

.8788 

.8785 

.8783 

.8780 

.8777 

.8774 

.8771 

.8769 

.8766 

.8763 

.8760 

.8757 

•8755 

.8752 

.8749 

.8746 

.8743 

.8741 

.8738 

.8735 

.8732 

.8729 

.8726 

.8724 

.8721 

.8718 

.8715 

.8712 

.8709 

.8706 

.8704 

.8701 

.8698 

.8695 

.8692 

.8689 

.8686 

.8683 

.8681 

.8678 

.8675 

.8672 

.8669 

.8666 

.8663 

.8660 

.4695 
.4700 
.4705 
.4710 
.4715 
.4720 
.4726 
.4731 
.4736 
.4741 
.4746 
.4751 
.4756 
.4761 
.4766 
.4772 
.4777 
.4782 
.4787 
.4792 
.4797 
.4802 
.4807 
.4812 
.4818 
.4823 
.4828 
.4833 
.4838 
4843 
.4848 
.4853 
.4858 
.4863 
.4868 
.4874 
.4879 
.4884 
.4889 
.4894 
.4899 
.4904 
.4909 
.4914 
.4919 
.4924 
.4929 
.4934 
.4939 
.4944 
.4950 
.4955 
.4960 
.4965 
.4970 
.4975 
.4980 
.4985 
.4990 
.4995 
.5000 ( 

62°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

61^0 

58 

56 

51 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

rf) o 0' 

• 

Dep. 

or 

K. W. 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 

Lat. 

or 

N. S. 










































































:o°o' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

1 ° 0 ' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

41 

46 

48 

50 

52 

54 

56 

58 

!° 0 - 


56°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

55°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

54°0' 


TRAVERSE TABLE 


Traverse Table for a Distance = 1. (Continued.) 


Dep. 

or 

E. W. 



Lat. 

or 

N.S. 

Dep. 

or 

E. W. 



Lat. 

or 

N.S. 

.5000 

60°0' 

32°0' 

• 8480 

.5299 

58°0' 

34°0' 

.8290 

.5005 

58 

2 

•8477 

.5304 

58 

2 

.8287 

.5010 

5o 

4 

•8474 

•5309 

56 

4 

.8284 

.5015 

54 

6 

•8471 

• 5314 

54 

6 

•8281 

.5020 

52 

8 

•8468 

•5319 

52 

8 

•8277 

.5025 

50 

10 

•8465 

•5324 

50 

10 

•8274 

.5030 

48 

12 

•8462 

•5329 

48 

12 

.8271 

.5035 

46 

14 

•8459 

• 5334 

46 

14 

.8268 

.5040 

44 

16 

•8456 

•5339 

44 

16 

.8264 

.5045 

42 

18 

•8453 

•5344 

42 

18 

.8261 

.5050 

40 

20 

•8450 

•5348 

40 

20 

.8258 

.5055 

38 

22 

•8446 

•5353 

38 

22 

.8254 

.5060 

36 

24 

•8443 

•5358 

36 

24 

.8251 

.5065 

34 

26 

•8440 

•5363 

34 

26 

.8248 

.5070 

32 

28 

•8437 

•5368 

32 

28 

.8245 

.5075 

30 

30 

•8434 

•5373 

30 

30 

.8241 

.5080 

. 28 

32 

•8431 

•5378 

28 

32 

.8238 

.5085 

26 

34 

•8428 

•5383 

26 

34 

.8235 

.5090 

24 

36 

•8425 

•5388 

24 

36 

.8231 

.5095 

22 

38 

•8421 

•5393 

22 

38 

.8228 

.5100 

20 

40 

•8418 

•5398 

20 

40 

.8225 

.5105 

IS 

42 

•8415 

•5402 

18 

42 

.8221 

.5110 

16 

44 

•8412 

•5407 

16 

44 

.8218 

.5115 

14 

46 

•8409 

•5412 

14 

46 

.8215 

.5120 

12 

48 

•8406 

•5417 

12 

48 

.8211 

.5125 

10 

50 

•8403 

•5422 

10 

50 

.8208 

.5130 

8 

52 

•8399 

•5427 

8 

52 

.8205 

.5135 

6 

54 

•8396 

•5432 

6 

54 

.8202 

.5140 

4 

56 

•8393 

•5437 

4 

56 

.8198 

.5145 

2 

58 

•8390 

• 5442 

2 

58 

.8195 

.5150 

59°0' 

33°0' 

•8387 

•5446 

57°0' 

35°0' 

.8192 

.5155 

58 

2 

•8384 

•5451 

58 

2 

.8188 

.5160 

56 

4 

•8380 

• 5456 

56 

4 

.8185 

.5165 

54 

6 

•8377 

•5461 

54 

6 

.8181 

.5170 

52 

8 

•8374 

• 5466 

52 

8 

.8178 

.5175 

50 

10 

•8371 

•5471 

50 

10 

.8175 

.5180 

48 

12 

•8368 

•5476 

48 

12 

.8171 

.5185 

46 

14 

•8364 

•5480 

46 

14 

.8168 

.5190 

44 

16 

•8361 

•5485 

44 

16 

.8165 

.5195 

42 

18 

•8358 

•5490 

42 

18 

.8161 

.5200 

40 

20 

•8355 

.5495 

40 

20 

.8158 

.5205 

38 

22 

•8352 

•5500 

38 

22 

.8155 

.5210 

36 

24 

•8348 

.5505 

36 

24 

.8151 

.5215 

34 

26 

•8345 

•5510 

34 

26 

.8148 

.5220 

32 

28 

•8342 

.5515 

32 

28 

.8145 

.5225 

30 

30 

•&339 

.5519 

30 

30 

.8141 

.5230 

28 

32 

•8336 

•5524 

28 

32 

.8138 

.5235 

26 

34 

•8332 

.5529 

26 

34 

.8134 

.5240 

24 

36 

•8329 

.5534 

24 

36 

.8131 

.5245 

22 

38 

•8326 

.5539 

22 

38 

.8128 

.5250 

20 

40 

•8323 

.5544 

20 

40 

.8124 

.5255 

18 

42 

•8320 

.5548 

18 

42 

.8121 

.5260 

16 

44 

•8316 

.5553 

16 

44 

.8117 

.5265 

14 

46 

• 8313 

.5558 

14 

46 

.8114 

.5270 

12 

48 

•8310 

.5563 

12 

48 

.8111 

.5275 

10 

50 

•8307 

.5568 

10 

50 

.8107 

.5279 

8 

52 

•8303 

.5573 

8 

52 

.8104 

.5284 

6 

54 

•8300 

.5577 

6 

54 

.8100 

.5289 

4 

56 

.8297 

.5582 

4 

56 

.8097 

.5294 

2 

58 

.8294 

.5587 

2 

58 

.8094 

.5299 

58°0' 

34=0' 

.8290 

.5592 

56°0' 

36°0' 

.8090 

Lat. 

or 

N.S. 


/ 

Dep. 

or 

E. IV. 

Lat. 

or 

N.S. 



Dep. 

or 

E. W. 





















































2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

7°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

ion- 


4)°0 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

Doo 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 


TRAVERSE TABLE, 


Lat. 

or 

N. S. 

Dep. 

or 

E. W. 

.8090 

.5878 

.8087 

.5883 

.8083 

.5887 

.8080 

.5892 

.8076 

.5897 

.8073 

.5901 

.8070 

.5906 

.8066 

.5911 

.8063 

.5915 

.8059 

.5920 

.8056 

.5925 

.8052 

.5930 

.8049 

.5934 

.8045 

.5939 

.8042 

.5944 

.8039 

.5948 

.8035 

.5953 

.8032 

.5958 

.8028 

.5962 

.8025 

.5967 

.8021 

.5972 

.8018 

.5976 

.8014 

.5981 

.8011 

.5986 

.8007 

.5990 

.8004 

.5995 

.8000 

.6000 

.7997 

.6004 

.7993 

.6009 

.7990 

.6014 

.7986 

.6018 

.7983 

.6023 

.7979 

.6027 

.7976 

.6032 

.7972 

.6037 

.7969 

.6041 

.7965 

.6046 

.7962 

.6051 

.7958 

.6055 

.7955 

.6060 

.7951 

.6065 

.7948 

.6069 

.7944 

.6074 

.7941 

.6078 

.7937 

.6083 

.7934 

.6088 

.7930 

.6092 

.7926 

.6097 

.7923 

.6101 

.7919 

.6106 

.7916 

.6111 

.7912 

.6115 

.7909 

.6120 

.7905 

.6124 

.7902 

.6129 

.7898 

.6134 

.7894 

.6138 

.7891 

.6143 

.7887 

.6147 

.7884 

.6152 

.7880 

.6157 


Lat. 

or 


N. S. 


Table for a Distance = 1. (Continue 




Lat. 

or 

N. 8. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

54=0' 

38°0' 

.7880 

.6157 

52°0 

40 °0' 

.7660 

58 

2 

.7877 

.6161 

58 

2 

.7657 

56 

4 

.7873 

.6166 

56 

4 

.7653 

54 

6 

.7869 

.6170 

54 

6 

.7649 

52 

8 

,7Nf>6 

.6175 

52 

8 

.7645 

50 

10 

.7862 

.6180 

50 

10 

.7642 

48 

12 

.7859 

.6184 

48 

12 

.7638 

46 

14 

.7855 

.6189 

46 

14 

.7634 

44 

16 

.7851 

.6193 

44 

16 

.7630 

42 

18 

.7848 

.6198 

42 

18 

.7627 

40 

20 

.7844 

.6202 

40 

20 

.7623 

38 

22 

.7841 

.6207 

38 

22 

.7619 

36 

24 

.7837 

.6211 

36 

24 

.7615 

34 

26 

.7833 

.6216 

34 

26 

.7612 

32 

28 

.7830 

.6221 

32 

28 

.7608 

30 

30 

.7826 

.6225 

30 

30 


28 

32 

.7822 

.6230 

28 

32. 

.7600 

26 

34 

.7819 

.6234 

26 

34 

.7596 

24 

36 

.7815 

.6239 

24 

36 

.7593 

22 

38 

.7812 

.6243 

22 

38 

.7589 

20 

40 

.7808 

.6248 

20 

40 

.7585 

18 

42 

.7804 

.6252 

18 

42 

.7581 

16 

44 

.7801 

.6257 

16 

44 

•7578 

14 

46 

.7797 

.6262 

14 

46 

.7574 

12 

48 

.7793 

.6 256 

12 

48 


10 

50 

.7790 

.6271 

10 

50 


8 

52 

.7786 

.6275 

8 

52 

.7562 

6 

54 

.7782 

.6280 

6 

54 

.7559 

4 

56 

.7779 

.6284 

4 

56 

.7555 

2 

58 

.7775 

.6289 

2 

58 

.7551 

53°0- 

39 >0' 

.7771 

.6293 

51 °0 

41°0' 

•7547 

58 

2 

.7768 

.6298 

58 

2 

.7543 

56 

4 

.7764 

.6302 

56 

4 

.7539 

54 

6 

.7760 

.6307 

54 

6 

.7536 

52 

8 

.7757 

.6311 

52 

8 


50 

10 

.7753 

.6316 

50 

10 

.7528 

48 

12 

.7749 

.6320 

48 

12 

.7524 

46 

14 

.7746 

.6325 

46 

14 

.7520 

44 

16 

.7742 

.6329 

44 

16 

.7516 

42 

18 

.7738 

.6334 

42 

18 

.7513 

40 

20 

.7735 

.6338 

40 

20 

.7509 

38 

22 

.7731 

.6343 

38 

22 

.7505 

36 

24 

.7727 

.6347 

36 

24 

.7501 

34 

26 

.7724 

.6352 

34 

26 

.7497 

32 

28 

.7720 

.6356 

32 

28 

.7493 

30 

30 

.7716 

.6361 

30 

30 

.7490 

28 

32 

.7713 

.6855 

28 

32 

.7486 

26 

34 

.7709 

.6370 

26 

34 

.7482 

24 

36 

.7705 

.6374 

24 

36 

.7478 

22 

38 

.7701 

.6379 

22 

38 

.7474 

20 

40 

.7698 

.6383 

20 

40 

.7470 

18 

42 

.7694 

.6388 

18 

42 

.7466 

16 

44 

.7690 

.6392 

16 

44 

.7463 

14 

46 

.7687 

.6397 

14 

46 

.7459 

12 

48 

.7683 

.6401 

12 

48 

.7455 

10 

50 

.7679 

.6406 

10 

50 

.7451 

8 

52 

.7675 

.6410 

8 

52 

.7447 

6 

54 

.7672 

.6414 

6 

54 

.7443 

4 

56 

.7668 

.6419 

4 

56 

.7439 

2 

58 

.7664 

.6423 

2 

58 

. 7485 

52°0' 

10°0' 

.7660 

.6428 

')0°0' 

42°0' 

.7431 



Dep. 

Lat. 



Dep. 



or 

or 



or 



E. W. 

N. S. 



E. W. | 

i 































































TRAVERSE TABLE, 


187 


Traverse Table for a Distance = 1. (Concluded.) 


{ 

Lilt. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 

n 

42°0' 

.7431 

.6691 

48°0' 

13°0' 

.7314 

.6820 

47°0' 

44°0' 

.7193 

.6947 

4<P0' 

2 

.7428 

.6696 

58 

2 

.7310 

.6824 

58 

2 

.7189 

.6951 

58 

4 

.7424 

.6700 

56 

4 

.7306 

.6828 

5B 

4 

.7185 

.6955 

56 

6 

.7420 

.6704 

54 

6 

.7302 

.6833 

54 

6 

.7181 

.6959 

54 

8 

.7416 

.6709 

52 

8 

.7298 

.6837 

52 

8 

.7177 

.6963 

52 

10 

.7412 

.6713 

50 

10 

.7294 

.6841 

50 

10 

.7173 

.6967 

50 

12 

.7408 

.6717 

48 

12 

.7290 

.6845 

48 

12 

.7169 

.6972 

48 

14 

.7404 

.6722 

46 

14 

.7286 

.6850 

46 

14 

.7165 

.6976 

46 

16 

.7400 

.6726 

44 

16 

.7282 

.6854 

44 

16 

.7161 

.6980 

44 

18 

.7396 

.6730 

42 

18 

.7278 

.6858 

42 

18 

.7157 

.6984 

42 

20 

.7392 

.6734 

40 

20 

.7274 

.6862 

40 

20 

.7153 

.6988 

40 

22 

.7388 

.6739 

38 

22 

.7270 

.6867 

38 

22 

.7149 

.6992 

38 

24 

.7385 

.6743 

36 

24 

.7266 

.6871 

36 

24 

.7145 

.6997 

36 

26 

.7381 

.6747 

34 

26 

.7262 

.6875 

.34 

26 

.7141 

.7001 

34 

28 

.7377 

.6752 

32 

28 

.7258 

.6879 

32 

28 

.7137 

.7005 

32 

30 

.7373 

.6756 

30 

30 

.7254 

.6884 

30 

30 

.7133 

.7009 

30 

32 

.7369 

.6760 

28 

32 

.7250 

.6888 

28 

32 

.7128 

.7013 

28 

34 

.7365 

.6764 

26 

34 

.7246 

.6892 

26 

34 

.7124 

.7017 

26 

36 

.7361 

.6769 

24 

36 

.7242 

.6896 

24 

36 

.7120 

.7021 

24 

38 

.7357 

.6773 

22 

38 

.7238 

.6900 

22 

38 

.7116 

.7026 

22 

40 

.7353 

.6777 

20 

40 

.7234 

.6905 

20 

40 

.7112 

.7030 

20 

42 

.7349 

.6782 

18 

42 

.7230 

.6909 

18 

42 

.7108 

.7034 

18 

44 

.7345 

.6786 

16 

44 

.7226 

.6913 

16 

44 

.7104 

.7038 

16 

46 

.7341 

.6790 

14 

46 

.7222 

.6917 

14 

46 

.7100 

.7042 

14 

48 

.7337 

.6794 

12 

48 

.7218 

.6921 

12 

48 

,7096 

.7046 

12 

50 

.7333 

.6799 

10 

50 

.7214 

.6926 

10 

50 

.7092 

.7050 

10 

52 

.7329 

.6803 

8 

52 

.7210 

.6930 

8 

52 

.7088 

.7055 

8 

54 

.7325 

.6807 

6 

54 

.7206 

.6934 

6 

54 

.7083 

.7059 

6 

56 

.7321 

.6811 

4 

56 

.7201 

.6938 

4 

56 

.7079 

.7063 

4 

58 

.7318 

.6816 

2 

58 

.7197 

.6942 

2 

58 

.7075 

.7067 

2 

43°0' 

.7314 

.6820 

47°0' 

t4°0' 

.7193 

.6947 

46°0' 

45 °0' 

.7071 

.7071 

45°0' 


Dep. 

or 

E. W. 

Lat. 

or 

N.S. 



Dep. 

or 

E. W. 

Lat. 

or 

N.S. 



Dep. 

or 

E. W. 

Lat. 

or 

N. S. 



When the angle exceeds 45°, the lats and deps are read upward from the bottom. 

Rem.— Since these lats and deps are for a dist 1, we may proceed as follows for greater dists. 
Thus, let the dist be 856.1. Add together 800 times, 50 times, 6 times, and .j-L. time the correspond¬ 
ing lats and deps of the table. 

Ex.—What is the lat and dep for 856.1 feet; the angle being 43° ? 

Here for 43° we have from the table, lat .7314; dep .6820. 

Hence, .7314 X 800 = 585.12; and .6820 X 800 = 545.60 

.7314 X 50= 36.57; and .6820 X 50 = 34.10 

.7314 X 6 = 4.39; and .6820 X 6 = 4.09 

.7314 X .1 = .07 ; and .6820 X .1 = .07 

Lat 626.15 Dep 583.86 

These multiplications may be made mentally. Or we may, with a little more trouble, mult the lat 
and dep of the table by the given dist. Thus, 

.7314 X 856.1 = 626.15 lat; and .6820 X 856.1 = 583.86 dep.* 


* Inasmuch as the engineer but rarely needs a traverse table, we have thought it best to give a 
correct one, rather than the common one for % degrees. The first involves more trouble in using it; 
but the last is entirely unfit for o'her than the rude calculations for oornmon surveying with compass 
courses taken to the nearest % degree. 

To divide a scale of one mile into feet, first cut off one-sixth of it; 

then divide the remainder into four equal parts. Each of these parts will be 1100 feet. 













































188 


THE ENGINEER'S TRANSIT. 


THE ENGINEER’S TRANSIT. 

























































THE ENGINEER’S TRANSIT. 


189 


r 



i 




T IIE details of the transit, like those of the level, are differently arranged by 

dlff m ^ k K rS ’u n n t0 SU i t , PFf lc , ular purposes. We describe it in its modern form 
as made by Heller and Brightly, of Philada. Without the Ion;; bubble-* u be 

tpantlif 1 ' U wmf n ie telescop and t he graduated arc g, it is their plaiit 
transit. With these appendages, or rather with a graduated circle in place of 
the arc , it becomes virtually a Complete Theodolite.* 1 

B D D, Hg 2, is the tripod*liea«l. The screw-threads at v receive the screw 
of a wooden tripod-head-cover when the instrument is out of use. S S A is the 
lower parallel plate. After the transit has been set very nearly over the 
center ot a stake, the shiftiug'-plate. dd cc, enables us, by slightly loosening 
the leyellms-screws K, to shift the upper parts horizontally a trifle, and 
thus bring the plumb-bob exactly over the center with less trouble than by the 
older method of pushing one or two of the legs further into the ground, or spread- 
lng them more or less. The screws, K, are then tightened, thereby pushing up¬ 
ward the upper parallel plate m mmxx , and with it the half-hall ft, thus 
pressing c c tightly up against the under side of S. The plumb-line passes 



through the vert hole in b. Screw-caps, f, g, protect the levelling-screws from 
dust, <fcc. The feet, i, of the screws, work in loose sockets, j, made flat at bottom, 
to preserve S from being indented. The parts thus far described are generally 
4 lef t attached to the legs at all times. Fig 1 shows the method of attachment. 

To set the upper parts upon the parallel plates. Place the 
lower end of U U in x x, holding the instrument so that the three blocks on m m 
(of which the one shown at F is movable) may enter the three corresponding 


*The price of a first-class plain transit with shifting-plate and plumb-bob, by Hel¬ 
ler & Brightly, is $185. One with vertical arc g and Jong bubble-tube J? F, $220. 

13 






























































































































190 


THE ENGINEER S TRANSIT. 


recesses in a, thus allowing a to boar fully on m, upon which the upper parts 
then rest. (The inner end of the spring-catch, /, in the meantime enters a groove 
around U, just below a, and prevents the upper parts from falling off, if the in¬ 
strument is now carried over the shoulder.) Revolve the upper parts horizontally 
a trifle, in either direction, until they are stopped by the striking of a small lug 
on a against one of the blocks F. The recesses in a are now clear of the blocks. 
Tighten q, thereby pushing inward the movable block F, which clamps the 
bevelled flange a between it and the two fixed blocks on m m, and confines the 
spindle U to the fixed parallel plates. It remains so clamped while the instrument 
is being used. 

To remove the upper parts from the parallel plates. Loosen 
q, bring the recesses in a opposite the blocks F. Hold back l, and lilt the upper 
parts, which are then held together by the broad head of the screw inserted into 
the foot of the spindle w. 

T T is the outer revolving: spindle, cast in one with the support- 
iu^-plate Z Z T to which is fastened the graduated limb O O. The limb 
extends beyond the compass-box, and thus admits of larger graduations than 
would otherwise be obtainable, w tv is the inner revolving: spindle. At 
its top it has a broad flange, to which is fastened the vernier plate P P. To 
the latter are fastened the com pass-box C, one of the bubble-tubes M M 
(the one shown in Fig 2), the dust-box W W, the standards V V, supporting 
the telescope, &c. Each bubble-tube is supported and adjusted by two capstan- 
screws, one at each end. One is shown at r. The bent strip curving over the 
tube protects the glass. 

The clamp-screw, H, presses the split collar, l l, tightly against the fixed spindle, 
U, but not against Z or T. The set-screws, G G, working in nuts that are cast in 
one with Z, hold between them the tongue, y, which projects from It, and the 
graduated limb is thus held fast, except that by moving the screws, G, it may be 
made to revolve slightly. 

In Fig 1, the tangent-screw, b , is seen passing through two towers, in which it 
works. One of the towers is fast to the lower one of the two small pieces at the 
foot of the clamp-screw e, Fig 2. When e is tightened, it draws the two small 
pieces together, confining between them an edge of the graduated limb, which is 
thus made fast to the'above-mentioned tower. The other tower is fast to the 
vernier-plate; and the tangent-screw, b, holds the towers at. a fixed dist apart. 
The clamping ot> thus prevents the vernier-plate from revolving over the gradu¬ 
ated limb, except that it may be moved slightly by turning b, and thus changing 
the dist apart of the towers. In Heller and Brightlv’s instruments, the screw', b, 
is provided with means for taking up its “wear,” or “lost-motion.” 

There are two verniers. One is shown at p, Fig 1. Both may be read, and 
their mean taken, w hen great accuracy is required. Ivory reflectors, c, facilitate 
their reading. Before the instrument is moved from one place to another, the 
coin|>as**-nee<lle. A;, Fig 2,should always be pressed up against the glass cover 
of the compass-box by means of the upright milled-head screw seen on the ver¬ 
nier-plate in Fig 1, just to the right of the nearest standard. The pivot-point is 
thus protected from injury. 

R, Fig 1, is a ring with a clamp (the latter not shown) for holding the telescope 
in any required position. It is best to let the eye-end, e, of the telescope revolve 
downward , as otherwise the shade on 0, if in use, may fall off. The tangent-screw, 
d , moves a vert arm attached to R, and is thus used for slightly changing the 
elevation of the telescope. In the arm is a slit like that seen in the vernier-arm 
l. When 0° of the vernier is placed at 30° on the arc, g , and the index of the 
opposite arm is placed over a small notch on the horizontal brace (not seen in our 
figs) of the standards, the tw r o slits will be opposite each other, and may be used 
for laying otf offsets, Ac, at right-angles to the line of sight. 

One end, R, of the telescope axis rests in a movable box, under which is a screw. 
By means of the screw, the box may be raised or lowered, and the axis thus ad¬ 
justed for very slight derangeinehts of the standards. For E, B, O, and A, see 
Level, p 201. a is a dust-guard for the object-slide. 

Stadia Hairs. Immediately behind the capstan-screw, p, Fig 1, is seen a 
smaller one. This and a similar one on the opposite side of the telescope, w ork 
in a ring inside the telescope, and hold the ring in position. Across the ring are 
stretched two additional horizontal hairs, called stadia hairs, placed at such a 
distance apart, vertically, that they will subtend say 10 divisions of a graduated rod 
placed 100 ft from the instrument, 15 divisions at 150 ft, &c. They are thus used for 
measuring hor and sloping distances. 

The long biibble-tiibe. F F, Fig 1, enables us to use the transit as a level, 
although it is not so well adapted as the latter to this purpose. 











THE ENGINEER’S TRANSIT. 


191 


To*adjust a plain Transit. 

wW»' e H eitll 9 r a level or a transit is purcliased, it is a good precaution (but one 
wh cb the writer has never seen alluded to) to first screw the object-glass firmly home 
tu its place .and then make a short continuous scratch upon the ring of the glass, and 
8 ale; 80 as to,,e abl e to see at any time when at work, that the glass is 
always in the same position with regard to the slide. For if, after all the adjustments 
ompleted, the position of the glass should become changed, (as it is apt to bo if 
unscrewed, and afterward not screwed up to the same precise spot,) the adjustments 
b0C °T “aerially deranged; especially if the object-glass is eccentric, 
> lot truly ground, which is often the case. Such scratches should be prepared bv 
ie maker. In making adjustments, as well as when using a transit or level, bo 
careful that the eye-glass and object-glass are so drawn out that there shall be ne 
parallax. 1 he eye-glass must first be drawn out so as to obtain perfect distinctness 
ot the cross-hairs; it must not be disturbed afterward; but the object-glass must 
be moved for different distances. 

Tirst. to ascertain that the bubble-tubes, M Mf, are placed 

parallel to the vernier-plate, and that therefore when both bubbles are in 
the centers of their tubes the axis of the inst is vert. By means of the four levelling- 
screws, K, bring both bubbles to the centers of their tubes in one position of the 
inst; then turn the upper parts of the inst half-way round. If the bubbles do not 
remain in the center correct half the error by means of the two capstan-screws 
r r; and the other half by the levelling-screws K. Repeat the trial until both 
bubbles remain in the center while the inst is being turned entirely around on 
its spindle. 

Second, to see that the standards have suffered no derange¬ 
ment : that is, that they are of equal height and perpendicular to the vernTer- 
plate, as they always are when they leave the maker’s hands. Level the inst 
perfectly; then direct the intersection of the hairs to some point of a high object 
(as the top of a steeple) near by; clamp the inst by means of screws H and e 
and lower the telescope until the intersection strikes some point of a low object’ 
(If there is none such drive a stake or chain-pin, Ac, in the liue.) Then’un- 
clamp either H or e, and turn the upper parts of the inst half-way round • fix the 
intersection again upon the high point; clamp; lower the telescope to the low 
point. It the intersection still strikes the low point, the standards are in order 
If not, correct one-quarter of the ditf (same principle as in Fig 4) by means of the 
adjusting-block and screw at the end, R, of the telescope axis, Fig 1, and repeat 
the trial de novo , resetting the stake or chain-pin at each trial. If the inst has no 
adjusting-block for the axis, it should be returned to the maker for correction of 
any derangement of the standards. 

A transit may be used for running straight lines, even if the standards become 
Slightly bent, by the process described at the end of the fourth adjustment. 

Thir<l. to see that the cross-hairs are truly vert and hor 
when the inst is level. When the telescope inverts, the cross-hairs are 
nearer the eye-end than when it shows objects erect. The maker takes care to place 
the cross-hairs at right-angles to each other in their ring, or diaphragm • and gene¬ 
rally he so places the ring in the telescope, that when levelled, tliev shall be vert 
and hor. Sometimes, however, this is neglected; or the ring may by accident be¬ 
come turned a little. To be certain that one hair is vert, (in which case the other 
must, by construction, be hor,) after having adjusted the bubble-tubes, level the in¬ 
strument carefully, and take sight with the telescope at a plumb-line, or other vert 

straight edge. If the vert hair coincides with this object, 
it is, so far, in adjustment; but if not, then loosen slightly 
only two adjacent screws of the four, pp i i, Fig 1; and 
with a knife, key, or other small instrument, tap very 
gently against the screw-heads, so as to turn the ring a 
little in the telescope; persevering until the hair be¬ 
comes truly vertical. When this is done, tighten the 
screw r s. In the absence of a plumb-line, or vert straight 
edge, sight tho cross-hair at a very small distinct 
point; and see if the hair still cuts that point, when 
the telescope is raised or lowered by revolving it on 
its axis. 

The mode of performing the foregoing will be readily 
understood from this Fig, which represents a section across the top part of the tele¬ 
scope, and at the cross-hairs. The hair-ring, or diaphragm, a ; vert hair, v; tele¬ 
scope tube, g ; ring outside of telescope.tube, d ; b is one of the four capstan-headed 
screws which hold the hair-ring, a, in its place, and also serve to adjust it. The 
lower ends of these screws work in the tnickness of the hair-ring; so that when 
they are loosened somewhat, they do not lose their hold on the ring. Small loose 








192 


THE ENGINEER’S TRANSIT. 


washers, c, are placed under the heads b cf the screws! A space y y is left around 
each screw where it passes through the telescope tube, to allow the screws and ring 
together to be moved a little sideways when the screws b are slightly loosened. 

Fourth, to see that the vertical hair is in the line of eolli- 
mation. Plant the tripod firmly upon the ground, as at a. Level the inst ; 
clamp it; and direct the \ert hair* by means of tangent-screws G (figs. 1 and 2) 
upon some convenient object b ; or if there is none such, drive a thin stake, or a 
chain-pin. Then revolving the teUscope vert on its axis, # c 

observe some object, as c, where the vert hair now strikes; ^ a 

or if there is none, place a second pin. Unclamp the instru- .--«o 

ment by the clamp-screw H; and turn the whole upper • %v 

part of' it around until the vert hair again strikes b. . Fig-, 4, 

(Jlamp again; and again revolve the telescope vert on its 

axis. If the vert h ur now strikes c, as it did before, it shows that c is really 
at o; and that b, a , c, are in the same straight line; and therefore this adjustment 
is in order. If not, observe where it does strike, say at w, (thedist a m being 
taken equal to a c,) and place a pin there also. Measure me; and place a pin 

at v, in the line in c, making m v = one-fourth of m c. Also put a pin at o, half¬ 

way between m and c, or in range with a and b. By means ot the two hor 
screws that move th- ring carrying the cross-hairs, adjust the vert hair until it 
cuts v. Now repeat the entire operation ; and persevere until the telescope, after 
being directed to b, shall strike the same object o, both times, when revolved on 
its axis. See whether the movement of the ring in this 4th adjustment has dis¬ 
turbed the verticality of the hair. If it has, repeat the 3d adjustment. Then re¬ 
peat the 4th, if necessary ; and so on until both adjustments are found to be right 
at the same time. Thus a straight line may be run, even if the hairs are out of 
adjustment; but with somewhat more trouble. For at each station, as at a, two 
back-sights, and two fore sights, a c and a rn , may be taken, as when making the 
adjustment; and the point o, half-way between Cand m, will be in the straight line. 
Tlie inst may then be moved to o, and the two back-sights be taken to a ; and so on. 

Angles measured by the transit, whether vert or hor, will evidently not be 
affected by the hairs being out of adjustment, provided either that the vert 
hair is truly vert, or that we use the intersection of the hairs when measuring. 

Tlie foregoing are all the adjustments needed, unless the tran¬ 
sit is required for levelling, in which case the following one must be attended to: 



To adjust the Iona: hnhblP«tTl'bp, F F, Fig. 1, we first place the line 

of sight of the telescope hor, and then make the bubble-tube hor, so that the 
two are parallel. Drive two pegs, a and b Fig. 5, with their tops at precisely 
the same level (see Rem. p. 193) and at least about 100 ft. apart; 300 or more 
will be better. Plant the inst firmly, in range with them, as at c, making b c 
an aliquot part of a b. and as short as will permit focusing on a rod at b. The 
inst need not be leveled. Suppose the line of sight to cut e and d. Take the 
readings b e and a d. Their diff is be — ad — an — a d = dn\ and ab : a c : : 
d n: d s; s being the height of the target at a when the readings (a s, b o ) on the 

d ix X a c 

two stakes are equal. as = ad + ds = ad-\ -——• If the reading on a 

exceeds that on b (as when the line of sight is v / g) theditFof readings is = a g — 

Q ?* X Cl C 

bf=ag — ai = gi; and as = ag — gs — ag— l 


a b 


Sight to s , bring the 

bubble to the cen of its tube by means of the two small nuts n n at one end of the 
tube, Fig. 1, and assume that the telescope and tube are parallel.* The zeros of 


* This neglects a small error due to the curvature of the earth; for a hor line at v is v h, tan¬ 
gential to the curved (or “ level”) surface of still water at v, whereas v s is tangential to water surf 
at a point midway between a and b. Hence if the telescope at v points to s it will not be parallel to 
the level bubble-tube. To allow for this, and for the refraction by the air, which diminishes the 
error, raise the target on a to a point h above s. /is = .0000000205 X square of a c in ft ; but when 
a c is 650 ft, h s is only about one tenth of an inch and barely covers the apparent thickness of the 
cross-hair in the telescope. 

























THE ENGINEER’S TRANSIT. 


193 


1 


® ircl + ?’ and ? f its vernier, may now be adjusted, if they require it, 
eide g the vernier screws and then moving the vernier until the two coin- 

bv^heTr;nl? ulmeVn at ^veiling the two pegs « and b, it may be done 

leveln hUS: Carefull y level ,h e two short bubbles, by means of the 

E b a l ' eg V l' from . 100 t0 300 feet from instrument o. 

men pUcm 0 a target-iod on m, clamp the target tight at whatever height, as ev, 

the hor hair happens to cut it; it being of no im- 
y s portauce whether the telescope is level or not; 
w although it might as well be as nearly so as can 
JJ conveniently be guessed at. Clamp the telescope 
in its position by the clamp-ring R, Fig 1. Re¬ 
volve the inst a considerable way round; say 
nearly or quite half way. Place another peg n. 



1 


n 


o 

Fig-. 6. 

, ,, oi ouiie nan wav. L' I ace another neo-« 

iiU I™™ the instrument that m is; and continue to drive Run- 

i'n e heSht S ‘ r S, r gGt plaCed ? n jt ’ and stm kept clamped to the rod, at the 
same height as when it was on m. W hen this is done, the tops of the two Dees are 

on a Rvel with each other and are rea<!y to be used as befor/directed. S 

When a transit is intended to be used for surveying farmf, Ac, or for retracing 
hnes of old surveys, it is very useful to set the compass so as to allow for the “va¬ 
riation during the interval between the two surveys. For this purpose a 
variation-vernier” is added to such transits; and also to the compass. 
W'hen the graduations of a transit are figured, or numbered, so as to read both 

Ip 0 10 

ways from zero, thus, l l l 11 i i i I i i i i I i i i 11 i i ii I i i i the vernier also is made 


double; that is, it also is graduated and numbered from its zero both ways. In this 
case, if the angle is measured from zero toward the right hand, the reading must he 
made from the right hand half of the vernier; and vice versa. If the figuring is 
single, or only in one direction, from zero to 360°, then only the single vernier is 
necessary, as the angles are then measured only in the direction that the figuring 
counts. Engineers differ in their preferences for various manners of figuring the 
graduations. The writer prefers from zero each way to 180°, with two double ver¬ 
niers. 

To replace cross-hairs ini a level, or transit. Take out the tubo 

from the eye end of the telescope. Looking in, notice which side of the cross¬ 
hair diaphragm is turned toward the eye end. Then loosen the four screws which 
hold the diaphragm, so as to let the latter fall out of the telescope. Fasten on new 
hairs with beeswax, varnish, glue, or gum-arabic water, Ac. This requires care. 
Then, to return the diaphragm to its place, press firmly into one of the screw-holes" 
on the circumf of the diaphragm itself, the end of a piece of stick, long enough to 
reach easily into the telescope as far as to where the diaphragm belongs. By this 
stick, as a handle,insert the diaphragm edgewise to its place in the telescope, and hold 
it there until tw-o opposite screws are put, in place and screwed. Then draw the stick 
out of the hole in the diaphragm; and with it turn the diaphragm until the same 
side presents itself toward the eye end as before; then put in the other two screws. 

The so-called cross-hairs are actually spider-web, so fine as to he barely visible to 
the naked eye. Heller A Brightly use very fine platina wire, which is much better. 
II liman hair is entirely too coarse. 

To replace a spiral-level, or bubble-glass. Detach the level from 
the instrument; draw off its sliding ends; push out the broken glass vial, and the 
cement which held it; insert the new one. with the proper side up (the upper side 
is always marked with a file by the maker); wrapping some paper around its ends, 
if it fits loosely. Finally, put a little putty, or melted beeswax over the ends of the 
vial, to secure it against moving in its tube. 

In purchasing instruments, especially when they are to be used far from a maker, 
it is advisable to provide extras of such parts as may be easily.broken or lost; such 
as glass compass-covers, and needles; adjusting pins; level vials; magnifiers, Ac. 


Theodolite adjustments are performed like those of the level and transit. 
1st. That of the cross-hairs; the same as in the level. 

2d. The long bubble-tube of the telescope; also as in the level. 

3d. The two short bubble-tubes: as in the transit. 

4th The vernier of the vert limb; as in the transit with a vert circle. 

5th. To see that the vert hair travels vertically; as in the fourth adjustment 
of the transit. In some theodolites, no adjustment is provided for this; but in 
large ones it is provided for by screws under the feet of the standards. 
Sometimes a second telescope is added; it is placed below' the hor limb, and is 








194 


THE BOX OR POCKET SEXTANT. 


called a watcher. It has its own clamp, and tangent-screw. Its use Is to ascertain 
whether the zero of that limb has moved during the measurement of lior angles. 
When, previously to beginning the measurement, the zero and upper telescope are 
directed toward the first object, point the lower telescope to any small distant 
object, and then clamp it. During the subsequent measurement, look through it, 
from time to time, to be sure that it still strikes that object; thus proving that no 
slipping has occurred. 


THE BOX OR POCKET SEXTANT. 



The portability of the pocket sextant, and the fact that it reads to single minutes, 
render it at times very useful to the engineer* by it, angles can be measured while 
in a boat, or on horseback; and in many situations which preclude the use of a 
transit. It is useful for obtaining latitudes, by aid of an artificial horizon. When 
closed, it resembles a cylindrical brass box, about 3 inches in diameter, and 1 % 
inches deep. This box is in two parts; 
by unscrewing which, then inverting 
one part, and then screwing them to¬ 
gether again, the lower part becomes a 
handle for holding the instrument. 

Looking down upon its top when thus 
arranged, we see, as in this figure, a 
movable arm I C, called the index, 
which turns on a center at C, and car¬ 
ries the vernier V at its other end. G 
G is the graduated arc or limb. It 
actually subtends about 73°, but is di¬ 
vided into about 146°. Its zero is at 
one end. Its graduations are not shown 
in the Fig. 

Attached to the index is a small mov¬ 
able lens, (not shown in the figure,) 
likewise revolving around C, for read¬ 
ing the fine divisions of the limb. When 
measuring an angle, the index is moved 
by turning the milled-head P of a 
pinion which works in a rack placed within the box. The eye is applied to a cir¬ 
cular hole at the side of the box, near A. A small telescope, about 3 inches long 
accompanies the instrument; but may generally be dispensed with. When so the 
eye hole at A should be partially closed by a slide which has a very small eye-hole 
in it; and which is moved by the pin h, moving in the curved slot. Another slide, 
at, the side of the box, carries a dark glass for covering the eye-hole when observing 
the sun. When the telescope is used, it is fastened on by the milled-head screw 1\ 
Ihe top part shown in our figure, can be separated from the cylindrical part bv 
removing 3 or 4 small screws around its edge; and the interior can then be exam¬ 
ined, and cleaned if necessary Like nautical, and other sextants, this one has 

pf them mirrors. One, the index-glass. is attached 

Uvo lot wl £ 1 1 ", dex> & c . ; lts - upper ed « e bein g indicated by the 

two dotted lines. The other, the horizon-class, l because when meas- 

uring the vert angles of celestial bodies, it is directed toward the horizon,) is also 
within the box; the position of its upper edge being shown by the dotted lines at 
It. Ihe horizon-glass is silvered only half-way down; so that one of the observed 
objects may be seen directly through its lower half, while the image of the other 
object is seen in the upper half, reflected from the index-glass. That the instrument 
may be in adjustment, ready tor use, these two glasses must be at right angles to the 
p ane of the instrument; that is, to the under side of the top of the box, to which thov 
are attached; and must also be parallel to each other, when the zeros of the vernier 
and of the limb coincide. The index-glass is already permanently fixed bv the 

if* 1, ^Quires no other adjustment. But the horizon-glass has two adjust¬ 
ments, which are made by a key like that of a watch, and having a milled-head K 
It is screwed into the top of the box, so as to be always at hand for use. When" 
needed, it is unscrewed. This key fits upon two small square-heads, (like that for 


^ Price, with telescope, about $50. Made by Stackpole & Bro., 41 Fulton St., New York 

























THE COMPASS, 


195 



winding a watch;) one of which is shown at S; while the other is near it, hut on tho 
side of the box. These squares are the heads of two small screws. If the 
horizon glass H should, ae in this sketch, (where it is shown endwise,) not heat 
right angles to the top U i!J of the box, it is brought right by turning the square¬ 
head S ot the screw S T; and if, after being so far rectified, it still is not parallel to 

the index-glass when the zeros coincide, it is moved 
a little backward or forward by the square head 
JJ at the side. 

To ad just a box sextant, bring the two 
zeros to coincide precisely; then look through the 
eye-hole, and the lower or unsilvered part of the 
horizon-glass, at some distant object. If the instru¬ 
ment is in adjustment, the object thus seen directly, 
will coincide precisely with its reflected image, 
seen at the same time, at the same spot. But if it 
is not in adjustment, the two w ill appear separated 
either hor or vert, or both, thus, * *; in which case 
apply the key Iv to the square-head S; and by turning it slightly in whichever direc¬ 
tion may be necessary, still looking at the object and its image , bring the two into a hor 
position, or on a level with each other, thus, * *. Then apply the key to the square- 
head in the side of the box; and by turning it slightly, bring the two to coincide 
perfectly The instrument is then adjusted. 

in some instruments, the hor glass has a hinge at r, to allow it play while being 
adjusted by the single screw ST; hut others dispense with this hinge, and use two 
screws like S on top of the box, in addition to the one in the side. 

if a sextant is used lor measuring vert angles by means of an Artificial 
horizon, the actual altitude will be but one-half of that read oif on the 
limb; because we then read at once both the actual and the reflected angle. The 
great objection to the sextant for engineering purposes, is that it does not measure 
angles horizontally, as the transit does; unless when the observer, and the two ob- 

^ jects happen to be in the same hor plane. 
Thus an observer with a sextant at A, if 
measuring the angle subtended by the 
mountain-peaks B and C, must hold the 
graduated plane of the sextant in the 
plane of A B C; and must actually meas¬ 
ure the angle BAC; whereas what he 
wants is the hor angle n A m. This is 
greater than BAC, because the dists A n 
and A ni are shorter than A B and A C. 

The transit gives the hor angle n A m, be¬ 
cause its graduated plane is first fixed hor by the levelling-screws: and the subse¬ 
quent measurement of the angle is not affected by his directing merely the line of 
sight upward, to any extent, in order to fix it upon B and C. For more on this sub¬ 
ject; and for a method of partially obviating this objection to the sextant, see the 
note to Example 2, Case 4, of “ Trigonometry,” page 113. 

The nautical sextant, used on ships, is constructed on the same principle 
as the box sextant; and its adjustments are very similar. In it, also, the index- 
glass is permanently fixed by the maker: and the horizon-glass has the two adjust¬ 
ments of the box sextant. It also lias its dark glasses for looking at the sun; and 
a small sight-hole, to be used when the telescope is dispensed with. 



THE COMPASS. 


To adjust a Compass. 

The first adjustment is that of the bubbles. Plant firmly; and level the 
instrument, in any position; that is, bring the bubbles to the centers of their tubes. 
Then turn the instrument half-way round. If the bubbles then remain at the cen¬ 
ters, they are in adjustment; but if not, correct one-half the diff in each bubble, 
by means of the adjusting-screws of the tubes. Level tho instrument again; turn 
it half round; and if the hubbies still do not remain at the center, the adjusting- 
screws must be again moved a little, so as to rectify half the remaining dill. Oenci- 









196 


THE COMPASS. 


ally several trials must be thus made, until the bubbles will remain at the center, 
while the compass is being turned entirely around. 

Second adjustment. Level the compass, and then see that the needle is 
hor; and if not, make it so by means of the small piece of wire which is wrapped 
around it; sibling the wire toward the high end. A needle thus horizontally ad¬ 
justed at one place, will not remain so if removed far north or south from that place. 

If carried to the north, the north end will dip down; and if to the south, the south 
end will do so. The sliding wire is intended to counteract this. 

Thir<l adjustment. This is always fixed right at first by the maker; that 
is, the sights, or slits for sighting through, are placed at right angles to the compitss 
plate; so that when the latter is levelled by the bubbles, the sights 
are vert. To test whether they are so, hang up a plumb-line; and 
having levelled the compass, take sight at the line, and see if the 
slits coincide with it. If one or both slits should prove to be 
out of plumb, as shown to an exaggerated extent in this sketch, 
it should be unscrewed from the compass, and a portion of its foot 
on the high side be filed or ground off, as per the dotted line; or 
as a temporary expedient, a small wedge may be placed under the 
low side, so as to raise it. 

Fourth adjustment, to straighten the needle, if it should become bent. 
The compass being levelled, and the needle hor, and loose on its pivot, see whether 
its two ends continue to point to exactly opposite graduations, (that is, graduations 
180° apart;) while the compiiss is turned completely around. If it does, the needle 
is straight; and its pin is in the center of the graduated circle ; hut if it does not, 
then one or both of these require adjusting. First level the compass. Then turn it 
until some graduation (say 90°) comes precisely to the north end of the needle. If 
the south end does not then point precisely to the opposite 90° division, lift off the 
needle, and bend the pivot-point until it does; remembering that every time said 
point is bent, the compass must be turned a liairsbreadth so as to keep the north end 
of the needle at its 90° mark. Then turn the compass half-way round, or until the 
opposite 90° mark comes precisely to the north end of the needle. Make a fine pen¬ 
cil mark where the *>«</» end of the needle now points. Then take off the needle, 
and bend it until its south end points half-way between its 90° mark and the pencil 
mark, while its north end is kept at 90° by moving the compass round a hairshreadth. 
The needle will then be straight, and must not be altered in making the following 
adjustment, although it will not yet cut opposite degrees. 

Fif til a<lJiiHtiueut, of the pivot-pin. After being certain that the needle is 
straight, turn the compass around until a part is arrived at where the two ends of the 
needle happen to cut opposite degrees. Then turn the compass quarter way around, 
or through 90°. If the needle then cuts opposite degrees, the pivot-point is already 
in adjustment; but it the needle does not so cut, bend the pivot-point until it does. 
Repeat, it necessary, until the needle cuts opposite degrees while being turned entirely 
around. 

Care and nicety of observation are necessary in making these adjustments properly ; 
because the entire error to be rectified is, in itself,a minute quantity; and the novice 
is very apt to increase his trouble by not knowing how to use his magnifier, ' 
when looking at the endof the needle and the corresponding graduations. The mag¬ 
nifier must always be held with its center directly over the point to he examined: and . 
it must be held parallel to the graduated circle. Otherwise annoying errors of 
several minutes will he made in a single observation; and the accumulation of two 
or three such errors, arising from a cause unknown to him, may compel him to 
abandon the adjustments in despair. This suggestion applies also to the reading of 
angles taken by the transit, Ac; although the errors are not then likely to be so 
great as in the case of the compass. In purchasing a magnifier for a compass, see 
that no part of it, as hinges, or rivets, are made of iron; lor such would change the 
direction of the needle. 

II the sight-slits of a compass are not fixed by the maker In line with the two 
opposite zeros, the engineer cannot remedy the defect. This can be ascertained by 
passing a piece ot fine thread through the slits, and observing whether it stands 
precisely over the zeros. 


Variation of tlie C’onipass.* 

The numerous disturbing influences to which the compass is subject, render its 

9 " ' “ ---- 

* For full information on this subject see that useful little book “ Magnetic Variation in the US" 
by J It Stone, C E» 1878. It is invaluable in retracing old lines. 








CONTOUR LINES. 


197 


K 

Indications of bearings or courses very unreliable. The daily variation itself some¬ 
times amounts to % of a degree; and always to at. least several minutes. It is almost 
incessantly changing the direction of the needle, to one side or the other, at the rate 
of I or 2 minutes per hour, especially in summer. Local attraction, from iron in the 
v 8oi 1, fen urinous gravel, trap rocks, &c, is another source of inaccuracy; as are also 
the annual and the secular variations. Electricity, either atmospheric, <>r excited 
by rubbing the glass cover, sometimes gives trouble. It may be removed by touch¬ 
ing the glass with the moist tongue, or finger. It. is plain that none of these causes 
(except the last) win affect the measurement of angles by the compass, 
v - In 1885 the line of no variation enters the U. S. near the N end of 
jLake Superior; passes through the Straits of Mackinaw; 40m W of Detroit, Mich • 
50 m o E ^° f C ' )1 ' ,nibns ’ 0: and caches the Atlantic about half way between Charles¬ 
ton, S C, and Wilmington, N C. A compass placed anywhere in the vicinity of that 
line, will point nearly due north and south. To the eastward of this line, the varia¬ 
tion is westward; and vice versa; becoming greater, the farther the place is from 
the line; until in some parts of Maine and along the Pacific coast it is as great as 
18° to 21°. This line is moving westward, at an average rate of about 3 or 4 min¬ 
utes per year. 

The needle, if of soft metal, sometimes loses part'of its magnetism,and consequently 
does not work well. It may be restored by simply drawing the north pole of a 
common magnet (either straight or horseshoe) about a dozen times, from the center 
to the end ot the south half of the needle: and the south pole, in the same way, along 
the north half; pressing the magnet gently upon the needle. After each stroke,- 
remove the magnet several inches from the needle, while bringing it back to the 
center for making another stroke. Each half of the needle in turn, while being thus 
operated on, should be held flat upon a smooth hard surface. Sluggish action of the 
needle is, however, more generally produced by the dulling or other injury of the 
point of the pivot. Demagnetizing will throw the needle out of balance; which must 
be counteracted by the sliding wire. 

En order to prevent mistakes by reading- sometimes from one end, 
and sometimes from the other end of (he needle, it is best to always point the N of 
the compass-box toward the object whose bearing is to be taken; and to read otf 
from the north end of the needle. This is also more accurate. 




CQNTQUE LINES. 


A contour line is a curved hor one, every point in which represents the same level; 
thus each of the contour lines 88c, 9lc, 94c, &c, Fig 1, indicates that every point in 
the ground through which it is traced is at the same level; and that that level or 
height is everywhere 88, 91, or 94 ft above a certain other level or height called 
datum; to which all others are referred. 

Frequently the level of the starting point of a survey is taken as being 0, or zero, 
or datum; and if we are sure of meeting with no points low er than it, this answers 
every purpose. But if there is a probability of many lower points, it is better to 
'assume the starting point to be so far above a certain supposed datum, that none of 
these lower points shall become minus quantities, or below said supposed datum or 
zero. The only object in this is to avoid the liability to error which arises w hen 
some of the levels are +, or plus; and some —, or minus. Hence we may assume 
the level of the starting point to be 10, 100, 1000, Ac, ft above datum, according to 
circumstances. 

The vert dists between each two contour lines are supposed to be equal; and in 
railroad surveys through well-known districts, w-here the engineer knows that his 
actual line of survey will not require to bo much changed, the dist may bo 1 or 2 ft 
only ; and the lines need not be laid down for widths greater than 100 or 200 ft on 
each side of his center-stakes. But in regions of which the topography is compara¬ 
tively unknown; and where consequently unexpected, obstacles may occur which 
, require the lino to be materially changed for a considerable dist back, the observa¬ 
tions should extend to greater widths; and for expedition the vertical dists apart 
may bo increased to 3, 5, or even 10 ft, depending on the character of the country, 
Ac. Also, when a survey is made for a topographical map of a State, or of a county, 
vert dists of 5 or 10 ft will generally suffice. 

Let the line A B, Fig 1, starting from O, represent three stations (S 1, S 2, S 3,) of 
the center line of a railroad survey; and lot the numbers 100, 103, 101, 104, along 
that line denote the heights at the stakes above datum, as determined by levelling. 
Then the use of the contour lines is to show in the office what would bo the effect 
of changing the surveyed center lino A B, by moving any part of it to the right or 






198 


CONTOUR LINES. 


left hand.* Thus, if it should be moved 100 ft to the left, the starting point 0 would 
be on ground about 0 ft higher than at present; inasmuch as its level would then 
be about 106 ft above datum, instead of 100. Station 1 would bo about 7 ft higher, 
or 110 ft instead of 10:1 Station 2 would be about 7 ft higher, or IDS ft instead of 
101.. If the line be thrown to the right, it will plainly be on lower ground. 

The field observations for contour lines are sometimes made with the spirit-level; 
but more frequently by a slope-man, with a straight 12-ft graduated rod, and a slope 
instrument, or clinometer. At each station he lays his rod upon the ground, as 



B 


j 

: 


Fig. I. 


nearly at right angles to the center lino A B as he can judge by eye; and placing 
the slope instrument upon it, he takes the angle of the slope of the ground to the 
nearest ^ of a degree. He also observes how far beyond the rod the slope continues 
the same; and with the rod he measures the (list. Then laying down the rod at that i 
point also, he takes the next slope, and measures its length ; and soon as far as may 1 
be judged necessary. Ilia notes are entered in his field-book as shown in Fig 2; the j 
angles of the slopes being written above the lines, and their lengths below; and I 
should be accompanied by such remarks as the locality suggests; such as woods, l 
rocks, marsh, sand, field, garden, across small ruu, Ac, Ac. j 


* In thus using the words right and left we are supposed to have our hacks turned to the starting 

point of the survey. In » river, the rig-lit hank or shore is that which 
is on the right hand as we descend it, that is, in speaking of its right or left 
bank, we are supposed to have our backs turned towards its head, or origin; and so with a survey. 
































CONTOUR LINES. 


199 


the slopes may be writ¬ 
ten in a straight line, 
as in Fig 2%. 


It is not absolutely necessary to represent the slopes roughly i 
in Fig 2; for by using tho sign + to signify “up;” —“down 


roughly in 
— “down; 


the field-hook, as 
” and ~ “level,” 



course office-work ; and 
is usually done at the 
same time as the draw¬ 
ing of the map, &c. The 
fielil observations at each 
station are then sepa¬ 
rately drawn by protrac¬ 
tor and scale, as shown 


The notes having been 
taken, the preparation 
of the contour lines by 
means of them, is of 



91 58 


IG + 10 + 


20 + #* 15 - 4 -~ 

-!-J—-L 



4 -- 26 - 


J- 


84 - 70 


in Fig 3 for the starting ^ 

point 0. ihe scale should not be less than about ■p'^’ inch to a ft, if anything like 
accuracy is aimed at. Suppose that at said station the slopes to the right, taken in 
their order, are, as in Fig 2, 15°, 4°, and'26°; and those to the left, 20°, 10°, and 16°; 
and their lengths as in the same Fig. Draw a hor line ho, Fig 3; and consider tlio 
center of it to be the station-stake. From this point as a center, lay off these angles 
with a protractor, as shown on the arcs in Fig 3. Then beginning say on the right 
hand, with a parallel ruler draw the first dist ac, at its proper slope of 15°; and of 
its proper length, 45 ft, by scale. Then the same with cy and y t. Do the same with 
those on the left hand. We then have a cross-section of the ground at Sta 0. Then 
on the map, as in Fig 1, draw a line as m n, or h w, at right angles to the line of road, 
and passing through tb 3 station-stake. On this line lay down the hor dists ad, d s,s v, 
a ffi V k, marking them with a small star, as is done and lettered in Fig 1, at Sta 0. 

When extreme accuracy is pretended to, these hor dists must be found by measure 
on Fig 3; but as a general rule it will be near enough, when the slopes do not ex¬ 
ceed 10°, to assume them to be the same as the sloping dists measured in the field. 
Next ascertain how high each of the points cytlni is above datum. Thus, measure 
) by scale the vert dist d c. Suppose it is found to be 5 ft; or in other words, that c 
is 5 ft below station-stake O. Then since the level at stake O is 100 ft above datum, 
that at c must be 5 ft less, or 100 — 5 = 95 ft above datum ; which may be marked in 
light lead-pencil figures on the map, as at d. Fig 1. Next for the point y, suppose 
we find s y to be 11 ft, or y to be 11 ft below stake 0; then its height above datum 
must be 100 — 11 = 89; which also write in pencil, as at s. Proceed in the same 
way with t. Next going to the left hand of the station-stake, we find el to be say 
2 ft; but l is above the level of the station-stake, therefore its height above datum is 



O 


t 


100 + 2 = 102 ft, as figured at e on the map. Let ng bo 5 ft; then is n, 100 -(- 5 = 
105 ft above datum, as marked at g ; and so on at each station. When this has been 
done at several stations, we may draw in the contour lines of that portion by hand 
thus: Suppose they are to represent vert heights of 3 ft. Beginning at Station O 
(of which the height above datum is 100 ft) to lay down a contour line 103 ft above 
datum, we see at once that the height of 103 ft must bo at t , or at )/, the dist from e 
to g. Make a light lead-pencil dot at t; and then go to the next Station 1. Here 
we see that tho height of 103 ft coincides with the station-stake itself; place a dot 
there, and go to Sta 2. The level at this stake is 101; therefore the contour for 103 









200 


CONTOUR LINES, 


ft must evidently be 2 ft higher, or at % of the dist from Sta 2 to +101; therefore 
make a dot at i. Then go to Sta 3. Here the level being 104 above datum, the con¬ 
tour of 103 must be at y, or - 5 - of the dist from Sta 3 to +99; put a dot at y. Finally 
draw by hand a curving line through t, SI, i, and y ; and the contour lino of 103 ft 
is done. All the others are prepared in the same way, one by one. The level of each 
must be figured upon it at short intervals along the map, as at 103 c, 106c, &c. 

Or, instead of first placing the + points on the map, to denote the slope dists actu¬ 
ally measured upon the ground, we may at once, and with less trouble, find and show 
those only which represent the points t, S 1, t, y, &c, of the contours themselves. 
Thus, say that at any given station-stake, Fig 4, the level is 104; that the cross-sec ¬ 
tion c s of the ground has been prepared as before ; and that we want the lior dists 
from the stake, to contour lines for 94, 97,100 ft, &c, 3 ft apart vert. 



Draw avert lino v Z, through the station-stake, and on it by scale mark levels of 
94, 97, 100, &c. It. This is readily done, inasmuch as we have the level 104 of the 
stake already given. Through those levels draw the lior lines a, b, m, n, &c. to the 
ground-slopes. Then these lines, measured by the scale, plainly give the required 
dists. 

When the ground is very irregular transversely, the cross-sections must be taken 
in the field nearer together than 100 ft. The preparation of contour lines will be 
greatly facilitated by the use of paper ruled into small squares of not less than about 

xV inch to a side, for drawing the cross-sections upon. 

When the ground is very steep, it is usual to shade such portions of the map to 
represent hill-side. The closer together the contours come, the steeper of course is 
the ground between them; and the shading should be proportionally darker at such 
portions. But for working maps it is best to omit the shading. 

In surveys of wide districts, the transit instrument with a graduated vertical cir¬ 
cle or arc, g , p. 188, is used for measuring the angles of slope, instead of the common 
slope-instrument.* 


* The preparing of contour lines is a slow and tedious office-work ; and the writer considers them 
of but little value in many cases; as when they are taken lor only about 100 ft on each side of the 
line, with reference to slight changes of direction, lie conceives that, ordinarily, every useful nurnose 
is fulfilled if the leveller or the topographer enters into his field-book at each station, notes sin : lar 
to the following: 


Sta GO.-3. 1R. +2.1L. 

G1.+2.2R. — 1.3L. 

62 .= 1. R. +4. 2 L. 

63 .•< “ 


Which means that at station 60, the slope of the ground on the right, as nearly as he can iudae hv 
eye, or by his hand-level, is about 8 ft downward, for 1 chain, or 100 ft; and on the left, about '> ft 
upward in 1 chain. At 61, 2 ft. up, in 2 chains to the right; and 1 ft down in 3 chains to the left I 
At 62, level for 1 chain to the right; and ascending 4 ft in 2 chains to the left. At 63, the same as at 
62. At some spots it will he well to add a sketch of a cross-section, like Fig *2 ; only, instead of the 
angles, use ft of rise or fall, to indicate the slopes, as judged by eye, or by a hand-level. Bv this 
method, the result at every station will be somewhat in error; but these small errors will balance 
each other so nearly that the total may he regarded as sufficiently correct for all the purposes of a 
preliminary estimate of the cost of a road. When the final stakes for guiding the workmen are placed 
the slopes should be carefully taken, in order to calculate the quantity of excavation accurately for 















THE LEVEL. 


201 




THE LEVEL. 


Although the levels of different makers vary somewhat in their details, still theif 
principal parts will be understood from the following figure.* The telescope T T 
rests upon two supports YY, called Ys; out of which it can be lifted, first removing 
the pins s s which confine the semicircular clips c c, and then opening the clips 
The pins should be tied to the Ys, by pieces of string, to prevent their being lost’. 
. ■‘■lie slide of the object-glass 0,is moved backward or forward by a rack and pinion 
v f by means of the milled head A. The slide of the eye-glass E, is moved in the same 
way by the milled head e. A cylindrical tube of brass, called a. shade, is usually 
lurnished with each level. It is intended to be slid on to the object-end O of the 
telescope, to prevent the glare of the sun upon the object-glass, when the sun is 
| ow - At B is an outer ring encircling the telescope, and carrying 4 small capstan- 
headed screws; two of which ,p p, are at top and bottom; while the other two 
of which i is one, are at the sides, and at right angles top p. Inside of this outer 
ring is another, inside of the telescope, and which has stretched across it two 
spider-webs, usually called the cross-hairs. These are much finer than they ap¬ 
pear to be, being considerably magnified. They are at right angles to each oilier; 
and, in levelling, one is kept vert, and the other hor. They are liable at times to be 



thrown out of this position by a partial revolution of the telescope, when carrying 
the level, or when setting the tripod down suddenly upon the ground; but since, in 
levelling, the intersection of the hairs is directed to the target-rod, this derangement 
does not affect the accuracy of the work. Still it is well to keep them nearly vert 
and hor, by keeping the bubble-tube D D as nearly directly over the bar V 1’ as can 
be judged by eye. This enables the leveller to see that the rod-man holds his rod 
nearly vert, which is absolutely essential for correct levelling. If perfect verticality 
is desired, as is sometimes the case, when staking out work, it may be obtained (if 
the. instrument is in perfect adjustment, and levelled) by sighting at a plumb-line, or 
other vert object, and then turning the telescope a little in its Ys, so as to bring the 
hair to correspond. When this is done, a short continuous scratch may be made on 
the telescope and Y, to save that trouble in future. Heller & Brightly, however, 
provide their levels with a small projection inside of the Ys, and a corresponding 
stop on the telescope, the contact of which insures the verticality of the hair. 
Should the hairs be broken by accident, they may be replaced as directed here¬ 
after. 

The small holes around the heads of the 4 small capstan-screws p, i, just referred to, 
are for admitting the end of a small steel pin. or lever, for turning them. If first 
the upper screw p he loosened, and then the lower one tightened, the interior ring 
will be lowered, and the horizontal hair with it. But on looking through the tele- 


* The price of a first-class level, by Heller & Brightly, is $145. It is 

bad economy to buy inferior instruments. 




































202 


THE LEVEL. 


scope they will appear to be raised. If first the lower one be loosened, and the upper 
one tightened, the hor hair will he actually raised, but apparently lowered. This is 
because the glasses in the eye-piece E reverse the apparent position of objects inside 
of the telescope; which effect is obviated, as regards exterior objects, by means of 
the object-glass 0. This must be remembered when adjusting the cross-hairs ; for if a 
hair appears to strike too high, it must be raised still higher; if it appears to be 
already too far to the right or left, it must be actually moved still more in the same 
direction. 

This remark, however, does not apply to telescopes which make objects appear 
inverted. 

There is no danger of injuring the hairs by these motions, inasmuch as the four 
screws act against the ring only, and do not come in contact with the hairs them¬ 
selves. 

Under the telescope is the bubble-tube D D. One end of this tube can be raised of 
lowered slightly by means of the two capstan-headed nuts n n, one of which must 
be loosened before the other is tightened. On top of the bubble-tube are scratches 
for showing when the bubble is central in the tube. Frequently these scratches, or 
marks, are made on a strip of brass placed above the tube, as in our fig. There are 
several of them, to allow for the lengthening or shortening of the bubble by changes 
of temperatuie. At the other end of the bubble-tube are two small capstan-screws, 
placed on opposite sides horizontally. The circular head of one of them is shown 
near t. By means of these two screws, that end of the tube can be slightly moved 
hor, or to right or left. Under the bubble-tube is the bar V F; at one end of which, 
as at V,are two large capstan-nuts w w, which operate upon a stout interior screw 
which forms a prolongation of the Y. The holes in these nuts are larger than the 
others, as they require a larger lever for turning them. If the lower nut is loosened 
and the upper one tightened, the Y above is raised; and that end of the telescope 
becomes farther removed from the bar; and vice versa. Some makers placea similar 
screw and nuts under both Ys; while others dispense with the nuts entirely, and 1 

substitute beneath one end of the bar a largo circular milled head, to be turned by 
the fingers. This, however, is exposed to accidental alteration, which should bo 
avoided. 

When the portions above m are put upon m, and fastened by the screw Y, all 
the upper part may be swung round hor, in either direction, by loosening the 
clamp-screw H ; or such motion may be prevented by tightening 'that screw. 

It frequently happens, after the telescope has been sighted very nearly upon an 
object, and then clamped by H, that we wish to bring the cross-hairs to coincide 
more precisely with the object than we can readily do by turning the telescope by 
hand; and in this case we use the tangent-screw ft, by means of which a‘ 
slight but steady motion may be given after the instrument is clamped. For 
fuller remarks on the clamp and tangent-screws, see “Transit.” 

The paraltel plates m and S are operated by four levelling-screws; 
three of which are seen in the figure, at K K. The screws work in sockets R; 
which, as well as the screws, extend above the upper plate. When the instrument 
is placed on the ground for levelling, it is well to set it so that the lower parallel 
plate S shall be as nearly horizontal as can be roughly judged by eye; in order 
to avoid much turning of the levelling screws K E in making the* upper plate 
in hor. The lower plate S, and the brass parts below it, are together called the 
tripod-head; and, in connection with three wooden legs Q Q Q,'constitute , 
the tripod. In the figure are seen the heads of wing-nuts J which confine the 
legs to the tripod-head. Under the center of the tripod-head should always be 
placed a small ring, from which a plumb-bob may be suspended. This is not 
needed in ordinary levelling, but becomes useful when ranging center-stakes, &c. 

To adjust a Level. 

This is a quite simple operation, but requires a little patience. Be careful to avoid 
straining any of the screws. The large Y nuts ww sometimes require some force to 
start them; but it should be applied by pressure, and not by blows. Before begin¬ 
ning to adjust, attend to the object-glass, as directed in the first sentence under “To 
adjust a plain transit,” p. 191. 

Three adjustments are necessary; and must be made in the following order: 

First, that of tlie cross-hairs; to secure that their intersection shall 
continue to strike the same point of a distant object, while the telescope is beiim 
turned round a complete revolution in its Ys. This is called adjusting the line 
of col I i illation, or sometimes, the line of sight; but it is not strictly the line 
of sight until all the adjustments are finished; for until then, the line of collimation 
will not serve for taking levelling sights. If cross-hairs break, see p 193. 

Second, that of the bubble-tube D D, to place it parallel to the line 





THE LEVEL. 


203 




of collimation. previously adjusted; so that when tho bubble stands at the centre of 
its tube, indicating that it is level, we know that our sight through the telescope is 
hor. To replace broken bubble tube, see p 193. 

Thi r«l, that of the Ys, by which the telescope and bubble-tube are supported; 
(to that the bubble-tube, and line of sight, shall be perp to the vert axis of the instru¬ 
ment; so as to remain hor while the telescope - is pointed to objects in diff directions, 
as when taking back and fore sights. 

To make the first adjustment, or that of the cross-hairs, plant'the 
< tripod firmly upon the ground. In this adjustment it is not necessary to level tho 
'l instrument. Open the clips of the Ys; unclamp; draw out the eye-glass E, until 
the cross-hairs are seen perfectly clear; sight the telescope toward some clear dis¬ 
tant point of an object; or still better, toward some straight line, whether vert or 
not. Move the object-glass 0, by means of the milled head A, so that the object shall 
be clearly seen, without parallax, that is, without any apparent dancing 
about of the cross-hairs, if the eye is moved a little up or down or sideways. To 
secure this, the object-glass alone is moved to suit different distances; the eye-glass 
is not to be changed after it is once properly fixed upon the cross-hairs. The neglect 
of parallax is a source of frequent errors in levelling. Clamp; and, by means of the 
tangent-screw b, bring either one of the cross-hairs to coincide precisely with tho 
object. Then gently, and without jarring, revolve the telescope half-way round in 
its Ys. When this is done, if the hair still coincides precisely with the object, it is 
' in adjustment; and we proceed to try the other hair. But if it does not coincide, 
then by means of the 4 screws p, i, move the ring which carries the hairs, so as to 
rectify, as nearly as can be judged by eye, only one-half of the error; remembering 
that the ring must be moved in the direction opposite to what appears to be the 
right one; unless the telescope is an inverting one. Then turn the telescope back 
again to its former position; and again by the tangent-screw bring the cross-hair to 
coincide with the object. Then again turn the telescope half-way round as before. 
The hair will now be found to be more nearly in its right place, but, in all probabil¬ 
ity, not precisely so; inasmuch as it is difficult to estimate one-half the error accu¬ 
rately by eye. Therefore a little more alteration of the ring must be made; and it 
may be necessary to repeat the operation several times, before the adjustment is 
perfect. Afterward treat the other hair in precisely the same manner. When both 
are adjusted, their intersection will strike the same precise spot while the telescope 
is being turned entirely round in its Ys. This must be tried before the adjustment 
can be pronounced perfect; because at times the adjustment of the second hair, 
slightly deranges that of the first one; especially if both were much out in the be¬ 
ginning. 


To make the second adjustment, or to place the bubble-tube parallel 
to the line of collimation. This consists of two dis¬ 
tinct adjustments, one vert, and one hor. The first 
of these is effected by means of the two nuts n n on 
the vert screw at one end of the tube ; and tlie second 
by the two hor screws at the other end, t, of the tube. 

Looking at the bubble-tube endwise, from t in the 
foregoing Fig, its two hor adjusting-screws t t are 
seen as in this sketch. The larger capstan-headed 
nut below , has nothing to do with the adjustments; 
it merely holds the end of the tube in its place. 

To make the vert adjustment of the bubble-tube, by means of the two nuts Tin. Place 
the telescope over a diagonal pair of the levelling-screws K K; and clamp it there. 
Open the clips of the Ys; and by means of the levelling-screws bring the bubble to 
the center of its tube. Lift the telescope gently out of the Ys, turn it end for end, and 
put it back again in its reversed position. This being done, if the bubble still remains 
at the center of its tube, this adjustment is in order ; but if it moves toward one end, 
that end is too high, and must be lowered; or else the other end must be raised. 
First, correct half the error by means of the levelling-screws K K, and then the re¬ 
maining half by means of the two small capstan-headed nuts nn. To raise the end 
«, first loosen the upper nut and then tighten the lower one; to do which, turn each 
nut so that the near side moves toward your right. To lower it, first loosen the lower 
nut, then tighten the upper one, moving the near side of each nut toward your left. 
Having thus brought tho bubble to the middle again, again lift the telescope out of 
its Ys; turn it end for end, and replace it. The bubble will now settle nearer tho 
center than it did before, but will probably require still further adjustment. If so, 
correct half the remaining error by the levelling-screws, and half by the nuts, as be¬ 
fore; and so continue to repeat the operation until the bubble remains at the center 
in both positions. For another method, see “ To adjust the long bubble-tube,” p 192. 

Horizontal adjustment of bubble-tube; to see that its axis is in the same plane 
with that of tho telescope, as it usually is iu new instruments. It is not easily do- 








204 


THE LEVEL. 


ranged, except by blows. Have the bubble-tube, as nearly as may be, directly under 
the telescope, or over the center of the bar V F. Bring the telescope over two of the 
levelling-screws KK; clamp it there; center the bubble with said screws; turn the 
telescope in its Ys, say about % iuch, bringing the bubble-tube out from over the 1 
center of the bar, first on one side, then on the other. If the bubble stays centered 
while so swung out, this adjustment is correct. If it runs toward opposite ends of its 
tube when swung out on opposite sides of the center, move the end t of the tube by 
the two horizontal screw’s tt until the bubble stays centered when the tube is swung 
out on either side. If the bubble runs toward the same end of its tube on both side • | 
the tube is not truly cylindrical, but slightly conical* so that if the telescope ( 
turned in its Ys the bubble will leave the center, even when the horizontal adjust- ; 
ment is correct. It is known to be correct, in such tubes, if the bubble runs the same . 


ment is correct. It is known to be correct, in such tubes, if the bubble runs the same , 
distance from the center when swung out the same distance on each side. 

Having made the horizontal adjustment, turn the telescope back in its Ys until the 
bubble-tube is over the bar. Repeat the vertical adjustment (p 203), which may have 
become deranged in making this horizontal one. Persevere until both adjustments 
are to and to be correct at the same time. 

To make the third adjustment, or to adjust the heights of the Ys so 

as to make the line of collimatiou parallel to the bar V F, or perp to the vert axis 
ot the instrument. The other adjustments being made, fasten down the clips of the 
Ys. Make the instrument nearly level by means of all four of the levelling-screws 
Jv. l lace the telescope over two of the levelling-screws which stand diagonally - 
and leave it there unclamped. Then bring the bubble to the center of its tube bv 
the two levelling-screws. Swing the upper part of the instrument half-way around 
so that the telescope shall again stand over the same two screw's; but end for end 
this done, if the bubble leaves the center, bring it half-way back by the large car- 
Stan nuts «>, iv ; and the other half by the two levelling-screw’s. Remember that t. 
raise the Y, and the end of the bubble over mi, w, the lower mi must be loosened ; and 
the upper one tightened ; and vice versa. Now place the telescope over the other 
diagonal pair of levelling-screws: and repeat the whole operation with them. Hav¬ 
ing completed it, again try with the first pair; and so keep on until the bubble re¬ 
mains at the center of its tube, in every position of the telescope 

Correct level Hug may be performed even if all the foregoing adjustments are 
(lit of order, provuled each fore-sight be taken at precisely the same distance from 
the instrument as the back-sight is. But a good leveller will keep his instrument always 
in adjustment; and will test the adjustments at least once a day when at work As 
much, however, depends upon the rodman, or target-man, as upon the leveller A rod- 
man who is careless about holding the rod vert, or about reading the sights correct! v 
should be discharged without mercy. 6 wrretuy, 

. ; !, lie ^veiling-screws in many instruments become very hard to turn if dirtv. Clean 
With water aud a tooth-brush. Use no oil on field instruments 

Forms for level note-books. When the distance is short, so as not to 

require two sets ot books, the following is perhaps as good as any. 


Diff. Level. Grade. I Cut. I Fill. 


No. of I Back I Fore 
Station. | sights. | sights. 

But on public works generally the original field-books have only the first five cols 
Afu i the grades have been determined by means of the profile drawn from these 
fhe results are placed in another book, which has only the first col and the last four 
In both cases, the right-hand page is reserved for memoranda. The wri ter cmiside* 

amllfln'? W } th , th r ! e , ve and with the transit, to consider the term “ Station ” 
apply to the whole dist between two consecutive stakes; and that its number si * 

from staked to s°take 6 IL* S ? ith the «"»*. Station 6 m™ns the d/ 

irorn static b to stake 6 , that it has a bearing or course of so and so; and its lemrt 

is so and so. And with the level, Station 6 also means the dist from stake r > to stab. 

Si «•$ }f*?£ gh .‘ fo f tha V llSt *** w*™ >“ a.*. 5, and „ n s!ak 

and t , t , tbe leve i’ S rade > cut, or fill is that at stake 6 . The starting-point of the 
survey, whether a stake, or any thing else, we call and mark simply 0 . 


sli ami * wi . be . re ! ne ff ,ml ; V by removing the tube and inserting a correctly- 

siapel one, and this is best done by an instrument-maker; but correct work can 

T° v^tv 111 - 8pi + e ° f lt; ’ tliiis ; Make all the adjustments as nearly correct as possible 
Level the instrument. By turning the telescope in its Ys, makrthrrJrtSfl.il.* 

with that on the adjoining Y^vheiAhe 1 bubble-tu be i'a under the^elesTOpe^ 















THE HAND-LEVEL, 


205 


THE HAXD-LEVEL. 



This very useful little instrument, as arranged by Professor Locke, of Cincinnati, is 
but about five or six inches long. Simply holding it in one hand, and looking through 
it in any direction, we can ascertain at once, approximately, what objects are at the 
same level with the eye. E is the eye end: and 0 the object end. L is a small 
level, enclosed in a kind of brass boxing t g , the bottom of which is open, with a cor¬ 
responding opening under it, through the top of the main tube E 0. Immediately 
at the bottom of the small level L, is a cross-wire, stretched across said opening, and 
carried by a small plate, which, for adjusting the wire, can be pushed backward a 
trifle by tightening the screw t , or pushed forward by a small spring within the box¬ 
ing, near g, when the screw t is loosened. At m is a small semicircular mirror a a , 
, ,lvered on the back m. This is placed at an angle of 45°, and occupies one-half the 
V idth of the tube E 0. Through the forementioned openings, the images of the 
.cross-wire and of the level-bubble are reflected down on the unsilvered face a a of 
the mirror, and thence to the eye, as shown by the single dotted lines c and w ; and 
when the instrument is adjusted, and held level, the wire will appear to be at the 
center of the bubble. At lc is one-half of a plano-convex lens, at the inner end of a 
short tube k p. which may be moved backward or forward by a pin n, projecting 
through a short slit in the main tube. By this means the image of the cross-wire is 
rendered distinct; and the half lens must be moved until, when viewing an object, 
the wire shall show no parallax; but appear steady against the object when the eyo 
is slightly moved up or down. At each end of the tube E 0 is a circular piece of 
plain glass for excluding dust. 


To adjust the hand-level, first fix two precisely level marks, say from 
50 feet to 100 yards apart. This being done, rest the instrument against one of the 
level marks, and take sight at the other. If, then, the wire does not appear to be 
precisely at the center of the bubble, move it slightly backward or forward, as the 
case may be, by the screw t, until it does so appear. 

The two level marks may be fixed by means of the 
hand-level itself, even if it is entirely out of adjust¬ 
ment, thus: First, by the pin n arrange the half lens 
’% so as to show the wire distinctly and without paral¬ 
lax. Then holding the level steadily, at any selected 
object, as a, so that the wire appears to cut the center 
of the bubble, see where it cuts any other convenient object, as b. Then go to b, 
' nd from it, in like manner, sight back toward a. If the instrument is in adjust- 
ent, the wire will cut a; but if not, it will strike either above it or below it, as at c. 
either case, make a mark m, half-way between c and a. Then b and m will be the 
• 6 level marks required. With care, these adjustments, when once made, will 
main in order for years. The instrument generally has a small ring r, for hanging 
t, around the neck: it is not adapted to very accurate work, but admirably so for 
Exploring a route. The height of a bare hill can be found by beginning at the foot, 
and sighting ahead at any little chance object which the cross-wire may strike, as a 
pebble, twig, &c ; then going forward, stand at that object, and fix the wire on 
another one still farther on, and so to the top. At each observation we plainly rise 
a height equal to that of the eye, say 5*4 feet, or whatever it may be. Whether 
going up or down it, if the hill is covered with grass, bushes, &c, a target rod must 
be used for the fore-sights; and the constant height of the eye may be regarded as 
the hack-sight at eacli station. An attachment may be made for screwing the level 
to a small ball and socket on top of a cane, or of a longer stick, for occasional use, 
when rather more accuracy is desired.* 



€ 


* Price, about $10 or $12; with about $3 more for attachable ball and socket. 

14 





















206 


LEVELS. 


To arijnst a 1>nil<Icr’s plumb- 
level, t b (l ; Stand it upon any two sup¬ 
ports m and n, and mark where the plumb- 
line cuts at o. Then reverse it, placing the 
foot t, upon n, and <1 upon w, and mark whore 
the line now cuts at c. Ilalf-way between o 
and c make the permanent mark. Whenever 
the line cuts this, the feet t and d are on a 
level. 



♦J T ? s^pe-instrnment, or clinometer. As usually madJ 

the bubble-tube is attached to the movable bar by a screw near each end, and th 
head of one of the screws conceals a small slot in the bar, which allows a slight ver 
motion to the screw when loose, and with it to that end of the tube. Therefore iif 
order to adjust the bubble, this screw is first loosened a little, and then moved’ui! 
or uown a trine, as may be re<jd. It is then tightened again. 











LEVELLING BY THE BAROMETER. 


207 


LEVELLING BY THE BAROMETER. 

1. Many circumstances combine to render the results of this kind of levelling un¬ 
reliable where great accuracy is required. This fact was most conclusively proved 
by the observations made by Captain T. J. Cram, of the U. S. Coast Survey. See 
Report of U. S. C. S., vol. for 1S54. It is difficult to read off from an aneroid (the 
kind of barorn generally employed for engineering purposes) to within from two to 
five or six ft, depending on its size. The moisture or dryness of the air affects the 
results; also winds, the vicinity of mountains, and the daily atmospheric tides, 
Which cause incessant and irregular fluctuations in the barom. A barom hanging 
quietly in a room will often vary yg of an inch within a few hours, corresponding 
Vo a diff of elevation of nearly 100 ft. No formula can possibly be devised that shall 
embrace these sources of error. The variations dependent upon temperature, lati¬ 
tude, &c, are in some measure provided for; so that with very delicate instruments, a 
skilful observer may measure the diff of altitude of two points close together, such 
as the bottom and top of a steeple, with a tolerable confidence that he is within two 
or three feet of the truth. But if as short, an interval as even a few hours elapses 
between his two observations, such changes may occur in the condition of the atmo¬ 
sphere that he may make the top of the steeple to be lower than its bottom ; or at 
least, cannot feel by any means certain that he is not ten or twenty ft in error; and 
this may occur without any perceptible change in the atmosphere. Whenever prac¬ 
ticable, therefore, there should be a person at each station, to observe at both points 
J <■ t the same time. Single observations at points many miles apart, and made on dif- 
c,hrent days, and in different states of the atmosphere, are of little value. In such 
cases the mean of many observations, extending over several days, weeks, or months, 
and made when the air is apparently undisturbed, will give tolerable approximations 
to the truth. In the tropics the range of the atmospheric pres is much less than 
in other regions, seldom exceeding % inch at any one spot; also more regular in 
time, and, therefore, less productive of error. Still, the barometer, especially either 
the aneroid, or Bourdon’s metallic, may be rendered highly useful to the civil engi¬ 
neer, in cases where great accuracy is not demanded. By hurrying from point to 
point, and especially by repeating, he can form a judgment as to which of two sum¬ 
mits is the lowest. Or a careful observer, keeping some miles ahead of a surveying 
party, may materially lessen their labors, especially in a rough country, by select¬ 
ing the general route for them in advance. The accounts of the agreement within 
a few inches, in the measurements of high mountains, by diff observers, at diff 
periods ; and those of ascertaining accurately the grades of a railroad, by means of 
an aneroid, while riding in a car, will be believed by those only who are ignorant 
of the subject. Such results can happen only by chance. 

When possible, the observations at different places should bo taken at the same 
time of day, as some check upon the effects of the daily atmospheric tides; and in 
■very important cases, a memorandum should be made of the year, month, day, and 
hour, as well as of the state of the weather, direction of the wind, latitude of the 
place, &c, to bo referred to an expert, if necessary. 

TS»e effects of lalitude are not included in any of our formulas. When 
reqd they may be found in the table page 209. Several other corrections must be 
made when great accuracy is aimed at; but they require extensive tables. 

In rapid railroad exploring, however, such refinements may be neglected, inas¬ 
much as no approach to such accuracy is to be expected; but on the contrary, errors 
of from 1 to 10 or more feet in 100 of height, will frequently occur. 
y^As a very rough average we may assume that the barometer falls 
/nch for every 90 feet that we ascend above the level of the sea, up to 1000 ft. But 
in fact its rate of fall decreases continually as we rise; so that at one mile high it 
falls inch for about 106 ft rise. Table 2 shows the true rate. 




208 


LEVELLING BY THE BAROMETER, 


To ascertain the diff of height between two points. 

Rule 1. Take readings of the barom and therm (Fall) in the shade at both 
stations. Add together the two readings of the barom, and div their sum by 2, for 
their mean ; which call b. Do the same with the two readings of the thermom, and 
call the mean t. Subtract the least reading of the barom from the greatest; and call 
the diff d. Then mult together this diff d; the number from the next Table No. 1, 
opposite t; and the constant number 30. Div the prod by b. Or 

Height _ Diff (d) of Tabular number opposite v. n . . on 
iu feet barom mean ( t) of therm om ^ onstan t ou. 

mean ( b) of barom. 

Example. Reading of the barom at lower station, 26.64 ins ; and at the upper; 
sta 20.82 ins. Thermom at lowest sta, 70°; at upper sta, 40°. W hat is the dill id, 
height of the two stations ? Here, 

Barom, 26.64 Therm, 70° 

“ 20.82 “ 4()o 

- Also, -- 

2)47.46 2)110 


23. <3 mean of bar, or b. 65° mean of 

. , , , therm, or t. 

The tabular number opposite 55°, is 917.2. 

Bar. Bar. 

Again, 26.64 — 20.82 = 5.82, diff of bar; or d. Hence, 
d, Tab No. Con. 

Height _ 5.82 X 917.2 X 30 160143.12 R>71S * 

‘“foot- 23.73 (or b) = - ^73“ “ ^ ft; anSWOr - 

Then correct for latitude, if more accuracy is reqd, by rule on next page. 

The screw at the back of an aneroid is for adjusting the index by a stand 
ard barom. After this has been done it must by no means be meddled with. Ii 
6ome instruments specially made to order with that intention, this screw mav be 
used also lor turning the index back, after having risen to an elevation so great that 
the index has reached the extreme limit of the graduated arc. After thus turninx 
it back, the indications of the index at greater heights must be added to that at 
tamed when it was turned back. 

TABLE 1. For Rule 1. 


Mean 

of 

Ther. 


0 ° 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 
29 


No. 


801.1 

803.2 

805.3 

807.4 

809.5 

811.7 

813.8 

815.9 
818.0 
820.1 
822.2 

824.3 

826.4 

828.5 

830.6 

832.8 

834.9 
837.0 

839.1 

841.2 

843.3 

845.4 

847.5 

849.6 

851.8 

853.9 
856.0 

858.1 

860.2 
862.3 


Mean 

of 

Ther. 


303 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 


No. 

Mean 

of 

Ther. 

No. 

Mean 

of 

Ther. 

No. 

864.4 

600 

927.7 

90° 

991.0 

866.5 

61 

929.8 

91 

993.1 

868.6 

62 

931.9 

92 

995.2 

870.7 

63 

934.0 

93 

997.3 

872.8 

64 

936.1 

94 

999.4 

874.9 

65 

938.2 

95 

1001.6 

877.0 

66 

940.3 

96 

1003.7 

879.2 

67 

942.4 

97 

1005.8 

881.3 

68 

944.5 

98 

1007 9 

883.4 

69 

946.7 

99 

1010.0 

885.4 

70 

948.8 

100 

1012.1 

887.5 

71 

950.9 

101 

1014.2 

889.6 

72 

953.0 

102 

1016.3 

891.7 

73 

955.1 

103 

1018.4 

893.8 

74 

957.2 

104 

1020.5 

896.0 

75 

959.3 

105 

1022.7 

898.1 

76 

961.4 

106 

1024.8 

900.2 

77 

963.5 

107 

1026.9 

902.3 

78 

965.6 

108 

1029.0 

904.5 

79 

967 7 

109 

1031.1 

906.6 

80 

969.9 

110 

1033.2 

908.7 

81 

972.0 

111 

1035.3 

910.8 

82 

974.1 

112 

1037.4 

913.0 

83 

976.2 

113 

1039.5 

915.1 

84 

978.3 

114 

1041.6 

917.2 

85 

980.4 

115 

1043 8 

919.3 

86 

982.6 

116 

1045.9 

921.4 

87 

984.7 

117 

1048.0 

923.5 

88 

986.8 

118 

1050.1 

925.6 

89 

988.9 

119 

1052.2 

































LEVELLING BY THE BAROMETER. 209 


Ri ; le 2. BelvIIle's short approx rule is the one best adapted to rapid 
ield use, namely, add together the two readings of the barom only. Also find the 
iitr between said two readings; then, as the sum of the two readings 
is to their diir, so is 55000 feet to the reqd altitude. 

Ex. Same as before. Here the sum = 26.64 -p 20.82 — 47.46 ins; and the diff = 
*6.64 — 20.82 = 5.82 ins. 

Sum. Diff. Feet. Feet. 

Hence, as 47.46 : 5.82 :: 55000 : 6744.6 reqd alt; instead of the 6748.5 of Rule 1. 

Correction for latit ude. The lat of the place affects the result to some 
xtent; but is usually omitted when great accuracy is not reqd. To apply the correction, first find 
he altitude by the rule, as before. Then divide it by the number iu the following table opp the lat 
f the place. Add the quotient to the alt if the lat is less than 45°; or subtract it if the lat is more 
ban 45°. If the two places are iu diff lats, use their mean. 


Table of corrections for latitude. 


Lat. 

0 ° 

352 

Lat. 

140 

399 

Lat. 

28° 

630 

L&ti 

42° 

3367 

Lat. 

54° 

1140 

Lat. 

68° 

490 

2 

354 

16 

416 

30 

705 

44 

10101 

56 

941 

70 

460 

4 

356 

18 

436 

32 

804 

45 

0 

58 

804 

72 

436 

6 

360 

20 

460 

34 

941 

46 

10101 

60 

705 

74 

416 

8 

367 

22 

490 

36 

1140 

48 

3367 

62 

630 

76 

399 

10 

375 

24 

527 

38 

1458 

50 

2028 

64 

572 

78 

386 

12 

386 

26 

572 

40 

2028 

52 

1458 

66 

527 

80 

375 


Levelling by Barometer; or by the boiling point. 

Rule 3. The following table. No. 2, enables us to measure heights either by means 
f boiling water, or by the barom. The third column shows the approximate alti- 
ude above sea-level corresponding to diff heights, or readings of the barom; and to 
he diff degrees of Fahrenheit’s thermom,at which water boils in the open air. Thus 
hen the barom, under undisturbed conditions of the atmosphere, stands at 24.08 
iches, or when pure rain or distilled water boils at the temp of 201° Fall; the place 
i about 5764 ft above the level of the sea, as shown by the table. It is therefore 
ery easy to find the diff of altitude of two places. Thus : take out from table No 2, 
tie altitudes opposite to the two boiling temperatures; or to the two barom readings, 
ubtract the one opposite the lower reading, from that opposite the upper reading, 
he rent will be the reqd height, as a rough approximation. To correct this, add 
jgether the two therm readings; and div the sum by 2, for their mean. From table 
>r temperature, p 211, take out the number opposite this mean. Mult the ap- 
roximate height just found, by this tabular number. Then correct for lat if reqd. 

Ex. The same as preceding; namely, barom at lower sta, 26.64; and at uppersta. 
1.82. Thermom at lower sta, 70° Fuh; and at the upper one, 40°. What is the diff 
r height of the two stations ? 

Alt. 


Here the tabular altitudes are, for 20.82. 9579 

and for 26.64. 3115 


6464 ft, approx height. 

70° 4- 40° 110° 

To correct this, we have -—-= „ - = 65° mean; and in table p 211, opp t* 

, wo find 1.048. Therefore 6464 X 1-018 = 6774 ft, the reqd height. 

This is about 26 ft more than by Rule 1 ; or nearly .4 of a ft in each 100 ft. 

At 70° Fah, pure water will boil at 1° less of temp, for an average of about 550 ft 
elevation above sea-level, up to a height of % a mile. At the height of 1 mile, 1° 
' boiling temp will correspond to about 560 ft of elevation. In table p 210 the 
ean of the temps at the two stations is assumed to be 32° Fah ; at which no correc- 
on for temp is necessary in using the table; hence the tabular number opposite 
!°, in table p 211, is 1. 

This diff produced in the temp of the boiling point , by change of elevation, must 
)t be confounded with that of the atmosphere, due to the same cause. The air be- 
.mes cooler as we ascend above sea-level, at the rate (very roughly) of about 1° Fah 
r every 200 ft near sea-level, to 350 ft at the height of 1 mile. See “ Air,” p 215. 
The following table. No. 2, (so far as it relates to the barom.) was de¬ 
iced by the writer from the standard work on the barom by Lieut.-Col. R. S. Wil- 
tmson, U. S. army.* 


* Published by permission of Government in 1868 by Van Nostraud, N. Y. 



























210 LEVELLING BY THE BAROMETER, ETC 


TABLE 2. 


Levelling- by Barometer; or by the boiling: point. 

Assumed temp in the shade 32° Fah. If not 32°, mult barom alt as per Table, p 211. 


Boil 
point 
in deg 
Fah. 

Barom. 

Ins. 

Altitude 
above 
sea level 
Feet. 

Boil 
point 
in deg 
Fah. 

Barom. 

Ins. 

Altitude 
above 
sea level 
Feet. 

Boil 
point 
in deg 
Fah. 

Barom. 

Ins. 

Altitude 
above 
sea level 
Feet. 

184° 

16.79 

15221 

.3 

19.66 

11083 

.6 

22.93 

7048 

.1 

16.83 

15159 

.4 

19.70 

11029 

.7 

22.98 

6991 

.2 

16.86 

15112 

.5 

19.74 

10976 

.8 

23.02 

6945 

.3 

16.90 

15050 

.6 

19.78 

10923 

.9 

23.07 

6888 

.4 

16 93 

15003 

.7 

19.82 

10870 

199 

23.11 

6843 

.5 

16 97 

14941 

.8 

19.87 

10804 

.1 

23.16 

6786 

.6 

17.00 

14895 

.9 

19.92 

10738 

.2 

23.21 

6729 

.7 

17 01 

14833 

192 

19.96 

10685 

.3 

23.26 

6673 

.8 

17.08 

14772 

.1 

20.00 

10(533 

.4 

23.31 

6617 

.9 

17.12 

14710 

.2 

20.05 

10567 

.5 

23.36 

6560 

186 

17.16 

14649 

.3 

20.10 

10502 

.6 

23.40 

6516 

• 1 

17.20 

14588 

.4 

20.14 

10450 

.7 

23 45 

6160 

.2 

17.23 

14543 

.5 

20.18 

10398 

.8 

23.49 

6415 

•3 

17.27 

14482 

.6 

20.22 

10346 

.9 

23 54 

6359 

.4 

17.31 

14421 

.7 

20.27 

10281 

200 

23.59 

$i04 

•5 

17.35 

14361 

.8 

20.31 

10230 

.1 

23 64 

6248 

•6 

17.38 

14315 

.9 

20.35 

10178 

.2 

23.69 

6193 

.7 

17.42 

14255 

193 

20.39 

10127 

.3 

23.74 

6137 

•8 

17.46 

14195 

.1 

20.43 

10075 

.4 

23.79 

6082 

.9 

17.50 

14135 

.2 

20.48 

10011 

.5 

23 84 

6027 

186 

17.54 

14075 

.3 

20.53 

9947 

.6 

23.89 

5972 

• 1 

17.58 

14015 

.4 

20.57 

9898 

.7 

23.94 

5917 

.2 

17.62 

13956 

.5 

20.61 

9845 

.8 

23.98 

5874 

•3 

17.66 

13896 

.6 

20 65 

9794 

.9 

24.03 

5819 

.4 

17.70 

13837 

.7 

20.69 

9743 

201 

24.08 

5764 

.5 

17.74 

13778 

.8 

20.73 

9693 

.1 

24.13 

5710 

.6 

17.78 

13718 

.9 

20.77 

9642 

.2 

24.18 

5656 

•T 

17.82 

13660 

194 

20.82 

9579 

.3 

24.23 

5602 

.8 

17.86 

13601 

.1 

20.87 

9516 

.4 

24.28 

5547 

.9 

17.90 

13542 

.2 

20.91 

9466 

.5 

24.33 

5494 

187 

17,93 

13498 

.3 

20 96 

9403 

.6 

24.38 

5440 

.1 

17.97 

13440 

.4 

21.00 

9353 

.7 

24.43 

5386 

•2 

18.00 

13396 

.5 

21.05 

9291 

.8 

24.48 

5332 

.3 

18.04 

13338 

.6 

21.09 

9241 

.9 

24.53 

5279 

.4 

18.08 

13280 

.7 

21.14 

9179 

202 

24.58 

5225 

•3 

18.12 

13222 

.8 

21.18 

9130 

.1 

24.63 

5172 

.6 

18 16 

13164 

•9 

21.22 

9080 

.2 

24.68 

5119 

.7 

18.20 

13106 

195 

21.26 

9031 

.3 

24.73 

5066 

•8 

18.24 

13049 

.1 

21.31 

8969 

.4 

24.78 

5013 

.9 

18.28 

12991 

.2 

21.35 

8920 

.5 

24.83 

4960 

188 

18.32 

12934 

.3 

21.40 

8859 

.6 

24.88 

4907 

.1 

18.36 

12877 

.4 

21.44 

8810 

.7 

24.93 

4855 

• 2 

18.40 

12820 

.5 

21.49 

8749 

.8 

24.98 

4802 

.3 

18.44 

12763 

.6 

21.53 

8700 

.9 

25.03 


.4 

18.48 

12706 

.7 

21.58 

8639 

203 

25.08 

4697 

.5 

18.52 

12649 

.8 

21.62 

8590 

.1 

25.13 

4645 

.6 

18.56 

12593 

.9 

21.67 

8530 

.2 

25.18 

4593 

.7 

18.60 

12536 

196 

21.71 

8481 

.3 

25.23 

4541 

.8 

18.64 

12480 

.1 

21.76 

8421 

.4 

25.28 

4489 

.9 

18.68 

12424 

.2 

21.81 

8361 

.5 

25.33 

4437 

189 

18.72 

12367 

.3 

21.86 

8301 

.6 

25.38 

4386 

.1 

18.76 

12311 

.4 

21.90 

8253 

.7 

25.43 

4334 

.2 

18.80 

12256 

.5 

21.95 

8193 

.8 

25.49 

4272 

.3 

18.84 

12200 

.6 

21 99 

8145 

.9 

25.54 

4221 

.4 

18.88 

12144 

.7 

22.04 

8086 

204 

25.59 

4169 

.5 

18.92 

12089 

.8 

22.08 

8038 

.1 

25.64 

4118 

.6 

18.96 

12033 

.9 

22.13 

7979 

.2 

25.70 

4057 

.7 

19.00 

11978 

197 

22.17 

7932 

.3 

25.76 

3996 

.8 

19.04 

11923 

.1 

22.22 

7873 

.4 

25.81 

3945 

.9 

19.08 

11868 

.2 

22.27 

7814 

.5 

25.86 

3894 

190 

19.13 

11799 

.3 

22.32 

7755 

.6 

25.91 

3844 

.1 

19.17 

11745 

.4 

22.36 

7708 

.7 

25.96 

3793 

.2 

19.21 

11690 

.5 

22.41 

7649 

.8 

26.01 

3742 

.3 

19.25 

11635 

.6 

22.45 

7602 

.9 

26.06 

3692 

.4 

19.29 

11581 

.7 

22.50 

7544 

205 

26.11 

3642 

.5 

19.33 

11527 

.8 

22.54 

7498 

.1 

26.17 

3582 

.6 

19.37 

11472 

.9 

22.59 

7439 

.2 

26.22 

3532 

.7 

19.41 

11418 

198 

22.64 

7381 

.3 

26.28 

3472 

.8 

19.45 

11364 

.1 

22.69 

7324 

.4 

26.33 

3422 

.9 

19.49 

11310 

.2 

22.74 

7266 

.5 

26.38 

3372 

191 

19.54 

11243 

.3 

22.79 

7208 

.6 

26.43 

3322 

.1 

19.58 

11190 

.4 

22.84 

7151 

.7 

2(».48 

3273 

.2 

19.62 

11136 

.5 

22.89 

7093 

.8 

26.54 

3213 


Boil 
point 
in deg 
Fah. 

Barom. 

Ins. 

Altituda 
above 
sea level 
Feet. 

.9 

26 59 

3164 

206 

26 64 

3115 

.1 

26 69 

3066 

.2 

26-75 

3007 ' 

.3 

26.80 

2958 

.4 

26 86 

2899 


.5 

26.91 

2850 

.6 

26.97 

2792 

.7 

27.02 

2743 

.8 

27.08 

2685 

.9 

27.13 

2637 

207 

27.18 

2589 

.1 

27.23 

2540 

.2 

27.29 

2483 

.3 

27.34 

2435 

.4 

27.40 

2377 

.5 

27.45 

2329 J 

.6 

27.51 

2272 i 

.7 

27.56 

2224 ’ 

.8 

27.62 

2167 

.9 

27.67 

2120 

208 

27.73 

2063 

.1 

27.78 

2016 

.2 

27.84 

1959 

.3 

27.89 

1912 

.4 

27.95 

1856 

.5 

28.00 

1809 

.6 

28.06 

1753 

.7 

28.11 

1706 

.8 

28.17 

1650 

.9 

28.23 

1595 

209 

28.29 

1539 

.1 

28.35 

1483 

.2 

28.40 

1437 

.3 

28.45 

1391 

.4 

28.51 

1336 

.5 

28.56 

1290 

.6 

28.62 

1235 

.7 

28.67 

1189 

.8 

28.73 

1134 

.9 

28.79 

1079 

210 

28.85 

1025 

.1 

28.91 

970 

.2 

28.97 

916 

.3 

29.03 

862 

.4 

29.09 

808 

.5 

29.15 

754 

.6 

29.20 

709 

.7 

29.25 

664 

.8 

29.31 

610 

.9 

29.36 

565 

211 

29.42 

512< 

.1 

29.48 

458^ 

.2 

29.54 

405 

.3 

29.60 

352 

.4 

29.65 

308 

.5 

29.71 

255 

.6 

29.77 

202 

.7 

29 83 

149 

.8 

29.88 

105 

.9 

29.94 

52 

212 

30.00 

sea lev=C 

Below sea level. 

.1 

30.06 

— 52 

.2 

30.12 

—104 

.3 

30.18 

—156 

.4 

30.24 

—209 

.5 

30.30 

—261 

.6 

30.35 

—304 

.7 

30.41 

—356 

.8 

30.47 

—40s 

.9 

30.53 

—459 

213 

30.59 

—611 


































SOUND 


211 


Corrections for temperature; to be used in connection with 
ltule 3, when greater accuracy is necessary. Also in con¬ 
nection with Table 2 when the temp is not 32°. 


Mean 


Mean 


Mean 


Mean 


temp 

Mult 

temp 

Mult 

temp 

Mult 

temp 

Mult 

iu the 

by 

in the 

by 

in the 

by 

in the 

by 

shade. 


shade. 


shade. 


shade. 


Zero. 

.933 

28° 

.992 

56° 

1.050 

84° 

1.108 

2° 

.937 

30 

.996 

58 

1.054 

86 

1.112 

4 

.942 

31 

1.000 

60 

1 058 

88 

1.117 

6 

.946 

34 

1.004 

62 

1.062 

90 

1.121 

8 

.950 

36 

1.003 

64 

1.066 

92 

1.125 

10 

.954 

38 

1.012 

66 

1.071 

94 

1.129 

12 

.958 

40 

1.016 

68 

1.075 

96 

1.133 

U 

.962 

42 

1.020 

70 

1.079 

98 

1.138 

16 

.967 

44 

1.024 

72 

1.083 

100 

1.142 

18 

.971 

46 

1.028 

74 

1.087 

102 

1.146 

20 

.975 

43 

1.032 

76 

1.091 

104 

1.150 

22 

.979 

50 

1.036 

78 

1.096 

106 

1.154 

24 

.933 

52 

1.041 

80 

1.100 

108 

1.158 

26 

.987 

54 

1.046 

82 

1.104 

110 

1.163 


SOUND. 


The velocity of sound in quiet open air, has been experimentally deter¬ 
mined to be very approximately 1090 feet per second, when the temperature is at 
freezing point, or 32° Fahrenheit. For every degree Fahrenheit ot increase of 
temperature, the velocity increases by from foot to feet per second, accoiding 
to different authorities. Taking the increase at 1 foot per second tor each degree 
(which agrees closely with theoretical calculations), we have 


at 

_ 30° 

Falir 1030 feet per sec 

= 0.1951 

a 

— 20° 


1040 

it 

tt 

= 0.1970 

a 

— 10° 

tt 

1050 

a 

a 

= 0.1989 

tt 

0 

tt 

1060 

tt 

a 

= 0.2008 

tt 

10° 

tt 

1070 

u 

a 

= 0.2027 

a 

20° 

it 

1080 

tt 

a 

= 0.2045 

a 

32° 

it 

1092 

ti 

a 

= 0.2068 

a 

4o° 

it 

1100 

tt 

a 

= 0.2083 

u 

50° 

tt 

1110 

tt 

a 

= 0.2102 

a 

60° 

tt 

1120 

tt 

a 

= 0.2121 

a 

70° 

tt 

1130 

tt 

a 

= 0.2140 

a 

80° 

tt 

1140 

ti 

a 

= 0.2159 

a 

90° 

tt 

1150 

n 

a 

= 0.2178 

a 

100° 

ti 

1160 

a 

a 

= 0.2197 

u 

110° 

tt 

1170 

a 

a 

= 0.2216 

a 

120° 

ti 

1180 

t i 

a 

= 0.2235 


« 

ii 

a 

a 

a 

u 

a 

tt 

a 

u 

it 

u 

a 

a 


u 

a 

u 

a 

tt 

tt 

a 

a 

u 

tt 

tt 

a 

tt 

tt 


1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 


44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

4 < 


5.08 

5.03 

4.08 

4.93 

4.88 

4.83 

4.80 

4.78 

4.73 

4.68 

4.63 

4.59 

4.55 

4.51 

4.47 


44 
44 
44 
44 
44 
44 
44 
44 
' 44 
it 
it 
44 
44 
44 
44 


If the air is calm, fog 


JI me an . D .v, & or rain does not appreciably affect the result; but winds do. 

Very loud sounds appear to travel somewhat faster than iow ones. The watchword 
>f sentinels .has been heard across still water, on a calm night, 10)4 miles, and a 
•annon 20 miles. Separate sounds, at intervals of jU of a second, cannot be c istin- 
ruished, but appear to be connected. The distances at which a speaker can he 
jndersto id in fiont, on one side, and behind him. are about as 4, 3, and . 

Dr Charles M. Cresson informs the writer that, by repeated trial", he fount i< 
n a Philadelphia gas main 20 inches diameter and 1 <>000 

n tin* i -Li th but empty of gas, and having one horizontal bend of 90 , and ot 4U teec 
-tdius the sound of pistol-shot travelled 16000 feet iu precisely 16 seconds, or 1000 
ect per second The arrival of the sound whs barely audible; but was rendered 
?ery apparent to tbe eye by its blowing off a diaphragm ot tissue-paper placed over 

;1 Tw d o 0 boatrancliore«l some distance apart may serve as a base line for 

■liangulating objects along the coast; the distance between them being hist found 

4708 fort per second, or *bout 4 that 

Iu woods, it is from 10 to 16 times; and lu metals, trorn 4 to 16 times 


eater than in air, according to some authorities. 























212 


HEAT. 

' 


Approximate expansion of solids by heat: and their melt* 
*ng points by Fahrenheit's thermometer. £ 


-r 


fi re-brick. 
Granite. J. 


5 from 

. / to 

Glass rod.;. 

Glass tube. 

*• crown. 

plate.». 

Platina.. j. 

Marble, glranular, white, dry,. 

“ moist. 

* black, compact 

Antimony . 

Cast iron... 

Slate.... 

Steel. 

“ blistered, .... 

“ untempered. 

“ tempered yellow.. 

“ hardened. 

4 ‘ annealed. 

Iron, rolled. 

“ soft, forged. 

“ wire. 

Bismuth... 

Gold, annealed. 

Copper...average 

Sandstone-} - .. 

Brass. average 

wire. 

Silver. 

Tin. average 

Lead. average 

Pewter. 

Zinc (most of all metals). 

White pine. 


For 1 degree. 

For 180 degrees.* 

1 part in 

% inch in 

1 part in 

h inch in 

365220 

3804 ft. 

2029 

21.14 ft. 

187560 

1954 

1042 

10.85 

228060 

2375 

1267 

13.20 

221100 

2306 

1230 

12.81 

214200 

2231 

1190 

12.40 

211500 

2203 

1175 

12.24 

209700 

2184 

1165 

12.13 

208800 

2175 

1160 

12.08 

173000 

1802 

961 

10.00 

128000 

1333 

711 

7.41 

405000 

4219 

2250 

23 44 

166500 

1722 

925 

9.63 

16200011 

1688 

900 

9.38 

173000 

1802 

961 

10.00 

151200 

1575 

810 

8.75 

159840 

1665 

888 

9.25 

167400 

1744 

930 

9.69 

131400 

1369 

730 

7.60 

146880 

1530 

816 

8.50 

147600 

1537 

820 

854 

149940H 

1562 

833 

8.68 

147420 

1536 

819 

8.53 

146340 

1524 

813 

8 55 

129600 

1350 

720 

7.50 

123120 

1282 

684 

7.12 

101400 

1088 

580 

6.04 

103320 

1076 

574 

5.98 

97740 

1018 

543 

5.66 

94140 

981 

523 

5.45 

95040 

990 

528 

5.50 

87840 

915 

488 

5.08 

63180 

658 

351 

3.66 

78840 

821 

438 

4.56 

61920 

645 

344 

3.58 

440530 

4588 

2447 

25.49 


Melting 
point 
in Deg.J 


4593 


J 

' 


955 

1920 to 
•2800 
2370 to 
2550 


3000 to 
3500 


506 

2016 

2000 


< 


1873 


1861 

444 

612 


680 to T72 




on^ou?^”^"^;®^’ Vriri0U3lye8timaled from 800 10 1140 de S- That of a charcoal 

Each 12° to 1«> 0 of heat produce in wrot iron an expansion equal to 

that produced by a teusmn of about 1 ton per sq inch of section ; vary mg with the quality of the iron. 

For temperature, expansion, conducting power, &c, of air, see p 216. 


* By adding part to the lengths in the two cols under 180°, 
a number of degrees less than 180°; or to 163°,63 deg; which 

Of t6MP ill thiG COl(ipr nnrtinna r»f the -r_ n. . . 


we get the lengths corresponding to 
may be taken as about the extreme* 


rif ii. ’ -- v* Liinuu as aoout tne extreme* 

part ° f ““ SUte9 * the Midd,e Statea the ext ^mes rarely reach 


No dependence whatever is to be placed on results obtained by Wedgewood’s pyrometer. 

fength 8 f sjsss, £’ri"5,rL”Hi sS*; Sr*’- 

tension within el* tic limR^aa* st^tchea^wk^^jiXM^rtT^ m0re than cast » whereaa uud e«- 























































THERMOMETERS. 


213 


THERMOMETERS. 


Below about 35® below zero of Fah, the mercurial thermometer and barometer become too irregular fa 
be depended on. Mercury begins to freeze at about -40° Fah. 

To change degrees of Fahrenheit to the corresnondinir de¬ 
grees Of Fentigrade; take a Fah reading .32° lower than thriven one* mult 
th s lower reading by o ; divide the prod by 9. Thus : +14° Fah = (14 -321 X 5 

^ ga ' u ’ £ a !l~ X 5 + 9 = —45 X 5 + 9 = —25° Cent. * ^ ^ ^ ® 

r ° to Reau; take a Fall reading 32° lower than the given 

one, multthjs lower reading by 4; div the prod by 9. Thus: +14° Fah — (14— 32, X 4 — 9 ——ih X 

4 ~i!~ - g 8 .° Reau - Again -13° Fah = (-13 -32) X 4 + 9 = -T5 X 4 + 9 =-20° R t *u *-* 

, change Fentto l 1 ah; mult the Cent reading l>v 9; divide liv 5 Taken 

'Again ^— 20 ° Cent ~^0X 9 M = (1 ° * 9 +32 = 18 +32 = +50° Fa* 

«*• Tlm ’ : s-wo-t-iox 

To change Reau to Fah ; mult by 9 ; div hy 4 

Th 1 u u s i+ 16 ° Reau -ae X9 + 4) + 82= 36 + 32 = +68° Fah. 

4) -f-oi — —18-f-32 = -f-14 0 ¥ ah. 

To change Reau to tent; mult hy 5 : div by 4. 

X°~4=+10°Cent. Again,—8° Reau = —8X5 + 4=—10° Cent. 

TABLE !. Fahrenheit compared with Centigrade and Reau¬ 
mur. In this table the Cent and Reau readings are given to the nearest decimal. 




Take a Fah reading 32° 
Again . — 8 ° Reau = (—8 X 9 + 

Thus : +8° Reau = + 8° 


F. 

/ - 

c. 

R. 

F. 

€. 

R. 

F. 

€. 

R. 

F. 

F. 

R. 

F. 

F. 

R. 

O 

0 

O 

O 

O 

0 

O 

O 

O 

O 

O 

O 

0 

O 


212 

100 

80.0 

158 

70.0 

56.0 

104 

40 0 

32.0 

50 

10.0 

8.0 

—8 

—19 4 

15 ft 

21 ! 

99.4 

79.6 

157 

69.4 

55.6 

103 

39.4 

31.6 

49 

9.4 

7.6 

—4 

—20 0 

16 0 

210 

98.9 

79.1 

156 

68.9 

55.1 

102 

38.9 

31.1 

48 

8.9 

7.1 

—5 

—20 6 

16 4 

209 

98.3 

78.7 

155 

68 3 

54.7 

101 

38 3 

30.7 

47 

8.3 

6.7 

—6 

—21 1 

1H Q 

208 

97.8 

78.2 

154 

67.8 

54.2 

100 

37.8 

30.2 

46 

7.8 

6.2 

—7 

— 21.7 

17 3 

207 

97.2 

77.8 

153 

67.2 

53.8 

99 

37.2 

29.8 

45 

7.2 

5.8 

—8 

— 22.2 

17 8 

206 

96.7 

77.3 

152 

66.7 

53.3 

98 

36.7 

29.3 

44 

6.7 

5.3 

—9 

—22 8 

18 2 

205 

96.1 

76.9 

151 

66.1 

52.9 

97 

36.1 

28.9 

43 

6.1 

4.9 

—10 

—23 3 

18 7 

204 

95.6 

76.4 

150 

65.6 

52.4 

96 

35.6 

28.4 

42 

6.6 

4.4 

—11 

—23 9 

_ 19 1 

203 

95.0 

76.0 

149 

65.0 

52.Q 

95 

35.0 

28.0 

41 

5.0 

4.0 

—12 

— 24.4 

— 19 6 

202 

94.4 

75.6 

148 

64.4 

51.6 

94 

34.4 

27.6 

40 

4.4 

3.6 

—13 

—25 0 

—90 0 

201 

93.9 

75.1 

147 

63.9 

51.1 

93 

33.9 

27.1 

39 

3.9 

S.l 

—14 

— 25 6 

90 4 

200 

93.3 

74.7 

146 

63.3 

50.7 

92 

33.3 

26.7 

38 

3.3 

2.7 

— 15 

— 26.1 

—20 9 

199 

92.8 

74.2 

145 

62.8 

50.2 

91 

32.8 

26.2 

37 

2.8 

2.2 

—16 

— 26.7 

—21 3 

198 

92.2 

73.8 

144 

62 2 

49.8 

90 

32.2 

25.8 

36 

2.2 

1.8 

—17 

- 27.2 

—21 8 

197 

91.7 

73.3 

143 

61.7 

49.3 

89 

31.7 

25.3 

35 

1.7 

1.3 

— 18 

— 27.8 

— 22.2 

196 

91.1 

72.9 

142 

61.1 

48.9 

88 

31.1 

24.9 

34 

1.1 

(Kft 

—19 

— 28.3 

— 22 7 

195 

90.6 

72.4 

141 

60.6 

48.4 

87 

30.6 

24.4 

33 

0.6 

0.4 

—20 

— 28.9 

— 23.1 

194 

90.0 

72.0 

140 

60.0 

48.0 

86 

30.0 

24.0 

32 

0.0 

0.0 

—21 

— 29.4 

— 23.6 

193 

89.4 

71.6 

139 

59.4 

47.6 

85 

29.4 

23.6 

31 

— 0.6 

— 0.4 

—22 

— 30.0 

— 24.0 

192 

88.9 

71.1 

138 

58.9 

47.1 

84 

‘ 28.9 

23.1 

SO 

— 1.1 

— 0.9 

—23 

— 30.6 

— 24.4 

191 

88.3 

70.7 

137 

58.3 

46.7 

83 

28.3 

22.7 

29 

— 1.7 

— 1.3 

—24 

— 31.1 

—24 9 

190 

87.8 

70.2 

136 

57.8 

46.2 

82 

27.8 

22.2 

28 

— 2.2 

- 1.8 

—25 

— 31.7 

— 25.3 

1S9 

87.2 

69.8 

135 

57.2 

45.8 

81 

• 27.2 

21.8 

27 

— 2.8 

— 2.2 

—26 

— 32.2 

— 25.8 

188 

86.7 

69.3 

134 

56.7 

45.3 

80 

26.7 

21.3 

26 

— 3.3 

— 2.7 

—27 

— 32.8 

— 26.2 

187 

86 1 

68.9 

133 

56.1 

44.9 

79 

26.1 

20.9 

25 

— 3.9 

— 3.1 

—28 

— 33.3 

— 26.7 

186 

85.6 

68.4 

132 

55.6 

44.4 

78 

25.6 

20.4 

24 

— 4.4 

— 3.6 

—29 

— 33.9 

— 27.1 

185 

85.0 

68.0 

131 

55.0 

44.0 

77 

25.0 

20.0 

23 

- 6.0 

— 4.0 

—30 

— 34.4 

— 27.6 

184 

81.4 

67.6 

130 

54.4 

43.6 

76 

24.4 

19 6 

22 

- 6.6 

— 4.4 

—31 

— 35.0 

— 28.0 

183 

83.9 

67.1 

129 

53.9 

43.1 

75 

23.9 

19.1 

21 

— 6.1 

— 4.9 

—32 

— 35.6 

— 28.4 

182 

83.3 

66.7 

128 

53.3 

42.7 

74 

23.3 

18.7 

20 

- 6.7 

— 5.3 

—33 

— 36.1 

— 28.9 

’81 

82.8 

66.2 

127 

52.8 

42.2 

73 

22.8 

18.2 

19 

- 7.2 

— 5.8 

—34 

— 36.7 

— 29.3 

' 10 

82.2 

65.8 

126 

52.2 

41.8 

72 

22.2 

17.8 

18 

— 7.8 

- 6.2 

—35 

— 37.2 

— 29.8 

179 

81.7 

65.3 

125 

51.7 

41.3 

71 

21.7 

17.3 

17 

— 8.3 

- 6.7 

—36 

— 37.8 

— 30.2 

178 

81.1 

64.9 

124 

51.1 

40.9 

70 

21.1 

16.9 

16 

- 8.9 

- 7.1 

—37 

— 38.3 

— 30.7 

177 

80.6 

64.4 

123 

50.6 

40.4 

69 

20.6 

16.4 

15 

— 9.4 

- 7.6 

—38 

— 38.9 

— 31.1 

176 

80.0 

64.0 

122 

50.0 

40.0 

68 

20.0 

16.0 

14 

— 10-0 

— 8.0 

—39 

— 39.4 

— 31.6 

175 

79.4 

63.6 

121 

49.4 

39.6 

67 

19.4 

15 6 

13 

— 10.6 

— 8.4 

—40 

— 40.0 

— 32.0 

174 

78.9 

63.1 

120 

48.9 

39.1 

66 

18.9 

15.1 

12 

— 11.1 

- 8.9 

—41 

— 40.6 

— 32.4 

173 

78.3 

62.7 

119 

48.3 

38.7 

65 

18.3 

14.7 

11 

— 11.7 

— 9.3 

—42 

— 41.1 

— 32.9 

172 

77.8 

62.2 

118 

47.8 

38.2 

61 

17.8 

14.2 

10 

— 12.2 

— 9.8 

—43 

— 41.7 

— 33.S 

171 

77.2 

61.8 

117 

47.2 

37.8 

63 

17.2 

13.8 

9 

- 12.8 

— 10.2 

—44 

— 42.2 

- 33.8 

170 

76.7 

61.3 

116 

46.7 

37.3 

62 

16.7 

13.3 

8 

— 13.3 

— 10.7 

—45 

— 42.8 

— 34.2 

169 

76.1 

60.9 

115 

46.1 

36.9 

61 

16.1 

12.9 

7 

— 13.9 

- 11.1 

—46 

- 43.3 

— 34.7 

168 

75.6 

60 4 

114 

45 6 

36.4 

60 

15.6 

12 4 

6 

— 14 . 4 , 

-ll.o! 

-47 

— 43.9 

- 35.1 

167 

75.0 

60.0 

113 

45.0 

36.0 

59 

15.0 

12.0 

5 

—15 0 ! 

— 12.0 

—48 

— 44.4 

— 35.6 

166 

74.4 

59.6 

112 

44.4 

35.6 

58 

14.4 

11.6 

4 

— 15.6 

— 12 . 4 ; 

—49 

— 45.0 

— 36.0 

165 

73 9 

59.1 

111 

43.9 

35.1 

57 

13.9 

11.1 

3 

— 16.1 

— 12.9 

—50 

- 45.6 

— 36.4 

164 

73.3 

58.7 

110 

43.3 

34.7 

56 

13.3 

10.7 

2 

— 16.7 

— 13.3 

—51 

— 46.1 

— 36.9 

163 

72.8 

58.2 

109 

42.8 

34.2 

55 

12.8 

10.2 

1 

— 17.2 

— 13.8 

—52 

- 46.7 

— 37.3 

162 

72.2 

57.8 

108 

42.2 

33 8 

54 

12.2 

9.8 

0 

— 17.8 

— 14.2 

—53 

— 47.2 

— 37.8 

161 

71.7 

57.3 

107 

41.7 

33.3 | 

53 

11.7 

9.3 

—1 

— 18.3 

- 14.7 

-54 

— 47.8 

— 38.2 

160 

71.1 

56.9 

106 

41.1 

32.9 

52 

11.1 

8.9 

—2 

- 18.9 ! 

- 15.1 

— 65 

— 48.3 

— 38.7 

159 

70.6 

56.4 

105 

40.6 

32.4 

61 

10.6 

8.4 


1 


























































214 


THERMOMETERS, 


TABLE 2. Centigrade compared with Fahrenheit and 

Reaumur. 


C. 

F. 

R. 

c. 

F. 


c. 

F. 

R. 

c. 

F. 

R. 

o 

O 

O 

o 

O 

o 

o 

O 

O 

o 

O 

O 


Exnct. 

Exnct. 


Exact. 

Exact. 


Exact. 

Exact. 


Exact. 

Exact. 

100 

212.0 

80.0 

62 

143.6 

49.6 

24 

75.2 

19.2 

—14 

6.8 

— 11.2 

9!) 

210.2 

79.2 

61 

141.8 

48.8 

23 

73.4 

18.4 

—15 

5.0 

— 12.0 

98 

208.4 

78.4 

60 

140.0 

48.0 

22 

71.6 

17.6 

—16 

3.2 

- 12.8 

97 

206.6 

77.6 

59 

138.2 

47.2 

21 

69.8 

16.8 

—17 

1.4 

—13.6 

96 

204 8 

76 8 

58 

136.4 

46.4 

20 

68.0 

16.0 

—18 

—0.4 

— 14.4 

95 

203.0 

76.0 

57 

134.6 

45.6 

19 

66.2 

15.2 

—19 

— 2.2 

—15.2 

94 

201.2 

75.2 

56 

132.8 

44.8 

18 

64.4 

14.4 

—20 

—4.0 

—16.0 

93 

199.4 

74 4 

00 

131.0 

44.0 

17 

62.6 

13.6 

—21 

—5.8 

—16.8 

92 

197.6 

73.6 

54 

129.2 

43 2 

16 

(91.8 

12.8 

—22 

—7.6 

—17.6 

91 

195.8 

72.8 

53 

127.4 

42.4 

15 

59.0 

12.0 

—23 

-9.4 

-18.4 

90 

194.0 

72.0 

52 

125.6 

41.6 

14 

57.2 

11.2 

—24 

— 11.2 

—19.2 

89 

192.2 

71.2 

51 

123.8 

40.8 

13 

55.4 

10.4 

-25 

—13.0 

— 20.0 

88 

190.4 

70.4 

50 

122.0 

40.0 

12 

53.6 

9.6 

—26 

—14.8 

—20 8 

87 

188.6 

69.6 

49 

120.2 

39.2 

11 

51.8 

8.8 

—27 

— 16.6 

— 21.6 

86 

186.8 

68.8 

48 

118.4 

38.4 

10 

50.0 

8.0 

—28 

—18.4 

—22.4 

85 

185.0 

68.0 

47 

116.6 

37.6 

9 

48.2 

7.2 

—29 

— 20.2 

—23.2 

84 

183.2 

67.2 

46 

114.8 

36.8 

8 

46.4 

6.4 

—30 

— 22.0 

—24.0 

83 

181.4 

66.4 

45 

113.0 

36.0 

7 

44 6 

5.6 

—31 

—23.8 

—24.8 

82 

179.6 

65.6 

44 

111.2 

35.2 

6 

42.8 

4.8 

—32 

—25.6 

—25 6 

81 

177 8 

64,8 

43 

109.4 

34.4 

5 

41.0 

4.0 

—33 

—27.4 

—26.4 

80 

176.0 

64.0 

42 

107.6 

33.6 

4 

39.2 

3.2 

—34 

—29.2 

—27.2 

79 

174.2 

63.2 

41 

105.8 

32.8 

3 

37.4 

2.4 

—35 

—31.0 

—28.0 

78 

172.4 

62.4 

40 

104.0 

32.0 

2 

35.6 

1.6 

—36 

—32.8 

—28.8 

77 

170.6 

61.6 

39 

102.2 

31.2 

1 

33.8 

0.8 

—37 

—34.6 

—29.6 ' 

76 

168.8 

60.8 

38 

100.4 

30.4 

0 

32.0 

0.0 

—38 

—36.4 

—30.4 

75 

167.0 

60.0 

37 

98.6 

29.6 

—1 

30.2 

— 0.8 

—39 

—38.2 

—31.2 

74 

165.2 

59.2 

36 

96.8 

28.8 

—2 

28.4 

— 1.6 

—40 

—40.0 

—32.0 

73 

163.4 

58.4 

35 

95.0 

28.0 

—3 

26.6 

—2.4 

—41 

—41.8 

— 32.8 

72 

161 6 

57.6 

34 

93.2 

27.2 

—4 

24.8 

—3.2 

—42 

—43.6 

—33.6 

71 

159.8 

56.8 

33 

91.4 

26.4 

—5 

23.0 

—4.0 

—43 

—45.4 

—34.4 

70 

158.0 

56.0 

32 

89.6 

25.6 

-6 

21.2 

—4.8 

—44 

—47.2 

—35.2 

69 

156.2 

55.2 

31 

87.8 

24.8 

—7 

19.4 

—5.6 

—45 

—49.0 

—36.0 

68 

154.4 

51.4 

30 

86.0 

24.0 

—8 

17.6 

—6.4 

—46 

—50.8 

—36.8 

67 

152.6 

53.6 

29 

81.2 

23.2 

—9 

15.8 

—7.2 

—47 

—52.6 

—37.6 

66 

150.8 

52 8 

28 

82.4 

22.4 

—10 

14-.0 

— 8.0 

—48 

—54.4 

—38.4 

65 

149.0 

52.0 

27 

80.6 

21.6 

— 11 

12.2 

— 8.8 

—49 

—56.2 

—39.2 

64 

147.2 

51.2 

26 

78.8 

20.8 

—12 

10.4 

—9.6 

—50 

—58.0 

—40.0 

63 

145.4 

50.4 

25 

77.0 

20.0 

—13 

8.6 

— 10.4 




TABLE 3. Reaumur compared with Fahrenheit and 
___ Ceil ii grade. 


R. 

F. 

c. 

R. 

F. 

c. 

R. 

F. 

c. 

R. 

F. 

c. 

0 

80 

79 

78 

77 

76 

75 

74 

73 

72 

71 

70 

69 

68 

67 

66 

65 

64 

63 

62 

61 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

■o 

Exact. 

212.00 

209.75 

207.50 

205.25 
203.00 

200.75 

198.50 

196.25 
194.00 

191.75 

189.50 

187.25 
185.00 

182.75 

180.50 
178 25 
176.00 

173.75 

171.50 

169.25 
167.00 

164.75 

162.50 

160.25 
158.00 

155.75 

153.50 

151.25 
149.00 

146.75 

144.50 

0 

Exact. 

100.00 

98.75 

97.50 

96.25 
95.00 

93.75 

92.50 

91.25 
90.00 

88.75 

87.50 

86.25 
85 00 

83.75 

82.50 

81.25 
80.00 

78.75 

77.50 

76.25 
75.00 

73.75 

72.50 

71.25 
70.00 

68.75 

67.50 

66.25 
65.00 

63.75 

62.50 

0 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

o 

Exact. 

142.25 
140.00 

137.75 
135 50 

133.25 
131.00 

128.75 

126.50 

124.25 
122.00 

119.75 

117.50 

115.25 
113.00 

110.75 
108 50 

106.25 
104.00 

101.75 

99.50 

97.25 
95.00 

92.75 

90.50 

88.25 
86.00 

83.75 

81.50 

79.25 
77.00 

o 

Exact. 

61.25 
60.00 

58.75 

57.50 

56.25 

55 00 

53.75 

52.50 

51.25 
50.00 

48.75 

47.50 

46.25 
45.00 

43.75 

42.50 

41.25 
40.00 

38.75 

37 50 

36.25 
35.00 

33 75 

32 50 

31.25 
30.00 

28.75 

27.50 

26.25 
25.00 

o 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

—1 

_2 

—3 

—4 

—5 

—6 

—7 

-8 

—9 

—10 

o 

Exact. 

74.75 

72.50 

70.25 
68.00 

65.75 

63.50 

61.25 
59.00 

56.75 
54 50 

52.25 
50.00 

47.75 

45.50 

43.25 
41.00 

38.75 

36.50 

34.25 
32.00 

29.75 

27.50 

25.25 
23.00 

20.75 

18.50 

16.25 
14.00 

11.75 
9.50 

o 

Exact. 

23.75 

22.50 

21.25 
20.00 

18.75 

17.50 

16.25 
15.00 

13.75 

12.50 

11.25 
10.00 

8.75 

7.50 

6.25 
5.00 

3.75 

2.50 

1.25 
0.00 

-1.25 
-2.50 
—3.75 
-5.00 
—6.25 
-7.50 
—8.75 
— 10.00 
—11.25 
—12.50 

0 

—11 
—12 
—13 
—14 
—15 
—16 
—17 
—18 
—19 
—20 
—21 
—22 
—23 
—24 
—25 
—26 
—27 
-28 
—29 
—30 
—31 
—32 
— 33 
—34 
-35 
—36 
-37 
—38 
—39 
—40 

0 

Exact. 
7.25 
5.00 
2.75 
0.50 
—1.75 
—4.00 
—6.25 
—8.50 
—10.75 
—13.00 
-15.25 
—17.50 
—19.75 
— 22.00 
—24.25 
—26.50 
—28.75 
—31.00 
—33.25 
—35.50 
—37.75 
—40.00 
—42.25 
—44.50 
—46.75 
—49.00 
—51.25 
—53.50 
—55.75 
—58.00 

0 

Exact. 

—13.75 

—15.00 

—16.25 

—17.50 

-18.75 

—20.00 

—21.25 

—22.50 

-23.75, 

-25.a 

—26.25 

—27.50 

—28.75 

—30.00 

—31.25 

—32.50 

—33.75 

—35.00 

—36.25 

—37.50 

—38.75 

—40.CO 

—41.25 

—42.50 

—43.75 

—45 00 

—46.25 

—47.50 

—48.75 

—50.00 





























































AIR. 


215 


AIR—ATMOSPHERE. 

Tlie atmosphere is known to extend to at least 45 miles 

above the earth. Its composition is about .79 measures of nitrogen gas 
and .21 of oxygen gas; or about .77 nitrogen, .23 oxygen, by weight. It gen¬ 
erally contains, however, a trace of water; carbonic acid, and carburetted 
hydrogen gases; and still less ammonia. 

When the barometer is at 30 inches, and the temperature 60° Fah, air 
weighs aboutone-815th part as much as water; or535grains = 1.224commercial 
ounces = .0765 commercial lb. per cubic foot. Or 13.072 cubic feet weigh 1 lb; 
and a cubic yard, 2.066 lbs. Or a cube of 30.82 feet on each edge, 1 ton. When 
colder it weighs more per cubic foot, and vice versa, at the rate of about a grain 

f >er degree of Fah. The average weight of the entire atmospheric column, (at 
east 45 miles high,) at sea level, is 14% ®>s avoirdupois per square inch ; or 2124 
lbs per square foot* = weight of a column of water 34 feet, or of mercury 30 
inches, high. This is what is usually called the “pressure of the air.” At % 
mile above sea level it is but 14.02 lbs per square inch; at % mile, 13.33; at % 
mile, 12.66; at 1 mile, 12.02; at 1% mile, 11-42; at 1% mile, 10.88; and at 2 miles, 
9.80 lbs. Therefore, a pump in a high region will not lift water to as great a 
height as in a low one. The pressure of air, like that of water, is, at any given 
point, equal in all directions. 

It is often stated that the temperature of tbe atmosphere lowers 
or becomes colder, at the rate of 1° Fah for each 300 feet of ascent above the 
earth's surface ; but this is liable to many exceptions, and varies much 
with local causes. Actual observation in balloons seems to show that up to the 
first 1000 feet, about 200 feet to 1°, is nearer the truth ; at 2000 feet, 250; at 4000 
feet, 300 ; and at a mile, 350. 

Ill breathing, a grown person at rest requires from .25 to .35 of a cubic 
foot of air per minute; which, when breathed, vitiates from 3% to 5 cubic feet. 
When walking or hard at work, he breathes and vitiates two or three times as 
much. About 5 cubic feet of fresh air per person per minute is required for the 
perfect ventilation of rooms in winter ; 8 in summer. Hospitals 40 to 80. 

Beneath the general level of*the surface of the earth in temperate 
regions, a tolerably uniform temperature of about 50° to 60° Fah exists at 
the depth of about 50 to 60 feet; and increases about 1° for each additional 50 
to 60 feet; all subject, however, to considerable deviations from many local 
causes. In the Rose Bridge colliery, England, at the depth of 2424 feet, the tem¬ 
perature of the coal is 93%° Fah ; aud at the bottom of a boring 4169 feet deep, 
near Berlin, the temperature is 119°. 

The air is a very slow conductor of heat; hence hollow walls 
serve to retain the heat in dwellings; besides keeping them dry. It rushes 
into a vacuum near sea level with a velocity of about 1157 feet per second ; 
or 13% miles per minute; or about as fast as sound ordinarily travels through 
quiet air. See Sound, p 211. 

Bike all other elastic fluids, it expands equally with equal 
increases of temperature. Every increase of 5° Fah, expands the bulk 
of any of them slightlv more than 1 per cent of that which it has at 0° Fah; 
or 500° about doubles its bulk at zero. The bulk of any of them diminishes 
inversely in proportion to the total pressure to which it is subjected. Thus, if 
we have a cylinder open at top, and 1 foot deep, full of air at its natural pres¬ 
sure of about 15 lbs per square inch; if by means of a piston we apply an addi¬ 
tional pressure of 15 lbs per square inch, making 30 lbs in all, or twice as much 
as the natural pressure, then the air will be compressed into 6 inches of depth 
of the cylinder, or one-half of what it occupied before. Or if we apply 45 lbs 
additional, making 60 lbs in all, or 4 times the natural pressure, then the air will 
be compressed into % of the depth of the cylinder. Experiment shows that 
this holds good with air at least up to pressures of about 750 lbs per square inch, 
or 50 times its natural pressure; the air in this case occupying one-fiftieth of 
its natural bulk. In like manner the bulk will increase as the total pressure is 
diminished ; so that, if we remove our additional 45 lbs per square inch, the air 
in the cylinder will regain its original bulk, and will precisely fill the cylinder. 
Substances which follow these laws, are said to be perfectly elastic. Under a 
pressure of about 5% tons per square inch, air would become as dense, or would 
weigh as much per cubic foot, as water. Since the air at the surface of the earth 
is pressed 14% lbs per square inch by the atmosphere above it, and since this 
is eaual to the weight of a column of water 1 inch square and 34 feet high, it 
follows that at the depths of 34, 68,102 feet, Ac, below water, air will becom- 


* = 1.033 kilogrammes per square centimetre. 





216 


WIND. 


pressed into y 2 , y, y, &c, of its bulk at the surface; because at those depths it 
is exposed to pressures equal to 2, 3, 4, «&c, times 14% ftis. per square inch, in as* 
much as the pressure of the atmosphere on the surface, is in eacli case to be 
added to that of the water. The pressure of the water alone at those depths 
would be but 1, 2, 3, &c, times 14% lbs per square inch. 

In a diving-bell, men, after some experience, can readilv work for sev- 
^f/rk OUrS at a de b t1 ' of 51 feet; or under a pressure of 2 % atmospheres; or 
37% Ibs^per square inch. But at 90 feet deep; or under 3.64 atmospheres - or 
nearly 00 tbs per square inch, they can work for but about an hour, without 
serious suffering from paralysis; or even danger of death. Still at the St Louis 

inch* 6 S ° me W ° rk WaS d ° Ue at a dGpth ° f Ui)lA feet 5 P ressure 63.7 P er square 
vapoJ 0 dCW ,>0iul is thafc tem P«rature (varying) at which the air deposits its 

heat of the air in the sun probably never exceeds 

145 Pah; nor the greatest cold —74° at night. About 130° above, and 40° below 

N W are H ll i eXtremeS 1,1 the ^’ S ' eas1, of the Mississippi; and 65° below in the 
S ''7 ’ u - a l T coai ™? n ? round Ieve] - It is stated, however, that —81° has been 
the Ini' YV^’ Sl - b f n ^i and +1 01 ° F al1 in the shade in Baris; and 4-153° iu 

ee^ed xmnoTY W - IC ^ 0 b u e 7 a ^° ry ’ bo,h in : Tu, . v - 1881. It has frequently ex- 
ed +100 1 ah id the shade in Philadelphia during recent years. 


WIND. 

The relation between the velocity of wind, and its nress. 

ure against an obstacle placed either at right angles to its course, or inclined 
^,‘L haS been wel1 determined ; and still less so its pressure against curved 
thnn Y '• T t 6 P resai . ire a 8 a *"St a lar ge surface is probably proportionally greater 
than against a small one. It is generally supposed to vary nearly as the squares 
of the velocities; and when the obstacle is at right angles to its direct on the 
pressure in lbs per square foot of exposed surface is considered to be e qi ?al to 
the square of the velocity in miles per hour, divided by 200 On this basis 

?re , ^d Pr0bablr qU “ e dereClive ' tU,! '“““"‘"S l “ ble . “ 8^™ by Seaton! fs 


Vel . in Miles 
per Hour . 


1 

2 

3 

4 

5 
10 
12 * 
15 
20 
25 
30 
40 
50 
60 
80 

100 


Vel . in Ft . 
per Sec . 

Fres . in Lbs . 
per Sq . Ft . 

1.467 

.005 

2.933 

.020 

4.400 

.045 

5.867 

.080 

7.33 

.125 

14.67 

.5 

18.33 

.781 

22 . 

1.125 

29.33 

2 . 

36.67 

3.125 

44 . 

4.5 

58.67 

8 . 

73.33 

12.5 

88 . 

18 . 

117.3 

32 . 

146.7 

50 . 



Remarks. 


Hardly perceptible. 
Pleasant. 


Fresh breeze. 


a 


n 


)/n 


Brisk wind. 

Strong wind. 

High wind. 

Storm. 

Violent storm. 

Hurricane. 

Violent hurricane, uprooting large trees 


The pres against 
a semicylindrical 
surface a c b n o m 
is about half that 
against the flat 
surf abnrn. 


pres of wiifd against if- but as roo"aw S’nftfucteYwIS Jo* ***** S< * ft ° f r °° f for 
the BUI force of the wind, this is plainly too much * M and consequently do not receive 

posed, even thus partially, to the windProbablvt'he f ' one - half of a roof is usually ex¬ 

sines of the angles of slopes. According to observational ?, SUch cas ® s vanea approximately as the 
hour, produced a pres of 14 lbs pe? sq f? againIt^a.foblc narnT 1 '?'’ '“A 860 ’ a wind of3 « Vesper 
(the severest gale on record : uthatcitvwYibsYYrsnfiY P tV ; and ,® ne of 70 miles per hour, 
more nearly equal to the square of the vel in milea M A wou d make the pres per sq ft, 

given in Smeatou-s table. We should ourselves g f v e HiVS r d " by ,0 ° : T or near| y twice as great as 
very violent gale in Scotland, registered bv an excellent P »n!fA 06 A 0 the L,ver l>ool observations. A 
ft. H is stated that as high as 55 lbs has been Sd r nometer, or wind-gauge. 45 lbs per sq 

•aesg&g "-w.**. .xssrwbw 

rs.'ajs, the 

cieut^anowancVfor'pres^f wmd? f00t ordtnar * double-sloping roofs, or 16 fts for shed-roofs, suffiT 

































WATER. 


217 


WATER. 


Pure water, as boiled and distilled, is composed of the two gases, hydro¬ 
gen and oxygen; in the proportions of 2 measures hydrogen to 1 of oxygen; 
or 1 weight of hydrogen to 8 of oxygen. Ordinarily, however, it contains sev¬ 
eral foreign ingredients, as carbonic and other acids; and soluble mineral, or 
organic substances. When it contains much lime, it is said to be hard; and will 
not. make a good lather with soap. The air in its ordinary state contains 
about 4 grains of water per cubic foot. 

The average pressure of the air at sea level, will balance a 
column of water 34 feet high ; or about 30 inches of mercury. At its boil¬ 
ing point of 212° Fab, its bulk is about one twenty-third greater than at 70°. 

Its weight per cubic foot is taken at 62% lbs,or 1000 ounces avoir; but 62% 
lbs would be nearer the truth, as per table below. It is about 815 times heavier 
than air, when both are at the temperature of 62°; and the barometer at 30 
inches. With barometer at 30 inches the weight of perfectly pure water is as 
follows. At about 39° it has its maximum density of 62.425 lbs per cubic foot. 


Temp, Fall. Lbs per Cub Ft. 

32°.62.417 

40°.62.423 

50°.62.409 

60°..62.367 


Temp, Fall. Lbs per Cub Ft. 

70°.62.302 

80°.62.218 

90°.62.119 

212°.59.7 


Weight of sea water 64.02 to 64.27 lbs per cubic foot, or say 1.6 to 1.9 
lbs more than fresh. . 

Water has its maximum density when its temperature is a little above 
39° Fall; or about 7 C above the freezing point. By best authorities 39.2 . From 
about 39° it expands either by cold, or by heat. When the temperature ot 32° 
reduces it to ice, its weight is but about 57.2 lbs. per cubic foot; and its specific 
gravity about .9175, according to the investigations of L. Dufour. Hence, as 
ice, it has expanded one-twelfth of its original bulk as water; and. the sudden 
ex'pasisive force exerted at the moment of freezing, is sufficiently great to 
split iron water-pipes; being probably not less than 30000 lbs per square inch. 
Instances have occurred of its splitting cast tubular posts of iron bridges, and 
of ordinary buildings, when full of rain water from exposure. It also loosens 
and throws down masses of rock, through the joints of which raiu or spring 
water has found its way. Retaining-walls also are sometimes overthrown, or 
at least bulged, by the freezing of water which has settled between their backs 
and the earth filling which they sustain ; and walls which are not founded at a 
sufficient depth, are often lifted*upward by the same process. 

It is said that in a glass tube % inch in diameter, water will not 
freeze until the temperature is reduced to 23°; and in tubes of less than 
inch, to 3° or 4°. Neither will it freeze until considerably colder than 32° in 
rapid running streams. Anchor ice, sometimes found at depths as great as 
25 feet, consists of an aggregation of small crystals or needles of ice frozen at 
the surface of rapid open water; and probably carried below by the force of the 
stream. It does not form under frozen water. For crushing strength of 
ice, see page 437. 

Since ice floats in water; and a floating body displaces a weight of the 
liquid equal to its own weight, it follows that a euhic foot of floating ice weighing 
57.2 lbs, must displace 57.2 lbs of water. But 57.2 lbs of water, one foot square, is 11 
inches deep: therefore, floating ice of a cubical or parallelopipedal shape, will 
have of its volume under water; and only fa above; and a square foot of ic® 
of any thickness, will require a weight equal to fa of its own weight to sink it 
to the surface of the water. In practice, however, this must he regarded merely 
as a close approximation, since the weight of ice is somewhat affected by en¬ 
closed air-bubbles. 

Pure water is usually assumed to boil at 212° Fah in the open air, at the 
level of the sea; the barometer being at 30 inches; and at about l c less for every 
520 feet above sea level, for heights within 1 mile. In fact, its boiling point 
varies like its freezing point, with its purity, the density of the air, the material 
of the vessel, Ac. In a metallic vessel, it may boil at 210°; and in a glass one, 
at from 212° to 220°; and it is stated that if all air be previously extracted, it 
requires 275°. For leveling by the boiling point, see page 209. 

It evaporates at all temperatures; dissolves more substances than any 
other agent: and has a greater capacity for heat than any other known substance. 

It i*s compressed at the rate of about one-21740th, (or aboutof an 
inch in 18 T r 5 feet,) by each atmosphere or pressure of 15 lbs per square inch. 
When the pressure is removed, its elasticity restores its original bulk. 











218 


WATER. 


Effect on metals. The lime contained in many waters, forms deposits in 
metallic water-pipes, and in channels of earthenware, or of masonry ; especially 
if the current be slow. Some other substances do the same; obstructing the 
flow of the water to such an extent, that it is always expedient to use pipes of 
diameters larger than would otherwise be necessary. The lime also forms very 
hard incrustations at the bottoms of boilers; very much impair¬ 
ing their efficiency; and rendering them more liable to burst. Such water is 
unfit for locomotives. We have seen it stated that the Southwestern R R Co, 
England, prevent this lime deposit, along their limestone sections, by dissolving 
1 ounce of sal-ammoniac to 90 gallons of water. The salt of sea water forms 
similar deposits in boilers; as also does mud, and other impurities. 

Water, either when very pure, as rain water; or when it contains carbonic 
ac id, (as most water does,) produces carbonate of lead in lead 
pipes; and as this is an active poison, such pipes should not. be used for such 
waters. Tinned lead pipes may be substituted for them. If, however, sulphate 
of lime also be present, as is very frequently the case, this eifect is not always 
produced; and several other substances usually found in spring and river 
water, also diminish it to a greater or less degree. Eresh water corrodes 
wrought iron more rapidly than cast; but the reverse appears to 
be the case with sea water; although it also affects wrought iron very 
quickly; so that thick flakes may be detached from it with ease. "The corrosion 
of iron or steel by seawater increases with the carbon. Cast-iron cannons 
lrom a vessel which had been sunk in the fresh water of the Delaware River 
lor more than 40 years, were perfectly free from rust. Gen. Fasley. who had 
examined the metals found in the ships Royal George, and Edgar, the first of 
which had remained sunk in the sea for 62 years, and the last for 133 years 
stated that the cast iron had generally become quite soft; and in some cases 
resembled plumbago. Some of the shot when exposed to the air became hot.• 
and burst, into many pieces. The wrought, iron was not so much injured’ 
except when m contact with copper , or brass gun-metal. Neither of these last, was 

T her ) , in co " ,act wi ’h iron. Some of the wrought iron 
was reworked by a blacksmith,and pronounced superior to modern iron.” “Mr 
Cottam stated that some of the guns had been carefully removed in their soft 
state to the lower of London ; and in time (within 4 years) resumed their aria - 
/mrdwess. Brass cannons from the Mary Rose, which had been sunk in the 
sea for -92 years, were considerably honeycombed in spots only ; (perhaps where 
ron had been in contact with them.) The old cannons, of wrought-ifon bare 
hooped together, were corroded about % inch deep; but had probab y been pro 

mS JKM?SSr"" S "°‘ be ° ame redh ° l “ posureto the ™ ® 

Unprotected parts of cast-iron sluice-valves on the sea (rates of the Cnle 
donian canal, were converted into a soft plumbaginous substance to a depth 

jss. texrxi ei p,lM * 

which^greaMy^'asGm^'decay ?n r eRher I f^d 1S Sai( l to induce a galvanic action 
recovered from a wreck which had boeif 8a f w , at . er - Some muskets were 
near New York. The brass nartswAro in s “ hm< ; r " ed V 1 . sea water for 70 years 
had entirely disappeared. Gal vaniyl»»^ r t f AnoH° nd ’ bl,t ' tl,e iron f mrfs 

servative to the iron, but at the exnens^f W1 ^.zone) acts as a pre- 
The iron then corrodes If iron h a w n i f V a zin which soon disappears, 
coal-tar. it will resist the action of Sa ^ a , nd , then coated wltl/ Hot 
It is very important that the tar be perfect!Jp urified wr 0 " 
ing, or one of paint., will not prevent . e J2 291, Such a coar_ 

attaching themselves to the* iron Asnhaltmn*' otl,er shells from 
coal-tar. Asphaltum, if pure, answers as well as 



TIDES. 




The most prejudicial exposure for iron, as well as for wood, is 
thai to alternate wet and dry. At some dangerous spots in Long Island Sound, 
it has been the practice to drive round bars of rolled iron about 4 inches diam¬ 
eter, for supporting signals. These wear away most rapidly between high and 
low water; at the rate of about an inch in depth in 20 years; in which time the 
4-inch bar becomes reduced to a 2-inch one, along that portion of it. Under 
fresh water especially or under ground, a thin coating of coal-pitch varnish, 
carefully applied, will protect iron, such as water-pipes, &c., for a long time. 
See page 292. The sulphuric acid contained in the water from coal mines 
corrodes iron pipes rapidly. In the fresh water of canals, iron boats 
have continued in service from 20 to 40 years. Wood remains sound for 
centuries under either fresh or salt water, if not exposed to he worn away by 
the action of currents; or to be destroyed by marine insects. 
f, Sea water differs a little in weight, at different places; but at the 
same place it is appreciably the same at all depths; and may be generally as¬ 
sumed at about 64 lbs; or lbs per cubic foot, more than fresh. The additional 
1% lbs, or one 36.6th of its entire weight, is chiefly common salt. Sea water 
freezes at 27° Fail: the ice is fresh. 

A teaspoonful of powdered alum, well stirred into a bucket of dirty water, 
will generally purify it sufficiently within a few hours to be drinkable. If a 
hole 3 or 4 feet deep be dug in the sand of the sea-shore, the infiltrating water 
will usually be sufficiently fresh for washing with soap; or even for drinking. 
It is also stated that water may be preserved sweet for many years by placing 
in the containing vessel 1 ounce of black oxide of manganese for each gallon 
of water. 

It is said that water kept in zinc tanks; or flowing through iron 
tubes galvanized inside, rapidly becomes poisoned by soluble salts of zinc 
formed thereby; and it is recommended to coat, zinc surfaces with asphalt 
varnish to prevent this. Yet, in the city of Hartford, Conn, service pipes of 
iron, galvanized inside and out, were adopted in 1855, at the recommendation 
of the water commissioners; and have been in use ever since. They are like¬ 
wise used in Philadelphia and other cities to a considerable extent. In many 
hotels and other buildings in Boston, the “Seamless Drawn Brass Tube” of the 
American Tube Works at Boston, has for many years been in use for service 
pipe; and has given great satisfaction. It is stated that the softest water may 
be kept in brass vessels for years without any deleterious result. 

The action of lead upon some waters (even pure ones) is highly poison¬ 
ous. The subject, however, is a complicated one. An injurious ingredient, may 
be attended by another which neutralizes its action. Organic matter, whether 
vegetable or animal, is injurious. Carbonic acid, when not in excess, is harm¬ 
less. See near bottom of page 419. 

Ice may be so impure that its water is dangerous to drink. 

The popular notion that hot w ater freezes more quickly 
than cold, with air at the same temperature, is erroneous. 

TIDES. 

The tides are those well-known rises and falls of the surface of the sea 
and of some rivers, caused by the attraction of the sun and moon. There are 
two rises, floods, or high tides; and two falls, ebbs, or low tides, every 24 hours 
„ and 50 minutes (a lunar day) ; making the average of 5 hours 12V£ minutes 
^between high and low water. These intervals are, however, subject to 
groat variations; as are also the heights of the tides; and this not only 
at different places, but at the same place. These irregularities are owing to the 
shape of the coast line, the depth of water, winds, and other causes. Usually at 
new and full moon, or rather a day or two after, (or twice in each lunar month, 
at intervals of two weeks,) the tides rise higher, and fall lower than at other 
times; and these are called spring- tides. Also, one or two days after the 
moon is in her quarters, twice in a lunar month, they both rise and fall less than 
at other times ; and are then called neap tides. From neap to spring they 
rise and fall more daily; and vice versa. The time of high water at any 
place is generally two or three hours after the moon has passed over either 
the upper or lower meridian; and is called the establishment of that 
place; because, when this time is established, the time of high water on any 
other day may be found from it. in most cases. The total height of spring tides 
is generally from 1 y, to 2 times as great as that of neaps. The great tidal 
wave is merely an undulation , unattended by any current, or progressive motion 
of the particles of water. Each successive high tide occurs about 24 minutes 
later than the preceding one; and so with the low tides. 




220 


RAIN. 


RAIN. 


The quantity that falls annually in any one place, varies 

greatly from year to year; the extremes being frequently greater than 2 to 1. 
In making calculations for collecting water in reservoirs,*whether for feeding 
canals, or forsupplying cities, we cannot safely assume more than the minimum 
fall observed for many years; or rather, somewhat less. And from even this 
inust be deducted the amount (a quite considerable one) lost by evaporation and 
leakage after it has been collected. The following table shows in some casce, 
the average annual falls; and in others, the least and the greatest ones observed 
at several places ; including snow water. It is highly probable that most of the f 
results are merely approximate. See Evaporation, p 222. 


Augusta, Georgia. 

Albany, N, York. 

Arkansas... 

Bath, Maine. 

Baltimore, Md. 

Boston, Mass. 

Charleston, S. C.. 

Canada... 

Carlisle, Penna. 

Detroit, Michigan. 

Frankford, Penna. 

Fort Gaston, California, in 9 months. 

Fort Yuma, Cal. 

Port (not Fort) Orford, Oregon.... . 

Fort Pike, Louisiana. 

Fort Pierce, E. Florida. 

Fort Conrad, New Mexico. 

Fort Kent, Maine. 

Fort Preble, “ . 

Fort Constitution, N. Hamp. 

Fort Adams, Rhode Island. 

Fort Hamilton, N. York Harbor. 

Fort Niagara, N. Y. 

Fort Monroe, Yirg. 

Fort Kearney, Nebraska. 


Inches 
per an. 

23 

31 to 51 
41 

30 to 50 
40 

25 to 4G 
40 to 7ti! 
36 
34 
30 

33 to 54 
129 

3% 

69 

72 

63 


6% 
36% 
45% 
35% 
52% 
43 % 
31 % 
51 

•a 


Fort Laramie, Nebraska. 

Fort Worth, Texas. 

Fort McIntosh, “ . 

Fort Dallas, Oregon.. 

Key West, Florida. 

Lebanon, Penna... 

Michigan. 

Monterey, Cat.’ 

Marietta, Ohio. 

New Orleans, Louisiana. 

New Fane, Vermont. 

New England. average.. 

Natchez, Miss. 

New York State............ averatre.. 

Ohio. “ 

Philadelphia, Penna. 

av for 54 years, to 1884... 

Pennsylvania. averarc.. 

Savannah, Georgia.?... 

Stow, Mass. 

St. Louis, Mo. 

Washington, D. C.* 

WestChester, Penna..* 

Williamstown, Mass.", 


Inches 
per an. 

20 * 

41 

18% 

14% 

30 t o 39 

34 to 45 

35 
12 % 

35 to 54 
51 

36 to 741 
47 

37 to 58 
36% 

36 

34 to 61 
45.2 * 

41 

30 to 60 
33 to 49 

42 
41 

39 to 54 
26 to 39 f 


The greatest fall recorded in one day in Philadelphia, was 

7.3 inches, Aug. 13, 1873. The greatest in any month, was 15.8 inches, Aug., 1857. 

■ "icy, 1842, C> inches fell in 2 hours. If has not readied 9 inches per month 
more than 7 or 8 times, in 25 years. During a tremendous rain at Norristown’ 
Pennsylvania, in 1865, the writer saw evidence that at least 9 inches fell within 

5 hours. At Genoa, Italy,on one occasion, 32 inches fell in 24 hours ; at Geneva, 
Switzerland, 6 inches in 3 hours; at Marseilles, France, 13 inches in 14 hours- 
in Chicago, Sept., 1878, .97 inch in 7 minutes. 

Near Loiidon, (.upland, the mean total fall for many years is 23 inches. 
Un one occasion, 6 inches fell in hours! In the mountain districts of the 
English lakes, tlie fall is enormous ; reaching in some years to 180 or 240 inches* 
or from 15 to 20 feet! while, in the adjacent neighborhood, it is hut 40 to 60 
inches. At Liverpool, the average is 34 inches; at Edinburgh, 30; Glasgow 22 • 
Ireland, 36; Madras, 47; Calcutta, 60; maximum for 16 vears, 82; Delhi’ 21'’- 
Gibraltar, 30; Adelaide, Australia, 23; West Didies, 36 to 96; Rome, 39. On the 
Khassya hills north of Calcutta, 500 inches, or 41 feet 8 inches, have fallen in the 

6 rainy months! In other mountainous districts of India, annual falls of 10 to 
20 feet are common. 

It requires a quite heavy rain, for 24 hours, to yield the depth of an inch: 
still, inasmuch as at rare intervals falls of as much as from 1 to 3, or even 6 
inches per hour occur, these latter depths must be considered in planning 
sewers, culverts, etc. See Art, 23, p 279e. 

As a general rule, more rain falls in warm titan in cold 
countries; and more in elevated regions than in low ones. Local peeuliar- 


} 


t 


* In 1869, during which occurred the greatest drought known in Philadelnhfa. 
for at least 50 years, it was 48.84 inches. ^ 


t 




















































SNOW. 


221 




ities, however, sometimes reverse this; and also cause great differences in the 
amounts in places quite near each other; as in the English lake districts just 
alluded to. It is sometimes difficult to account for these variations. In some 
lagoons in New Granada, South America, the writer has known three or four 
heavy rains to occur weekly for some months, during which not a drop fell on 
hills about 1000 feet high, within 10 miles’ distance, and within full sight. At 
another locality, almost a dead-level plain, fully three-quarters of the rains that 
fell for 2 years* at a spot 2 miles from his residence, occurred in the morning; 
while those which fell about 3 miles from it, in an opposite direction, were in 
the afternoon. 

“ The returns of several rain gauges in the Longdendale district, England, for 
1847, gave the rainfalls at different altitudes above the sea, as follows:” 


At 500 feet altitude. 46.0 ins. 

800 “ . 50.5 “ 

1700 “ . 52.1 “ 


At 1750 feet altitude. 56.5 ins. 

1800 “ . 62.1 “ 


“The annual average fall at Edinburgh, 200 feet above the sea, in three succes¬ 
sive years, was 30 inches. In the Pentland hills, a few miles south, and 700 feet 
abov'e sea, 37.4 inches: and at Carlops, similarly situated near the last, but 900 
feet above sea, 49.2 inches.” 

There are probably bnt few places in the United States, 

where an annual fall of 2 feet may not he safely relied on; and since, as an 
average, about half of it may be collected into reservoirs, a square mile of 
drainage (=27878400 square feet) should yield annually 27878400 cubic 
i'-'eet; equal to 76379 cubic feet per day. Allowing 4 cubic feet, or 30 gallons, per 
‘ Jay for each person; and making no deduction for evaporation and filtration, 
this would supply a population of 19095 persons ; or a square of 38% feet on a 
side, would in like manner suffice for one person. From two-tenths to eight- 
tenths of all the water annually resulting from rain and snow, passes off into 
the neighboring rivulets; and thence into the larger streams and rivers; or 
may be collected into reservoirs. Under ordinary circumstances of locality, 
about one-half may usually be thus secured. The difference is owing chiefly 
to the distance which the water may have to run; the rates of absorption of 
various soils; the rate of descent of the sides of the valleys leading to the 
streams; the season of the year, Ac, Ac. See p 222. 

An inch of rain amounts to 3630 cubic feet; or 27155 U. S. gal- 
. Ions; or 101.3 tons per acre; or to 2323200 cubic feet; or 17378743 U. S. gallons; 
or 64821 tons per square mile at 62% !bs per cubic foot. 

The most destructive rains are usually those which fall upon snow, under 
which the ground is frozen, so as not to absorb water. 


SNOW. 

Trials at different times by the writer, showed the weight of freshly 
fallen snow to vary from about 5 to 12 lbs per cubic foot; apparently depend¬ 
ing chiefly upon the degree of humidity of the air through which it had passed. 
On one occasion when mingled snow and hail had fallen to the depth of 6 inches, 
he found its weight to be 31 lbs per cubic foot. It. was very dry and incoherent. 
A cubic foot of heavv snow may, by a gentle sprinkling of water, be converted 
! nt.o about half a cubic foot of slush, weighing 20 lbs; which will not slide or 
!'•' un off from a shingled roof sloping 30°, if the weather is cold. A cubic 
block of snow saturated with water until it weighed 45 !bs per cubic foot, just 
slid on a rough board inclined at 45°; on a smoothly planed one at 30°; and on 
slate at 18°; all approximate. A prism of snow, saturated to 52 lbs per cubic 
foot; one inch square, and 4 inches high, bore a weight of 7 lbs ; which at 
first compressed it about one-quarter part of its length. European engineers 
consider 6 lbs per square foot of roof, to be sufficient allowance for the 
weight of snow; and 8 lbs for the pressure of wind; total, 14 lbs. The 
writer thinks that in the U. S. the allowance for snow should not be taken at 
less than 12 lbs; or the total for snow and wind, at 20 lbs. There is no danger 
that snow on a roof will become saturated to the extent just alluded to; because 
a rain that would supjply the necessary quantity of water, would also by its 
(‘■'violence wash away the snow; but we entertain no doubt whatever that the 
united pressures from snow and wind, in our Northern States, do actually at 
times reach, and even surpass, 20 lbs per square foot of roof. See Table 4, p581, 
of Trusses. The limit of perpetual snow at the equator is at the height 
of about 16000 feet, or say 3 miles above sea-level; in iat 45° north or south, it 
is about half that height; while near the poles it is about at sea-level. 







222 


HYDROSTATICS. 


EVAPORATION, FILTRATION, AND LEAKAGE. 


The amount of evaporation from surfaces of water exposed to 

the uatural effects of the opeu air, is of course greater in summer than in winter; although it is quite 
perceptible in even the coldest weather, it is greater in shallow water than iu deep, inasmuch as the 
bottom also becomes heated by the sun. It is greater iu ruuuiug, thau iu stauding water; on much 
the same priueiple that it is greater during winds than calms. It is probable that the average daily 
loss from a reservoir of moderate depth, from evaporation alone, throughout the 3 w armer months 
of the year, (June, July, August,) rarely exceeds about T 3 o inch, in any part of the United States. Or 
T*(T * uc ^ tlle 0 colder mouths; except in the Southern States. These two averages would give 

a daily one of .15 inch ; or a total annual loss of 55 ins, or 4 ft 7 ins. It probably is 3.5 to 4 ft. 

15y sonic trials by tlie writer, in tlie tropics, ponds of pure water 

8 ft deep, iu a stiff retentive clay, and fully exposed to a very hot suu all day, lost during the drv sea-f 
son, precisely 2 ins in 16 days; or % inch per day; while the evaporation from a glass tumbler was 
J4 iuch per day. The air in that region is highly charged with moisture; and the dews are heavy. 
Every day during the trial the thermometer readied from 115° to 125° in the sun. 

The total annual evaporation in several parts of England and Scotland is stated to average from 22 
to 38 ins; at Paris, 34; Boston, Mass, 32 ; many places in the U. S., 30 to 36 ins. This last would give 

a daily average of y^y inch for the whole year. Such statements, however, are of very little value, 
unless accompanied by memoranda of the circumstances of the case;'such as the depth, exposure, 
size and nature of the vessel, pond, Ac. which contains the water. Ac. Sometimes the total annual 
evaporation from a district of country exceeds the rain fall; and vice versa. 

Oil canals, reservoirs, Ac, it is usual to combine the loss by evaporation, 

with that by filtration. The last is that which soaks into the earth; and of which some portion 
passes entirely through the banks, (when in embankt;) and if in verv small quantitv. mat be dried! 
np by the sun and air as fast as it reaches the outside; so as not to ex’hibil itself as water; but if *■» 
greater quantity, it becomes apparent, as leakage. 

E. II. Gill, CE, slates Ihc average evaporation ami filtra¬ 
tion on the Sandy ami Beaver canal. Ohio, (38 ft wide at water sur¬ 
face ; 26 ft at bottom : and 4 ft deep,) to be but 13 cub ft per mile per minute, in a dry season Here 
the exposed water surf in one mile is 200640 sq ft; and iu order, with this surf, to lose 13 cub ft per ' 
min, or 18720 cub ft per day of 24 hours, the quantity lost must be ^ 3^7 2 0 =-0 933 ft, = 1 % inch in 

depth per day. Moreover, one mile of the canal contains 675840 cub ft; therefore, the number of days 
reqil for the combined evaporation and filtration to amount to as much as all the water in the canal, is 

”1 8 72 0 days. Observations ip warm weather on a 22 mile reach of the Chenango canal, N 

York, 740; 28; and 4 ft.) gave 65% cub ft per mile per min ; or 5 times as much as in the preceding 
case This rate would empty the canal in about 8 days. Besides this there was an excessive leakage 
at the gates of a lock, (of only 5% ft lift,) of 479 cub ft per min, 22 cub ft per mile per min ; and at 
aqueducts, and waste-weirs, others amounting to 19 cub ft per mile per min. The leakage at other 
locks with lifts of 8 ft, or less, did not exceed about 350 cub ft per min. at each. On other canals it 
hits *o be from 50, to 500 ft per min. On the Chesapeake and Ohio canal, (where 50 32 

and 6 ft.,) Mr. Fisk, C E, estimated the loss by evap and filtration in 2 weeks of warm weather, to be 

equal to an the water in the canal. Professor Ilankinc assumes 2 ins tier 

day, for leakage of canal lied, and evaporation, on English 

canals. J. B. Jervis, C E, estimated the loss from evap. filtration, and leakage through lock- 
gates, on the original Erie canal, (40, 28, and 4 ft,; at 100 cub ft per mile per miu : or 144000 cub ft 
1*4 4 CM) Wi4tCr SUrfa mile is 211200 S T ft i therefore, the daily loss would be equal to a depth of 

2 1 1 2 0 0 ~ 682 ft ’ = Say ins ‘ See end of Rain> P 221 - 


f he Delaware division of the Pennsylvania canals when 

filtration win t ? nporarily sh "t off from any long reach, the water falls from 4 to 8 ins per da'v 
filtration will of course be much greater on embankts, than in - - 1 

At. II 1 IT M D ttl n n nbln 1. • * _ .« . 


The 




ran • , " vvu vi l vaouUo 1 > ’ U • 


HYDROSTATICS.' 












HYDROSTATICS 


223 


point. At any given depth, the pres of water is equal in every direction; and is In 
direct proportion to the vert depth below the surf. In all cases whatever, the total pres 
of quiet water against, and perp to any surf, is equal to the wt of a uniform column 
ot water, the area of whoso cross-section parallel to its base, is everywhere equal to the 
area ot the surf pressed; and whose height is equal to the vert depth of the cen of 
grav of the surf pressed, below the hor surf of the water. This fact is one of those 
important ones ot frequent application, which the young student should impress 
firmly upon his memory. The wt of a cub ft of fresh water is usually assumed to 
be lbs avoir; which is sufficiently correct for ordinary engineering purposes; 
although 62J4 is nearer the truth for ordinary temperatures of about 70° Fall. Hence, 

To find the total pres of quiet water against, and perp to 
** ! *„V surf whatever, as a dam , embkt, lock-gate, die ; or the bottom, side, or top 
'?/ any cant lining vessel, water-pipe, dr,, whether said surf be vert, hor, or inclined at 
any angle whatever; or whether it bejlat, or curved; or whether it reach to the surf of 
the water, or be entirely below it: 

Rule. Mult together the area, in sq ft, of the surf pressed ; the vert depth in ft 
of its cen of grav below the surf of the water; and the constant number 62.5. The prod will be the 
reqd pres in pounds.* 

Ex. 1. The wall A. Fig 1, is 50 ft long ; and the depth, no, of water pressing against its vert back is 
uniformly 10 ft. What pres does the wall sustain? 

The area of surf pressed is 50 X 10 = 500 sq ft. And the vert depth of its cen of grav below the surf 
of the water is 5 ft; hence, 


500 X 5 X 62.5 = 156250 pounds, or about 70 tons, the pres reqd. 



FigJ 


Here the area of surf pressed is 50 X 15 = 750 sq ft. And 
the surf of the water is 5 ft. as before ; heuce. 


The pres in this case being perp to a 
vert surf, is horizontal; tcudiugeither 
to overturn the wall; or to make it 
slide on its base. The center of press¬ 
ure is at c; or one-third of the veit 
depth from the bottom. 

Ex. 2. As in the foregoing case, 
the wall B, Fig 114 1 is 50 ft long ; and 
the vert depth of water is 10 ft; but it 
presses against the sloping side of the 
wall; n o being 15 ft. What is the 
total pres, or the pres perp to no; or 
in the direction of the arrow ? 
the vert depth of its cen of grav below 


750 X 5 X 62.5 = 234375 pounds, or about 105 tons, the total pres reqd. 


The cen of pres as before, is at c, one-third of the depth from the bottom. 

In such cases, the total pres perp to n ©, may be considered as resolved into two pressures; one of 
them acting hor , either to overthrow the wall, or to make it slide; and the other actiug vert to hold it 
in its place. And if the sloped line n o be taken at any scale to represent the total pres, then will the 
vert line m o, measured by the same scale, represent the hor pres; aud the hor line m n, the vert 
one. Sfce Art 34, Force in Rigid Bodies. Therefore, so long as the vert depth of water remains the 
same, the hor pres remains the same, no matter what may be the slope of no; but the vert, as well 
as the total pres, will increase with n o. See Art 4. In Fig 2, the pres tends to lift the ^&11* 

Rkm 1 This total pres of the water is of course distributed over the entire depth of the wetted 
part of the back of the wall; being least at top. and gradually increasing toward the bottom ; but so 
far as regards the united action of every portion of it, in tending to overthrow' the wall, considered as 
a single mass of masonry, incapable of being bent or broken, it may all be assumed to be applied at 
c; dist from the bottom of tbe water one-third of its vert depth; or, which is the same thing, at 
one-third of the slop ng dist o n, Figs 1 $4 and 2. 


Rem. 2. It follows, from the foresroinjr rule, that the amount 
of pres of water against any surf is entirely independent of 
*he quantity of the water, so long as the area pressed, and the vert 
'-.nth oT its cen of grav below the level surf remain unchanged. The wall A or B would sustain as 
great a pres from a layer of water only an inch thick behind it. as if the water had extended back 
'or miles. From this cause, retaining-walls of mortar masonry carelessly backed, have been bulged, 
and cracked, by the infiltration of rain behind them ; while walls of dry masonry would have per¬ 
mitted the water to escape through the open joints; and would therefore have stood safely. 

Also in vessels a, h. Fig 214, of any size or shape whatever, if they contain 
1 the same vert depth of water ; and have equal bases o o, pressed bv said depths 

of water, the pressures on the bases will all be equal, without any regard to 
the quantity of water. Or, if we have two water-pipes of tbe same diam, both 
full or water, one standing vertically. 10 feet long; and the other 20 miles long, 
and laid at an inclination of H ft per mile, so as to make its vert depth of water 
also 10 ft. the pressures at the lower ends of the two pipes will be equal. 
This fact, that the pres on a given surf at a given depth is independent of 
the quantity of water, is called the hydrostatic paradox. In the vessel a, the 
pres on the base is much greater than the wt of the water; but in ft, it is less. 




n °° 

Fio-2 


% 


Uhm. 3. Since the pres of water against any point is at right angles to the surf at that point, it fol- 

* This is strictly true as regards the pres of the water alone ; and this is usually all that is required. 
Jt R mustbe taro” in mind that the surf of the water is itself pressed by the air; to the average 
v ent (near the level of the sea) of about 14.7 lbs per sq inch ; or 2117 lbs or nearly 1 ton per sq foot. 

to find the true total pres, we should mult the area in sq ft of the surf pressed by the water. 
J7n f r»s ;andadd the P?2d to P the water-pres given by the rule. But in ordinary engineering cases, 



















224 


HYDROSTATICS 


lows that props p p, for strengthening such structures as the sloping dam D, Fig 3, should be placed 
at right angles to them iu order to oppose the greatest possible resistauce to the pres. Other consid- j 
eratious may at times prevent our doing so; thus the outer prop, p. if so placed, would be in danger 
of being broken by ice, or logs tumbling over the dam; and therefore, had better be more nearly 
vertical. 

Rem. 4. It follows, from the foregoing rule, that in a cubical vessel, filled with water, the pres on 
the base is equal to the weight of the water; that on each of the four sides, to half the weight of the 
water; and that on the bottom and the 4 sides together, to 3 times the wt of the water. Iu a conical 
vessel, forming an entire cone, the pres on its hor base is equal to 3 times the w - t of the water; and so 
likewise in a pyramidal vessel; for in both cases the wt of the water is but that of a uniform column 
of water of the same height. In a full spherical vessel, the total pres against its entire interior surf, 
is also equal to 3 times the wt of the water, as in a cubical one. 

Since the pres increases with the depth, the props in the dam, Fig 3, 
should be closest together near the bottom ; also the hoops of a tank. 

The following: Table gives the pres to the nearest 

lb per sq ft at diff vert depths ; and also the total pres against a plane one 
foot wide extending vert from the surface to those depths. The first in¬ 
creases as the depths; the last as the squares of the depths. 



Eio3 


For the pres in lbs per sq inch at any given depth, mult the depth in ft 
by .434. For lbs per sq ft, mult by 62.5. For tons per sq ft, mult by .0279. For 
the depth in ft at which any given pres exists, divide the lbs per sq inch by 

.434; or the lbs per sq ft by 62.5: or the tous per sq ft by .0279. 


D 

in 

Ft. 

Per 

sq 

Ft. 

Tot 

P. 

D 

in 

Ft. 

Per 

sq 

Ft. 

Tot 

P. 

D 

in 

Ft. 

Per 

sq 

Ft. 

Tot 

P. 

D 

in 

Ft. 

Per 

sq 

Ft. 

Tot 

P. 

D 

in 

Ft. 

Per 

sq 

Ft. 

Tot 

P. 

1 

62. 

31 

11 

687. 

3781 

21 

1312. 

13781 

31 

1937. 

30031 

41 

2562. 

52531:.. 

2 

125. 

125 

12 

750. 

4500 

22 

1375. 

15125 

32 

2000. 

32000 

42 

2625. 

55125 

3 

187. 

281 

13 

812. 

5281 

23 

1437. 

16531 

33 

2062. 

34031 

43 

2687. 

57781 

4 

250. 

500 

14 

875. 

6125 

24 

1500. 

18000 

34 

2125. 

36125 

44 

2750. 

60500 

5 

312. 

781 

15 

937. 

7031 

25 

1562. 

19531 

35 

2187. 

38281 

45 

2812. 

63281 

6 

375. 

1125 

16 

1000. 

8000 

26 

1625. 

21125 

36 

2250. 

40500 

46 

2875. 

66125 

7 

437. 

1531 

17 

1062. 

9031 

27 

1687. 

22781 

37 

2312. 

42781 

47 

2937. 

69031 

8 

500 

2000 

18 

1125. 

10125 

VS 

1750. 

24500 

38 

2375. 

45125 

48 

3000. 

•'2000 

9 

562. 

2531 

19 

1187. 

11281 

29 

1812. 

26281 

39 

2437. 

47531 

49 

3062. 

75031 

10 

625. 

3125 

20 

1250. 

12500 

30 

1875. 

28125 

40 

2500. 

50000 

50 

3125. 

78125 


a 


Thus we see that at the depth of 36 ft, the pres of water against a single sq ft of surf, whether hor, 
vert, or oblique, is fully 1 tou ; requiriug great precaution to prevent leakage, or breaking. At 72 ft, 
it would be 2 tons. &c. A pres of 62^ lbs per sq ft gives a pres of .434 lbs per sq inch. 

Further; let a b, Fig 3)^, be a tube or 36 ft vert height; full of water; with a bore so 
small that the tube would contain say only one pound of water; aud let this tube open at 
its lower end into a vessel also full of water; the top and bottom of which are 8 ft apart. 

Then the 1 fit) of water in the tube, will cause each sq ft of the top of the vessel, (which 
is 36 ft below the surf of the water in the tube) to be pressed upward with a force of 2250 
lbs, as per table. Each sq ft of the bottom of the vessel (which is 44 ft below the surf 
of the water in the tube) will be pressed downward with a force of 2750 lbs; and any par¬ 
ticular sq ft of the sides of the vessel, will be pressed hor outward. with the force given 
in the table, opposite to the depth of the ceil of grav of said sq ft below- the same water 
surf of the top of the tube, whatever said depth may happen to be. Or. suppose, first 
only tho 'ower vessel to be filled with water, and its inner surf to be sustaining the pres 
arising therefrom; if we then fill the 36 ft tube with its 1 lb of water, this 1 fib will create 


n 


Ri3V 


an additional pres of 2250 lbs against every sq ft of said inner surf; so that if the 6 sides of the 
vessel be each 8 ft square ; or contain in all 384 sq ft of inner surf, this 1 ft of water will produce addi¬ 
tional pres of 864000 fts, or full 385 tons, against them. If we then press upon the top of the water 
vrith our thumb to the extent of 1 ft, we shall thereby redouble this enormous pres. This fact, how¬ 
ever, belongs to Art. 7, 

Art. 2. Surfaces pressed on both sides; and immersed. 

When two bodies of water of diff depths, press against two oppo¬ 
site sides of a plane which is completely immersed, whether vert or 
sloping; as, for instance, against the two sides a b, n o, Fig 4; or 
the two sides d e. c r, then, the total pres against i b, i e, a b, n o, or 
c r, &c, may still be found bv the foregoing rule, in Art 1 ; but the 
excess of pres against the part a b, or d e, of the immersed plane, 
beyond the counter-pres against the opposite part n o, or c r, will 
be equal to the wt of a column of water whose section is equal 
to the area of the part a b, or d e, (as the case may be ;) and 
whose vert height is equal to m n, or xp. the vert diff of level of the 
two bodies of water. Consequently, this excess of outward pres is 
found by mult together, the area of a b or d «. in sq ft; the vert 
height m, n or xp, in ft; and the constant 62.5 lbs wt of a cub ft of 
water. Thus, if a b is 10 ft high, and 20 ft long; and the vert height 



mn, 12 ft; then the excess of pres against a b, over that againstno, will be 10 X 20 X 12 X 62.5—15000® 
fts. The excess will be greater on d e, than on a b, although both are exposed to the same vert depths 
mn, xp: because the area of d e is greater than that of a b. Moreover, this excess of outward pres 
is equally distributed over the entire area of a b or d e ; being no greater at b and c, than at o or d ; 
in other words, every sq ft of area of a b or d e is pressed outward at right angles to its surf, by an 
excess of force equal to the wt of a column of water 1 ft sq ; and of a height equal to m n, or xp. 


this pres of the air may, and should be omitted; because it is counterbalanced by an equal pres of air 
against the opposite side, face, or surf of the pressed body. It becomes necessary, therefore, to take 
it into consideration only when the opposite face of the body is not exposed to a couuterbalancin; 
atmospheric pressure; as when there is a vacuum on that side. 












































HYDROSTATICS. 


225 


8 


62535 

i 

in. 

1875^ 

2 

u 

a 



13 

ISP 

437C 

4 

pS75 

5625^: 

5 

^3125 




Fid5 



W . 1 ' b . e understood by means of Pig 5. which may represent fire 
piautcs, i, i ,3, 4 , and o, forming a dam, and seen endwise; each one 1 ft 
in depth, and say 20 ft long hor; making the area of each surf pressed, 
equal to 30 sq ft. The pres in lbs agaiust each separate 20 sq ft of area, 
calculated by the rule in Art 1, is shown in the fig. Now, the outward 
pres against the upper immersed 20 ft area, or that of plank 3. is 3125 fts • 
while the coutiter-pres against it from the other side is 625 lbs; making 
the excess of outward pres equal to 3125 - 625 = 2500 lbs. Again, at till 
lowest plank^ number 5, the outward pres exceeds the inward one by 
ob “ a 312o _ 2o00 lbs, t he same as in the upper one. Aud so of auv other 
equal area of surf, at any depth whatever; the excess depending upon the 
vert height of m n, will be equally distributed over a b. It only remains 
to show that the total excess of outward pres agaiust a h, is equal in 
amount to the wt of a.uuilorm column of water with a base equal in area 
to a 6. aud with a height equal to mn. Thus, we have seen that in the 
instance before ns, the excess amounts to 3 times 2500 lbs, or to 7500 lbs. 
Now, the wt of the column or water will be 60 (or area of a b) X mn (or 
2 ft) X 62.5 lbs = 7500 lbs ; or the same as the excess pres on a b. 

.. t The excess of pres against the entire side s b, over that agaiust n o, is 

evidently the diff between those two pressures calculated respectively by the rule in Art 1. 

Art. 3. Surfaces, vert, as b tn c s, a n o t, Fig 6. or otherwise, of 
equal widths, b m, a n; commencing at the level, b a am. of 
the water, but extending- to ditr depths, me, no, measured 
vert; and having the same inclination to the surf of the 
water: sustain total pressures proportional to the squares 
of those depths. 

In Fig 6, let the two vert sides, a no t, and b m c s, of a vessel, 
1 have the same width a n, and b m : then if the depth m c, be 2, 3, 

b «. 4, 5, &c, times greater than the depth n o, the pres against the surf 

b m cs, will be 4, 9, 16, 25, &c, times greater than that against a n o t. 
This will be seen by referring to the pressures figured on the left side 
of Fig 5, where, as stated in Art 2, the surf of plank 1, exposed to 

the pres on the left side, is 20 sq ft; that of planks I aud 2, 40 sq ft; 

that of plauks 1, 2, and 3, 60 sq ft, &c. All these surfs commence at 
the level of the water; and all of them being vert, are of course at 
the same inclination with the water surf; but their depths are re¬ 
spectively 1, 2, aud 3 ft. The pres against the surf of 1, is 625 lbs; 
that against the surf of 1, 2, is 625 -j-1875 = 2500; and that against 
the surf of 1, 2, 3, is 625 -}-1875 -f-3125 = 5625. But 2500 is four times 
625; and 5625 is nine times 625. And the pres against the entire 
surf s h, (which is 5 times as deep as plank 1,) is 25 times as great as that against plank 1; or 
625 X 25 = 15625 lbs = the sum of all the pressures marked on the left side of Fig 5. 

This follows, from the Rule iu Art 1; for twice the area of surf, mult bv twice the vert depth of the 

cen of grav below the surf, must give 4 times the pres: three times the area, by three times the depth, 

most give 9 times the pres, <fcc. See third columns of table, p 224. 

It follows, also, that at any particular point , or against any given area placed at various depths, the 
pres will increase simply as the vert depth : thus, if there be three areas, each one sq ft, placed in 
the same positions, but with their centers of grav respectively 8, 16, and 24 ft below the surf, the pres 
against them will be respectively as 8, 16, and 24; or as 1, 2, and 3. See second columns table, p 224. 

Art. 4. The pressure of quiet water, in any one given di¬ 
rection, against any given surf, whether vert, hor, inclined, flat, or curved, is equal 
to the wt of a uniform column of water, the area of whose section, parallel to its base, is everywhere 
equal to the area of the projection of the pressed surf taken perp to the given direction; and the 
height of the column equal to the vert depth of the cen of grav of the pressed surf below the upper 
surf of the water. Hence the 

Rule. To find the pres in Il>s, mult together the area 

in sq ft of the projection taken at right angles to the given direction; the 
vert depth in ft of the cen of grav of the pressed surf below the upper surf 
of the water; and the constant 62.5 lbs wt of a cub ft of water. 

Ex. Let m c s n, Fig 7, be an inclined surf, sustaining the pres of water 
which is level with its top m c. Then the total pres against m c s n, and at 
right angles to it, as found by the rule in Art 1, is an illustration of the pres¬ 
ent rule; because the projection of m c sn, taken at right angles to the given 
direction, or parallel to m c s n, is in fact m c s n itself, or equal to it. Hence 
the rule in Art l is merely a simple modification of the present one, appli¬ 
cable to the case of total pres against any surf. 

But if it be reqd to find only the vert or downward pres 
against m e s n, in pounds, mult together the area of the hor projection an cm, 
in sq ft; the vert depth in ft of the cen of grav of m c s n below the surf; and 62.5. Or if only the 
hor pres against m c s n be sought, mult together the area of the vert projection a o s n; the vert 
depth of the cen of grav of m c s n; and 62.5. 

In Fig 8 also, the total pres against e/ g h- is found by rule in Art 1 ; while 
the hor and vert pressures against it are found as in Fig 7, by using the projec¬ 
tions efk i , and k i g h. In Fig 7 the vert pres is downward: while in Fig 8 
(j it is upward ; but this circumstance in no respect affects the rule. 

Rem. 1. At any given depth, the pres, perp to anv given surf, is the same 
in all directions; but Figs 7 and 8 show that the total pres oblique to a given 
surf will be less than the perp one at the same depth ; because an oblique pro¬ 
jection of a surf must be less than the surf itself, which last is the projection 
when the pres is perp to it. Thus, in a reservoir, the total pres perp to a 
sloping side, as m ns c, Fig 7, is greater than either the vert or the hor pres upon it. 



Fig 7 
































226 


HYDROSTATICS. 


its 


Again, let Fig 9 represent a conical vessel fill! of water; 

base 6 c, 2 ft diam; its vert height a n, 3 ft; then the circumf of the base will be 


6.2832 ft; ttie area of the base 3.1416 sq ft; the length of its slant side a b or a c, 3.16 

6.2832 X 3.16 „ „„ , . 

. =: 9.93 sq ft; and the 


ft; the area of its curved slanting sides will be 


2 



vert depth of the cen of grav of the slauting sides will be at two-thirds of the vert 
height a n from the apex a, or 2 ft. 

Here, to_ tiud the total pres against the base, we have by rule in Art 1, 3.1416 X 3 
X 62.5 — 589.05 lbs. For the total pres against the slaut sides, by the same rule, 

9.93 X 2 X 62.5 — 1241.25 lbs. For the vert pres upward against the entire area of the 
slaut sides, we have given the area of the base (which is here the hor projection of 
the slant sides) — 3.1416; and the vert depth of the cen of grav of the slant sides, 2 ft. 

3.1416 X 2 X 62.5 — 392.7 lbs, the upward vert pres. 

Finally, for the hor pres in any given direction against the slant sides of one half of the cone, w« 

Q I'll tllli I.,,, I... I »' «... _... J l i. . «. _• , • ... . . « .. 


Eg 9 


Therefore, 


have the vert projection ot that half, represented by the triangle a 1 c, with its base 2 ft, and its perp J 
height 3 ft; and consequently, with an area of 3 sq ft. The depth of its cen of grav is 2 ft; therefore. 1 
3 X 2 X 62.5 - 375 lbs, the reqd hor pres.* . 


In Fig 10, which represents a vessel full of water, the total pres 
against the semi-cylindrical surf a v e to d k, and perp to it, must be 
also hor, because the surf is vert; but inasmuch as the surf is curved, 
this total pres, us found by rule in Art 1, acts against it in many di¬ 
rections, which might be represented by an infiuite number of radii 
drawn from o as a center. But let it be reqd to find the hor pres in 
fts, in one direction only, say parallel to o e, or perp toad; which 
would be the force tending to tear the curved surf away from the flat 
sides a b n v, and d c s Ac, by producing fractures along the lines a v 
and d k ; or which would tend to burst a pipe or other cylinder. In 
this case, mult together the area of the vert projection a d k v in sq 
ft; the depth of the cen of grav of the curved surf in ft; (which, in 
the semi-cylinder would be half of e m, or of o i;) and 62.5. Since 
the resulting pres is resisted equally by the strength of the vessel 
along the two lines a v and d k , it is plain that each single thickness 
along those lines need only be sufficient to resist safely one half of it; 
and so in the case of pipes, or other cylinders, such as’hooped cisterns 
or tanks. See Art 17. 

Should the pres against only one half of the curved surf, as e dmk 
be sought, and in a direction parallel to o d, tending to produce frac- 


*1 


0 


1 1 

1 , 

1 I 

- f -^ 

• 

i ; 

v: : 

v .•. 

...in 

/ \ » 
m. \;i 

V" \ 

- lli 



FiolO 


tures along the lines e m, and d k, then use the vert projection oemi; with the same depth; and 62.5 
as before. 


It follows, that if the face of a metallic piston be made concave or convex, no more pres will be reqd 
to force the piston through any dist, than if it were flat; for the pres against the face of the piston, 
in the direction in which - -*-- J *• ■* - • ■ ... 


in which it moves, must be measured by the area of a projection of that face, taken 
at right augles to said direction; and the area of said projection will be the same in all three cases. 

Rem. 2. If a bridge pier, or other construction. 

Fig 10 U, be founded on Maud or gravel, or on any kind of 
foundation through which water may find its way underneath, even in a very thin 
sheet, then the upward pres of the water will take effect upon the pier; and will tend 
to lift it, with a force equal to the wt of the water displaced by the pier; (Arts 18, 

19- In other words, the effective wt of the submerged portion of the pier) 

will be reduced 62>4 lbs per cub ft; or nearly the half of the ordinary wt of masonry. 

Bnt if the foundation be on rook, covered with a layer 
of cement to prevent the infiltration of water beneath the masonry, no such effect 
will be produced ; but on the contrary, the vert pres downward, afforded by the bat- 

t P r 1 n C9 Q 1 t 1 O O A t f h O rt 1 O f* n 11 ,1 t\ . • «4n rfV 4TV* ♦ rt an 111 4n., A A. t-. .14 J 1 . . a 



Fig 10 2 


tering sides of the pier, and bv its offsets, will tend to hold it down, and thus increase its stshtlitr • 
which, in quiet water, will then actually be greater thau on land. ' ’ 

Art. 5, To divide a rectangular surf, 
whether vert as abed, or inclined as 
tn ti o p, Fig 11, whose top a b or m n is 
level with the surf of the water, by a 
hor line * 2, such that the total pres 
against the part above said hor line 
shall equal that against the part be¬ 
low it. 

Rule. 


Mult one half of the length of 6 c, or m p, as the case 
may be, by the constant number 1.4142; the prod will be 5 2 
or to x. * 

Ex. Let be— 12 ft. Then 6 X 1.4142 = 8.4852 ft; or 6 2 
Let mp = 16 ft. Then 8 X 1.4142 = 11.3136 ft, or to x. 

Rem. The line x 2, thus found, must not be confounded with 
the cen of pres, which is entirely diff. See Art 8. 

Art. 6. In a rectangular surf, whether vert as a b c d. or in. 
dined as mn op, I ig 11, whose top a b or wt n coincid«*w uiti. 



the...rror»he*kt e r:toBid'»uyS5mbeT«Tiomlr C i?r , 2 *i? 

the given surf into smaller rectangles, all of which 
shall sustain equal pressures. ’ wnicn 


Rule. First fix on the number of small rectangles reqd. Then for noint 1 fi-om tt> Q .. 

tmber 1, by this number of rectangles. Take the sq rt of the prod . MuUthis Jqrtby^the^ntire'length 

* III a sphere filled with a fluid the total inside pres 


3 times wt of fluid. 






















HYDROSTATICS, 


227 


b e or m p, as the case may be. Di v the prod by the number of rectangles. The quot will be the dist 
6 1 . or n 1 , as the case may be. 

For the dist 6 2, or n 2, proceed in precisely the same way; only instead of the number 1, use the 
number 2 to be mult by the number of rectangles: and so use successively the numbers 3 , 4, b , &e, 
if it be reqd to find that number of points. 

Ex. Let b c — 10 ft; and let it be reqd to find 2 points, 1 and 2, for dividiug the rectangular surf 
abed, into 3 rectangular parts, which shall sustaiu equal pressures. Here we have for point 1, 

17 

1X3 = 3. The sqrt of 3 = 1.732. And 1.732 X 10 (or 6 c) = 17.32. And — _. =5.773 ft = 61. 


3 rectangles 


24.40 
3 rectangles 


= 8.163 ft = 6 2. 


ICO— 


For point 2, we have 

2X3 = 6 . The sq rt of 6 = 2.449. And 2.449 X 10 (or 6 c) = 24.49. And 
And so for any number of poiuts. 

Rem. 1. This rule will be found useful in spacing the cross¬ 
bars of lock-grates; the hoops around cylindrical cisterns; 
ami the props to a structure, like Fig; 3. 

Rem. 2. For dividing: any surf, asofted, Fig; 12. which is not 

rectangular, in the same manner, 

with an accuracy sufficient for most practical purposes, per¬ 
haps the following method is as convenient as any. 

Hulk. First div the surf, as in Fig 12, into several small 
hor parts, equal or not, at pleasure. Then by Rule in Art 1, 
find tbe pres on each part separately, as is supposed to be 
done iu the numbers on the left hand of tbe fig. The sum of 
these (in this case 15510) is the total pres against the entire 
surf o b c d. Now suppose we wish to div this surf in 4 parts 
bearing equal pres; first div 15510 by 4 = 387,8. Then begin¬ 
ning at the top, add together a number of the separate 
pressures sufficient to amount to 3878: by this means find 
point 1. Then proceed with the addition until the sum 
amounts to twice 3878, or 7756, which will indicate point 2; 
and in the same manner find point 3. by adding up to three 
times 3878. or 11634. Then the hor dotted lines ruled through 
points 1, 2, and 3, wilt give the reqd divisions approximately. 
In this manner the hoops of conical, and other shaped ves¬ 
sels, may be spaced nearly enough for practical purposes. 


520- 


310 
740. 

960- 

1180. 

1400- 

1620.. 

1840- 

2060. 

2280— 
2500. 



Total = ,5510^^®^ 

Fig\ 12 . 



Fig:. 13. 


Art. 7. The transmission of pressure through water. Wa¬ 
ter, in common with other fluids, possesses the important 
property of transmitting pres equally in all directions. Tims, 

suppose the vessel, Fig 13, to be entirely closed, and filled with water; 
and suppose the transverse area of T,C, D, and E, to be each equal to one 
sq inch. Then, if by means of a piston, or otherwise, a pres of 1 lb. 1 
ton, or any other amount, be applied to the one sq inch of area of T, C, 
D, or E, every sq inch of the inner surf of the vessel, and of the pipe a, 
will instantly receive, at right angles to itself, an equal pres of 1 lb. or 
1 ton, &c; in addition to the pres which it before sustained from the 
water itself; and this will occur if the vessel consist of parts even miles 
asunder; as, for instance, if T were miles distant from E: and united 
to it by a long series of tubes. If the vessel were a strong steam boiler 
full of water, a single pres of a few hundred pounds at T, C, &c, would 
burst it. See also fig 3%. 

The hydrostatic press acts on this prin¬ 
ciple. Any body, within the vessel, would also receive 
an equal additional pres on each sq inch of its surf. 

If the top of T he open, tbe air will press upon the sq inch of the exposed surf of water to the extent 
of nearly 15 lbs; and the same degree of pres will also be transmitted to every sq inch of the interior 
surf of the vessel, and its connecting tubes; but no danger of bursting will result from this atmo- 
' >heric pres, because the air also presses every sq inch of the outside of the vessei to the same extent. 

Air. and other gaseous fluid*, transmit pres eqnally in all 
directions, like liquids; but not as rapidly. 

Art. 8. The center of pressure. Let Fir 14 

represent a vessel full of water, and suppose the side P to be perfectly 
loose, so as to tie thrown outward by the slightest pres of the water from 
within. Now, there is but one single point, P, in every surf so pressed, 
no matter what its shape may be. to which if we apply a force equal to 
the pres of the water, and in a direction opposite to said pres, the side P 
will be thereby prevented from yielding. Such point is called the cen¬ 
ter of pressure. It must not be understood by this that the actual 
amount of pres of the water against that part of the surface which is 
above the hor dotted line passing through P, is equal to that of the water 
below Raid line ; but that the sum of the products of the several pressures 
above it, mult by their several leverages, or vert dists from P, is equal 
to the sum of the products of the pressures below, mult by their lever¬ 
ages ; or, in other words, that the sum of the moments around the point 
P, of’the pressures above the line, is equal to the sum of the moments 
-pig. 14 of those below it; so that if a hor iron rod b b were passed entirely 

* through the side P, at the same level as the dotted line, as shown in the 

ig. so as to serve as a hinge for the side P to turn on, the side would have no tendency Lo turn. 


S 



•w 



I 


Nb 










































228 


HYDROSTATICS, 


Art. 9. To find tlie ceil of pres of a quiet 
fl 11 id, against a plane surface. Fig 15. 

1. The center of pressure of a quiet fluid against any plane surface 
whose width is uniform throughout its depth, whether said surface be 
vertical, as e o, or inclined, as c a, (or inclined in the opposite direction :) 
and whose top c, or e, coincides with the hor water surf; is distant vert 
below the water surf, two-thirds of the vert depth, s x, from said water 
surf to the bottom of the plane; as at n and i. Inasmuch as a hor line 
at % of the depth of sx, intersects both caaud eo at % of their lengths 
respectively, we might say at once that the center of pres against a plane 
of uniform width is at two-tbirds of its length below the water surface. 

Throughout Art 9 any measure, as yard, foot, or inch 
&c, may be used. 


2. Hut if the hor top a, or o, Fig 16, of the rectangular plane ag, or 
o ft, be covered to some depth with water, then the rert depth sm, of the 
ceu of pres d, or e, below the surf of the water, will be equal to 


o „ cube of »c — cube of stc 
0 square of s c — square of s to 

where sc Is the vert depth of the bottom, and sw the vert depth of the 
top, of the pressed surf, below the water surf. Or, in words; From the 
cube of sc, take the cube of stc; and call the rent a. Then, from the 
square of s c, take the square of stc; and call the rem b. Div a by 6, 
and take two-thirds of the quot for s m. 



i 


3. When a plane surf of any shape whatever, whether 
rectangular, triangular, or circular, &o; whether vert as 
op, Fig 17, or inclined as mn, is entirely immersed, so as to 
be pressed over the entire area of both sides • but by diff 
depths of water on its two sides; then the cen of pres coin¬ 
cides with the cen of grav of the pressed surf. 

In the 3 foregoing figures the supposed surfaces are shown 
edgewise, so that their widths do not appear. 



4. In any triangular plane surf, whether right-angled, or 
otherwise, as abc, Fig 18; whether vert, or inclined; the base 
a b of which coincides with the hor surf of the water; the cen 
of pres o, will be in the center of the line c r, which bisects the 
base a b. 


6. But if the triangle, as as e, vert, or inclined, have its 
apex, a, at the surf of the water; and its base sc, hor; then the 
cen of pres x. will also be in the line am which bisects the base; 
but ax will be % of am. 



0. If any plane triangle abc, Fig 19, base up, and hor; have its base 
a b covered to some depth nd, with water; then the cen of pres o, will 
be in the line cs which bisects the base; aud no will be equal to 


mx2 -j- (2bi X ma) -f- 3n»«2 
(m x -)- 2 m a) X 2. 


in n 



7. The center of pres against any 
plane rectangular surface, Fig 20, 
whether vert as mn, or inclined as 
po, or k; having its top coinciding 
with the surf of the water; and 
pressed by diff depths of water on 
its opposite sides, as shown in the 
fig; will be vert below the upper 
Water surf, a dist equal to 


Fig. 19. 

? > 



r of vert \ / 

area of surf m n.^depth ab _/ « 
or p o, or w x *--- J ^c n, 


area of surf 
or eo, or bx 


sq of vert \ / vert area 0 t t >ert \ 

X depth f^ J — ^devth X surf cu, X depth rb 


ar oreo.orsx 


( area of surf ( area of surf ^ . .. , ,\ 

Vmn, or po, or wx X ^ f of ) ^ C n, or eo, or at * half of rbj. 




































































HYDROSTATICS. 


229 


8. To find the cen of pres against either a circular, or an elliptic surf, pressed on 
one side only; whether vert, or inclined; and having its top either coinciding with 
the surf of the water, or below it. 

Call the vert depth of the cen of pres below the water surf, h. 

The vert (or inclined, as the case may be> semi diam of the surf, r. 

The vert dist of the cen of the pressed surf, below the water surf, d. 


Then, h = — + d. 
4 d 


In a vert circle with top at surf, h = 1% rad. 


Tl 

~Y 



~ - TIT*,—* 

e ’lX.l 

i J» 


=1 2 

-0—: 


%20 { 


Art. 10. Walls for resisting- the pres of quiet water. A study 

of what we have said on retaining-walls for earth, 
will be of service in this connection. It is of course 
I ' \ assumed that the water does not find its way under 

the wall; and that the wall cannot slide. In making 
calculations for walls to resist the pres of either earth, 
or water, it is convenient to assume the wall to be but 
one foot in length; (not height, or thickness;) for then 
the number of cub ft contained in it, is equal to that 
of the sq ft of area of its cross-section, or profile; so 
that these sq ft, when mult by the wt of a cub ft of 

the masonry, give the wt of the wall. In ordinary 

cases, it is well for safety to assume that the water 
extends down to the very bottom line of the Mall. 
Now, by Art 1, the total pres of quiet water, against 
♦.he rectilineal back of a wall, whether vert or sloping, is found in lbs, by mult to¬ 
gether the area in sq ft of the part actually pressed, (or in contact with the water;) 

half the vert depth of the water, in ft, (being the vert depth of the cen of grav of a 
rectilineal back, below the surf;) and the constant 62.5 lbs; and this total pres is 
always perp to the pressed area. 

When the hack of the wall is vert , as in Fig 20%, this pres p is of course less than 
M-hen it is battered; and is also hor; and it tends to overthrow the wall, by making 
it revolve around its outer toe, or edge t. The cen of pres is at c; cs being % the 
vert depth o n; in other words, the entire pres of the water, so far as regards over¬ 
throwing the wall as one mass, (see Force in Rigid Bodies,) nay be consid¬ 
ered as concentrated at the point c; where it acts with an overthrowing leverage 11 , 
(see Aits 46, 47, 49 of Force). The pres in ft>s, mult by this leverage in feet, gives the 

moment in ft-lbs of the overturning force; (see Art 49, 
Force in Rigid Bodies.) The wall, on the other hand, 
resists iu a vert direction g a, with a moment equal to its 
wt (supposed to be concentrated at its cen of grav g,) mult 
by the hor dist a t, which constitutes the leverage of the 
wt with respect to the point t as a fulcrum. If the mo¬ 
ment of the water is greater than that of the Mall, the 
latter w ill be overthrown; but if less, it will stand. 

Rf.m. 1. Art 49 of Force in Rigid Bodies, will sufficiently 
explain the subjects of moments and leverage; and make 
it evideut that the same principle applies also to sloping 
backs, as in Fig 21. Here the overturning moment of the 
water is equal to its calculated pres p X its leverage tl; 
while the moment of stability of the wall is equal to its 
M r t X its leverage a t. By aid of a drawing to a scale, we may on this principle ascer- 
iin whether any proposed wall will stand. For we have only to calculate the pres p ; 
.nen apply it at c, and at right angles to the back ; prolong it to l; measure 11 by the 
same scale. Then calculate the wt of wall; find its cen of grav g; draw g a vert, and 
measure the leverage a t. We then have the data for calculating the two moments. 
For finding the cen of grav, see Cen of Grav, Trapezoid, p. 351 c. 

Rem. 2. If the water, instead of being quiet, is liable to waves, the wall should 
be made thicker. 



2i 





















230 


HYDROSTATICS. 






Fig. 24. 


Art. 11. To find tlie thickness at base of a wall required to he^i 

afe against- overturning under llie pres of quiet- water level with its top, and 

Caution. See Art. 13, p 23L 


pressing against its entire vert back 

(1st) Vertical wall. Fig 22. 


Thickness __ Height / 
in feet — in feet X A/ 


Factor of safety * _ Height the proper decimal 

in feet X j n following table 


3 X sp grav of wall 
To change a vert wall into a battered one, see Art. 8, p 691. 


(2d) Right angled triangular wall. Fig 23. 

Thickness Height / 


at base = 


in feet 


in 


Factor of safety* Height the proper decimal 


O - I '' • ^ vuv |»i vj.ni uci/llliai 

feet * \ 2 X sp grav of wall = iu ,eet X iri following table 
= thickness, mo, of vertical wall X 1.225. 


Notwithstanding their greater thickness at- base, such triangular walls con¬ 
tain, as seen by the fig, not much more than half the quantity of masonrv reqd 
lor vert, ones of equal stability. This is owing to the fact that their cent-of 
grav is thrown farther back; thus increasing the leverage by which the wtof 
the wall resists overthrow. 


(3d) Wall with vertical back and sloping face. Fig 24. 

it. x factor of safety *) -f- (batter h w 2 , ft X sp grav of wall) 


Thickness 
at base 
in feet 


3 X specific gravity of wall 
Height in feet X the proper decimal in the following table. 


Fig. 22. 

Dressed Granite... 
Dressed Saudstone 

Mortar Rubble. 

Brickwork. 

Fig. 23. 
Dressed Granite... 
Dressed Sandstone 

Mortar Bubble. 

Brickwork. 


Sp. Gr. 


Lbs per 
Cub Ft. 


•2.5 

2.2 

2 . 

1.8 


2.5 

2.2 

2 . 

1.8 


158 

137 

125 

112 


156 

137 

125 

112 


Resist = 1.5 pres. 


.447 

.477 

.500 

.527 


.548 

.584 

.613 

.646 


Fig. 24. 


Dressed Granite... 
Dressed Sandstone 

Mortar Rubble. 

Brickwork. 


c. 

tn 


2.5 

2.2 

2 . 

1.8 


Jo 


156 

137 

125 

112 


Resist =1.5 pres. 


Resist = 2 pres. 

Resist = 3 pres. 

.516 

.623 

.550 

.674 

.578 

.707 

.609 

.746 r' 

.633 

.775 

.675 

.826 

.707 

.866 

.746 

.913 


Resist = 2 pres. 


Batter 
l in. to 
a foot. 

Batter 

2 ins. to 
a foot. 

Batter 

4 ins. to 
a foot. 

Batter 

6 ins. to 
a foot. 

Batter 

1 in. to 
a foot. 

Batter 

2 ins. to 
a foot. 

Batter 

4 ins. to 
a foot. 

Batter 

6 ins. t« 
a foot. 

.449 

.480 

.502 

.530 

.458 

.488 

.510 

.539 

.487 

.515 

.536 

.562 

.532 

.558 

.578 

.602 

.519 

.552 

.571 

.610 

.526 

.560 

.586 

.618 

.551 

.583 

.609 

.640 

,593~ 

.622 

.646 

.674 


* Factor of safety = Re q |ltr e4 moment of stability of wall 

overturning moment of water 


See p 229. 













































































HYDROSTATICS. 


231 


Art. 12. Table showing how the stability of a wall sustain* 
ing’ water is affected by a change in the form of the wall; 

the quantity of masonry remaining the same. Rem. When the base of a tri¬ 
angular wall, of sp grav 2, is less than £ the ht, the stability is greatest when 
the water presses the vert side; but if the base exceeds £ the ht, the stability 
is greatest with the water on the battered side. Caution. See Art. 13. 



All these walls contain precisely the same 
quantity of masonry. The masonry is supposed 
to be mortar rubble, weighing 125 lbs per cubic foot; or twice as much 
as water; or about the same as ordinary rough mortar rubble. If 
the sp gr of the masonry is actually greater or less than this, the 
safety also will be greater or less, in precisely the same proportion. 

Base in 
parts of 
height. 

Approx 
resist of 
wall. 

1 

Vertical wall.. 

.5 

1.5 

2 

Face vertical ; back batters one-tenth height. 

.55 

1.8 

3 

“ “ “ “ one-fifth “ . 

.6 

2.2 

4 

“ “ “ “ one-fourth “ . 

.625 

2.6 

5 

“ “ “ “ one-third “ . 

.667 

3.5 

6 

“ “ “ “ four-tenths “ . 

.7 

4.9 

7 

“ “ “ “ one-half “ . 

.75 

14.0 

8 

Back vertical; face batters one-tenth height. 

.55 

1.8 

9 

“ “ “ “ one-fifth “ . 

.6 

2.1 

10 

“ “ “ “ one-fourth “ . 

.625 

2.2 

11 

“ “ “ one-third “ . 

.667 

2.4 

12 

“ “ “ “ four-tenths “ . 

.7 

2.6 

13 

“ “ “ “ one-half “ . 

.75 

2.9 

14 

Back and face, each batter one-tenth height. 

.6 

2.2 

15 

“ “ “ “ one-fifth “ . 

.7 

3.4 

16 

** •* “ “ one-fourth “ .. 

.75 

4.6 

17 

“ “ “ “ one-third “ . 

.833 

9.0 

18 

“ “ “ “ four-tenths “ . 

.9 

36.0 


Art. 13. Liability of wall or foundation to crush under 
unequal distribution of pressure. Arts 11 and 12 apply only to 
the stability of a rigid wall resting upon a rigid base, and therefore incapable 
of failure except by overturning as a ivhole. They show that the stability is 
greatest when the water presses against the sloping side. But in practice the 
point where the resultant of all the pressures on the base of the wall cuts the 
base, must not be so near to either toe as to endanger a crushing of wall or 
of foundation. This consideration often makes it best to let the water press 
against the vert back, notwithstanding the consequent loss in stability. 

Thus, Fig 25 represents, to scale, a dam wall at Poona, India, designed by Mr. 

FifY* P F. F.ncrlnnH Tf. nf mortar ruhhlft. of 150 


rbx 


Fife, C. E., of. England. It is of mortar rubble, of 150 
lbs per cub ft. its total vert height is 100 ft; thickness 
uv at base, 60 ft 9 ins; at top, rx, 13 ft 9 ins. The front 
ru slopes 42 ft in 100 ft; and the back xv , 5 ft in 100 ft. 
Its foundation is 7 feet deep; but we here assume that 
the water presses against its expire back xv. Through 
the cen of grav G draw G s vert. From c, where the 
direction of the pres P of the water strikes Gs, lay off 
cn bv scale = 139.6 tons (of 2240 lbs) water pres against. 1 
ft in length of xv; and ct = 249.4 tons wt of 1 ft length 
of wall. Complete the parallelogram enmt of forces. 
Its diag cm represents the resultant of all the pressures 
upon the base uv, and cuts the base at a, 20 ft'back from 
the toe u. Doing the same with the 151.4 tons pres p 
against ru, we get the resultant oy, which is greater 
than cm, and cuts the base (at i) only 12.7 ft from the 
toe v, or 7.3 ft less than a is from u. 

Hence, when the water presses against xv the wall is 
less liable to fracture or crushing, and the earth foun¬ 
dation uv is more evenly loaded, and hence less liable to 
yield unequally so as to cause cracks in the wall. On 
this account xv is made the back of the wall, although 
the moment of stability of the wall is then only 2.2 (calling the overturning 
moment of the water 1), while if the water pressed against ru it would be 3, or 

For rules governing distribution of pressure, see Art 14, p 231a. 











































231a 


HYDROSTATICS. 


Art. 14. Distribution of pressure over the base uv, Figs 25, A. 

B, C and 1). Let 

uv = the length of the rectangular base, or of any rectangular surface common 
to two bodies which are pressed against each other; or (since the width of 
the surf is taken as 1) = the area of that surf. 

P = the resultant of all the extraneous forces pressing one of the two surfs 
against the other. The amount of its pres is represented bv the trapezoid 
uosv, Fig A ; triangie uov, Fig B; diff of triangles uo d — dvs, Fig C ; tri¬ 
angle uod, Fig D; or by the parallelogram untv in all four Figs. We 
confine ourselves to cases where P cuts the base iu the center of its width 
measured at right angles to uv. 

R = the resultant of all the resistances of the several points of the other surf. 
It is necessarily equal and opposite to P. 
un tv P 

u n — -- =-= the mean pressure 

uv uv 

u o — the maximum pressure 

vs— (Figs A, B, C) the minimum pressure 


j- per unit of area u v. 


This Art. applies equally whether the surf is hor, vert or inclined, and 
whether the forces are oblique to it (Figs A, B, C, D); or perp to it (Fig 71, p 357). 
If oblique, a portion of the resistance R is that of friction. See Art. 25, p. 318 e. 




as 1 it e wnn r |d 11 hi°^f a p re P r ? sents '*e pres uniformly distributed over uv, 

as it would be if P cut uv at its center, e. The intensity of the pres or its 
amount far unit of area of uv, would then be everywhere — un. ’ 

,n5 U -i ln ° Ur Figs ’ P cuts uv at an y other point, a, its pres is uneauallv 
distributed; the nearer toe, u, receiving the max pres, uo. unequally 

If (ligs A, B) ea does not exceed one-sixth uv, then 
maximum pressure uo = un (1 4- and 

\ U V / * 

minimum pressure vs = (2 un) — uo (See * and Figs.) 

If ea = one-sixth uv (Fig B), this becomes 

maximum pressure mo = 2 wn; and minimum pressure vs — 0. 

If ea exceeds one-sixth uv (Fig C), vs is less than 0, or minus • i e the 

riUlditv 8 r f ?h ndenCy u° TlSe ' a “2 actl)all > r d<>es so unless prevented either bv the 
rigidity of the pressed part, u d, of the base, or bv a tensile resistance (as bv the 

St h (Fig C) we“un’L , avV he ^ n4***&&* 

maximum pressure mo = un (l + • and 

. . \ at;/’ 

minimum pressure vs — (2 un) — uo, 

Iryswss-jja 

( le of the forces, as P, x their perpendicular distance from each other). This couple tends 


/ 




























HYDROSTATICS. 


2316 


vs being the tension, (or minus pres) per unit of area, at v. Tin’s rarelv hap¬ 
pens; for no ordinary mortar or cement can he depended upon to resist such 
tensions as might thus occur. u 


neu tral axis, d, or point of no pres; lay off' uo and vs, and draw 
d°v U ° d ' ° n Ud ' iS = * uo - ud = v P lus the tension, dvs, in 


But when (as usual) uv is practically incapable of resisting 
ply concentrated upon a portion, ud Fig D (= 3 ua) of uv: 3 
cally the base; the remainder, dv, being idle. We then have 


tension, P is sim- 
ua is then practi- 


*• 


mean pres on 3 u a — : 


3 ua 


maximum pressure uo = 2X mean pres on 3 ua = 2 P = unX 


3 ua 

pres at d, and from d to v, — 0. 


i(. 5 ~—) 

\ uv) 


In any case, the maximum pressure uo should evidently not exceed the safe 
strength of the masonry or soil. Therefore (if, as usual, no part of the base is 
to be relied upon for tension) ea in feet must not exceed 


uv, in ft X 



2 P 


3 uv X safe load per sq ft 


ft) 5 * 


Pand the safe load being in the same unit; as both in lbs, or both in tons, etc. 

First class rubble in cement mortar, or good cement concrete, should be safe 
with 8 tons per sq ft, which limit will rarelv be reached. Sound earth or gravel 
foundations, sunk to a depth sufficient to protect them from frost, rain sliding 
etc, should be safe with from 2 to 4 tons per sq ft. ’ 


I 


to press u downward and raise v. Tbis tendency causes (and is resisted by) a second couple, con¬ 
sisting of an increase, nco. of the pres on u e, and an equal decrease, tea, of the pres on ev; i e 
tes, instead of pressing on ev as before, would be called upon to help resist the first couple. The 
points, r and z, at which the resultants of the resisting forces nco and tes act, are opposite to 
the ceDS of grav, G and G', of those triangles. Each is therefore distant % of half uv from e; 
and their dist rz from each other, measd along uv, is twice this, or % u v. The resisting couple 
(f, p. 347 d) is equal and opposite to the first one. Hence each of its forces, nco and tes, must 

v,. _ first couple P .ea .. jj-.- » ... . 

be— ---t--- = -= the additional pressure, nco, on u e. (If the dist apart 

leverage of nco and tes %uv 

of the 2 forces is thus measd along it v in both couples, said dists will be in the same proportion to 
each other as the leverages.) The mean additional pres on u e, or the middle ordinate of the triangle 

n c o, is = ?«£? — PjJLff ; and the max additional pres, no, is = twice this, =IL:f a • 

ue %uv 1 %uv ' 

^ v = P xeax— X * =un— ea (^because is = u n\ And 
4 u v 2 u v uv\ u v / 


uo = itn + no = nn + (tin — un(\ 4- P e a Y 

\ uv / \ u v / 

t Fig. D. m=- # —to; 3 it a = 3 — e a 

P _„«» • „ 2mv — 2 

U 0 — 2 ■ — — 2 - —— U 71-— U M • 


3 u a 


3m a 


'(¥—) 


1 Fig D. Safe load — uo = tt nX 


3 y.5_i«\ 

\ uv/ 


(see t) or 3 { . 


(j-•-*)’ 

\ UV/ 

(.5-— a \ = 

\ uv/ 


safe load 


or 3 — = (3 X .5) — 

H V 

or € a — 


2 P 


uv X safe load 


e a 

or — = .5 — 
u v 


2 P 


ea — uv X (.5 -—- ) 

\ 3 m v X safe load / 


3 uv X safe toad 



















232 


HYDROSTATICS. 


Art. 15. The points a and i, Fig 25, are called centers of pressure 
upon the base, or centers of resistance of the base. If similar points, as 
d and z, be found in the same way for other lines, as / h, by treating a part (as 
rxhf) of the wall as if it were an entire wall; a slightly curved line joining 
these points is called the line of pressure. Thus, ba is the line of pres¬ 
sure when the water presses against xv. Each point, as d, in ba, shows where 
any joint, as fh, drawn through that point, is cut by the resultant of all the 
forces acting upon said joint, bi is the line of pres when the water presses 
against ru. These lines do not show the direction of the resultants. Thus, at a, 
the latter is cm, not ba. The angle between thedirection of the resultant and a 
line at right angles to the bed or joint, must be less than the angle of frietiou 
(p.‘155) of the materials forming the joint. 

If from the end or y of the resultant of the pressures upon any joint, we 
*.lraw m2 or y l hor, then c2 or ol (as the case may be) measures the entire vert 
pres on that joint; and m2 or yl measures the hor pres against the back of the 
wall, which tends to cause sliding at the same joint. If the direction of the re¬ 
sultant comes within the limit stated in the preceding paragraph, m2 or y l will 
be less than the frictional resistance to sliding, which last is = c2 (or ol) X the 
coetf of friction for the surfaces forming the joint. Hence sliding cannot take 
pb»ce. Sliding never occurs in the masonry of walls of ordinary forms. Good 
mortar, well set aids to prevent sliding, but it is better not to rely upon it. But 
entir • walls have s'idden on slippery foundations. (Art. 9, p. 692; Rem. 2,683.) 

Art. 16. In California is this dam of a mining reservoir, built of 

rough stone without mortar, founded on rock. Height, 70 feet; base, 50: top,6; 

water-slope, 30 feet; outer-slope, 14. To prevent leaking the 
14 water-slope is only covered with 3-inch plank bolted horizon- ' 
tally to 12 by 12 inch strings, built into the stone-work. All 
laid with some care by hand, except a core of about one-fifth of 
the mass, which was roughly thrown in. Cost about S3 per cubic 
yard. It has been in use since 1860. 

Rom. If a dam is compactly backed with earth 
at its natural slope, and in sufficient quantity to prevent the 
water from reaching the dam, the pressure against the dam will 
not be increased. 



Art. 17. To find the thickness of a cylinder to resist safely the 

pressure of water, steam, Ac, against its interior. If riveted, see next page. 

Where the thickness is less than one-thirtieth of the 
radius, as it is in most cases, the usual formula 
. Thickness pressure 

(1) in inches - itrength Xrad,us« 

is employed. It regards the material as being subjected only to a direct tensile 
strain, which is sufficiently correct in such thin shells. 

For somewhat greater pressures and thicknesses. Professor 
F. Reuleaux (Der Konstrukteur, p 52) gives 

Thickness _ pressure / pressure 

(2) in inches ~ 8afe strei ^th V + 2X safe strength ) X radlUS * 
For very sfreat pressures and thicknesses, as in hydraulic 

presses, cannons, Ac, Professor Reuleaux (Konstrukteur, p 53) gives Lame’s 
formula: 

Thickness_ / I safe strength -f pressure 

in inches \ \ safe strength — pressure 

The three formulae give results as follows, pressures and strengths in lbs per 
square inch: 1 


11 X radius.* 


Diameter. 

Radius. 

Pressure. 

Safe 

tensile 

strength. 

20 inches. 

10 inches. 

50 

10000 

it 

U 

500 

t( 

it 

a 

5000 

U 


Thickness, inches. 


Formula (1). 

Formula (2). 

Formula (3) 

.05 

.050125 

.05 

.50 

.5125 

.513 

5.00 

6.25 

7.32 


me lid m ii lie appropriate xo tne several pressures are 
printed in heavy type. It will be seen that in these cases the results differ 
but slightly, except for very great pressures. 


*In all three formulae take the 
in pounds per square inch. 


radius in inches, and the pressure and strength 








































HYDROSTATICS. 


233 


Rem... Want of uniformity iutlie cooling of thick castings makes 

them proportionally weaker than thin oues, so that in order to reduce thickness in important cases 
we should use only best iron remelted 3 or 4 times, by which mesas an ult cohesion of about 30000 

ins per sq inch may be secured. Hut even with this precaution no rule will 
»Pl*ly safely in practice to cast cylinders whose thickness exceeds either 

about 8 to 10 ins, or the inner rad however small. 

Under a pres of 8000 lbs per sq inch, water will ooze through east 
iron 8 or 10 ins thick; and under but 250 lbs per sq inch, through .5 inch 

Table of thicknesses of single-riveted wrought iron pines, 

tanks, standpipes, &c. by the above rule, to bear with a safety of 6 a quiet pressure of 1000 ft head 
of water, or 434 fbs | e • sq inch ; the ult coh of fair quality plate iron being taken at 480o0 tbs per sq 
inch or at 8000 lbs for a safety of 6; which is farther reduced to 8000 X .56 - 4480 lbs. to allow for 

weakening by rivet holes; for single-riveted cyls have hut about .56 of the 
strength of the solid sheet; and double-riveted ones about .7. With the 

1; ‘above pres and other data, the rule here leads to thickness — .1016 X inner rad in ins. 

Tor a similar table for tanks, see p 803 ; and for cast iron and 

1 lead pipes, foot of this, and top of next page. (Original.) 


Di. 

Ins. 

Ths. 

Ins. 


Di. 

Ins. 

.5 

.025 


5 

1.0 

.051 


6 

1.5 

.076 


8 

2.0 

.102 


10 

3.0 

.152 


12 

4.0 

.203 


14 


Ths. 

Ins. 


Di. 

Ins. 

Ths. 

Ins. 


Di. 

Ins. 

Ths. 

Ins. 

.254 


16 

.813 


30 

1.52 

.305 


18 

.914 


33 

1 68 

.406 


20 

1 016 


36 

1.83 

.508 


22 

1.117 


42 

2.13 

.609 


24 

1.219 


48 

2.44 

.711 


27 

1.371 


54 

2.74 


Di. 

Ins. 

Ths. 

Ins. 


Di. 

Ins. 

Di. 

.Ft. 

Ths. 

Ins. 

60 

3.05 


120 

10 

6.09 

66 

3.35 


132 

11 

6.70 

72 

3.66 


144 

12 

7.31 

84 

4.27 


192 

16 

9.75 

96 

4.88 


240 

20 

12.19 

108 

5.49 


288 

24 

14.63 


Fora less head or pressure, or for any safety less than 6, it is safe and 

j . .ear enough in practice, to reduce the thickness of wrought iron cyls in the same proportion as said 
head, pres, or safety is less than the tabular one. 


Double-rive led cylinders, Fairbairn says, are about 1.25 times as strong 
as single-riveted. Hence they may be one-fifth part thinner. Lap-welded 
ones are nearly 1.8 times as strong as single-riveted; and hence may be only 

.56 as thick. 


Many continuous miles of double-riveted pipes in California have 

been in use for years with safetys of but 2 to 2.6. In one case the head is 1720 ft, with a pres or 746 fi>s 
per sq inch ; diam 11.5 ins ; thickness, .34 inch ; safety, 2.6 by rule p 232 for such iron as in our table. 


Cast iron city water pipes must be thicker than required bv formula 
(1), p 232, in order to endure rough handling and the effects of “ water-ram ” 
j (due to sudden stoppage of flow, see second Rem, p 234), and to provide against 
1 irregularity of casting and the air bubbles or voids to which all castings are 
more or less liable. In the following table the ultimate tensile strength of cast 
iron is taken at 18,000 lbs per square inch. Column A gives thicknesses by Mr. 
J. T. Fanning’s formula (Hydraulic Engineering, p 454). 

Thickness | = (pres, lbs per sq in -j- 100) X bore, ins , / bore, ins \ 

in inches / .4 X ultimate tensile strength ' ' \ 100 / 

These correspond with average practice. The addition of 100 lbs to the pres is 
made in order to allow for water-ram. Column B gives thicknesses by formula 
(1), p 232, taking coefficient, of safety = 8 (thus making safe tensile strain = 2250 
lbs per square inch) and adding three-tenths of an inch to each thickness given 
by the formula: 


Head in feet 50 
Pressure, 


100 


200 


300 


500 


1000 


5s per sq in. 


21.7 


43.4 


86.8 


130 


217 


434 


e, ins. 



Thickness of 

pipe, in 

Inches. 




A 

B 

A 

B 

A 

B 

A 

B 

A 

B 

A 

B 

2 

.36 

.31 

.37 

.32 

.38 

.34 

.39 

.36 

.42 

.40 

.48 

.51 

3 

.37 

.31 

.38 

.33 

.40 

.35 

.42 

.40 

.45 

.45 

.54 

.60 

4 

.39 

.32 

.40 

.34 

.42 

.38 

.45 

.42 

.50 

.50 

.61 

.72 

6 

.41 

.33 

.43 

.36 

.47 

.42 

.50 

.48 

.57 

.60 

.75 

.94 

8 

.45 

.34 

.47 

.38 

.52 

.47 

.57 

.55 

.66 

.70 

.90 

1.15 

10 

.47 

.35 

.50 

.40 

.56 

.50 

.62 

.60 

.74 

.81 

1.04 

1.35 

12 

,49 

.35 

.53 

.42 

.60 

.54 

.67 

.66 

.82 

.91 

1.18 

1.57 

16 

.55 

.38 

.60 

.46 

.70 

.62 

.79 

.77 

.98 

1.10 

1.46 

2.00 

18 

.57 

.39 

.63 

.48 

.74 

.65 

.85 

.84 

1.06 

1.21 

1.60 

2.20 

20 

.61 

.40 

.67 

.50 

.79 

.68 

.91 

.90 

1.15 

1.31 

1.75 

2.50 

24 

.60 

.42 

.73 

.53 

.87 

.77 

1.02 

1.01 

1.30 

1.51 

2.03 

2.84 

30 

.74 

.45 

.83 

.59 

1.01 

.89 

1.19 

1.19 

1.55 

1.82 

2.46 

3 47 

36 

.82 

.47 

.93 

.65 

1.15 

1.01 

1.36 

1.37 

1.80 

2.12 

2.88 

4.11 

48 

.98 

.53 

1.13 

.77 

1.42 

1.24 

1.70 

1.73 

2.28 

2.73 

3.73 

5.38 









































234 


HYDROSTATICS 


Table of thickness of lead pipe to bear internal pressures with 

safety of 6; taking the ultimate cohesion of lead at 1400 lbs per sq inch. By rule on p '£32. 

Rem. Although these t hicknesses are safe againstquiet pressures.they might n^ | 

resist shocks caused by too sudden closing of stop cocks against running water. See Service pipes, p 29 




Heads in Feet. 




Heads in Feet. 

t 

© 

J3 

O 

o 

100 

200 

300 

400 

500 

00 

© 

JQ 

© 

100 

200 

300 

400 

500 

ja 

Pres in lbs per sq inch. 

.2 

Pres in lbs per sq inch. 

© 

u 

© 

43.4 

86.8 

130 

174 

217 

© 

u 

43.4 

86.8 

130 

174 

217 

pa 






« 







Thickness in Inches. 


Thickness in Inches. 

y< 

.026 

.055 

.089 

.128 

.171 

1 

.102 

.221 

.357 

.511 

.682 

% 

.038 

.083 

.134 

.192 

.256 

1* 

.127 

.276 

.447 

.639 

.S53 

y 

.051 

• 111 

.179 

.256 

.341 


.153 

.332 

.5:46 

.767 

1.02 

% 

.064 

.138 

.223 

.320 

.427 

w 

.178 

.387 

.626 

.895 

1.20 

X 

.076 

.166 

.268 

.383 

.512 

2 

.204 

.442 

.714 

1.02 

1.36 

% 

.089 

.193 

.313 

.447 

.597 








Beni. The valves of water-pipes must be closed slowly, and ‘ 

the necessity for this precaution increases with their diams. Otherwise the sud¬ 
den arresting of the momentum of the running water will create a great pressure against the pipes i 
in all directions, and throughout their eutire length behind the gate, even if it be many miles : thus ! 
endangering their bursting at any point. Hence stop-gates are shut by screws, " which pre- i 
vent any very sudden closing; but in large diams even the screws must be worked very s'owly to 1 
avoid bursting. 


Art. 18. Tlie buoyancy of liquids. When a body is placed in a 

liquid, whether it float or sink, it evidently displaces a bulk of the liquid equal to the bulk of the 
immersed portion of the body ; and the body in both cases, and at auy depth, and in any position 
whatever, is buoyed up by the liquid with a force equal to the wt of 
the liquid so displaced. Thus,if we immerse eutirely in waterapiece 
of cork c, c, I*'ig 26, or any body of less sp gr than water, the cork will 
by its wt, or force of gravity, tend to descend still deeper; but the 
upward buoyant force of the water, being greater than the downward 
force of gravity of the cork, will compel the latter to rise with a 
force equal to the diff between the two. in this case, the cork receives 
a total downward pres equal to the wt of tiie vert column of water 
above it. shown by the vert lines in vessel 1; and a total upward 
pres equal to the wt of the column shown in vessel 2. The diff be¬ 
tween these two columns is evidently (from the figs) equal to the 
bulk of the cork itself; therefore the diff between their wts or 
pressures, (or, in other words, the buoyancy of the water,) is equal 
to the wt or pres of the water which would have occupied the place 
of the cork : or, in other words, of the water which is displaced by 
the cork. This diff, or buoyancy, will plainly be the same at any 
depth whatever of entire immersion. Now the cork, if left to itself, will continue to rise until a por¬ 
tion of it reaches above the surf, as in vessel 3; so that the downward pressing column ceases to 
exist; and the cork is then pressed downward only by its own wt. But as it now remains station¬ 
ary, we know (from the fact that when two opposite forces keep a body at rest, they must be equal tr 
one another) that the upward pres of the water must be equal to the wt of the cork" But the npwni :, ‘ 
pres of the water arises only from the shaded columu shown in vessel 3; and this columu is ias i' 
the case of total immersion) equal to the bulk of water displaced. Therefore, tn all cases, the buo ft 
ancy is equal to the wt of water displaced ; and when the body floats on the surf, the buoyancy oi 
the wt of water displaced, is also equal to the wt of the body itself. 1 



If I lie immersed body c, c, be of iron, or any other substance spe¬ 
cifically heavier than water, the diff between the upward and downward pres will of course remain 
the same; or equal to the wt of water displaced. But the wt of the body is now greater than tha^ 
of the water which it displaces; or, in other words, the downward force of gravity of the body is 
greater than the upward buoyant force of the displaced water; and therefore the body descends, or 
sinks, with a force equal to the diff between the two. Thus, if the body he a cub ft of cast iron, 
weighing 450 lbs. while a cub ft of fresh water weighs 62)^ lbs, the iron will descend with an effective 
force of only 450 — 82^ = 387.5 lbs. 


If the immersed body has the same sp yr as the fluid, it will 

neither rise nor sink ; but will remain wherever it is placed ; because then the wt of the body, and 

the buoyancy of the water, are equal. 

The air also buoys bodies upward to an extent equal to Ihe 
wt of air displaced: therefore, although a pound of iron, and a pound of 

feathers, weighed in the air. will balance each other, yet in the exhausted bell glass of an nir-pumi 
the feathers will outweigh the iron, by as much as the bulk of air which they displaced outweighs 
the bulk of air displaced by the iron. 

A balloon rises in the air on the same principle that cork 
rises in water. Its ascending force is equal to the diff between its wt when 
full Of gas, and the wt of the hulk of air which it displaces. The balloon does not actually tend to 
rise, but to descend; but the air being, bulk for bulk, heavier than the balloon, pushes the latter 
upward with more force than the gravity, or the wt of the balloon, exerts to bring it down Bo also 
warm smoke has no tendency in itself to rise. It is pushed up by the heavier cold air. No substance 
tends to rise; but all tend downward toward the center of the earth. 

























































BUOYANCY, FLOTATION, METACENTER, ETC. 


235 



L 

G 

► 

; 

; 

= Wt 



The downwd force of grav may be regarded (p 348) as concentrated at tfie cen of 
•rav G of a floating body. The upwd pres, or buoyancy,f of the water may similarly 
. e regarded as acting at the cen of gr W of the displaced water.* W is also called 
Mie center of pressure, or of buoyancy, of the water; and a vert line 
frawu through it is called the axis, or vertical, of buoyancy, or of flo¬ 
at i<»n. Ordinarily,J W shifts its position with every change in that of the body, 
bus in L it is at the ceu of gr of the rectangle o o b b\ and in N at that of the tri¬ 
angle a a v. 

When a floating 
body, L, P or R, is at 
rest, and undisturbed 
by any third force, 
as F, it is said to be 
in equilibrium, 
and G and W are then 
in the same vert line 
11 Figs L and R. or 
e e Fig P ; which line 
is called the axis, 
or vertical, of 
equilibrium. § 

When a third torce, as F, causes the axis of equilib to lean, as in Figs N, 0 and S, 
then if a vert line be drawn upwd from the cen W of buoy, the point M where said 
line cuts said axis, is called the metacenter of the body. || G and W are then no 
longer in the same vert line;! and the two opp and vert forces, grav and buoy, act¬ 
ing upon those points respectively, form a “ couple ” (page Mid): and, when lb" 
third force F is removed, they no longer hold the body in equilib, but cause it to 
rotate. If (as in Figs 0 and S) the positions of G and W are then such that the 
metacenter M is above the cen of gr G, this rotation will tend to restore, the body to 
its former position , and the body is said to have been (before the application of the 
third force F) in stable equilibrium.^ But if (as in N) M is below G the direc¬ 
tion of rotation is such as to upset the body, by causing it to depart further f rom its 
former position , and the body is said to have been in unstable equilibrium.! 

F 



F 

t 


— —7 ivr 

_ 

. f t 

7 - — 

G 


/ cii 



ll 

it W 

- / 

Uv 1 

■ -A- l 

uV I 


H> 

The tendency or moment in ft-lbs of a floating body either to upset or to right itself, is, 

_ the wt of the body (or the equal v the hor dist between W M and G H, 
upwd pres of the water) in lbs Figs N, O and S, in ft. 

The third force F may of course be so great as to overpower the tendency of the body to right it¬ 
self. Thus, a ship may upset in a hurricane, although judiciously loaded and ballasted for ordinary 
-'winds. A hor section of a body at water-line is called its plane of flotation. 


} * The body is in fact acted upon by other forces, such as the hor 

-pressures of the water against its immersed portions; but as all of these in any one giveu direction 
are balanced by equal ones in the opposite direction, they have no effect upon the forces G and \V. 
It is also acted upon by the air, which presses it downwards with a force of 14-75 lbs per sq inch : hut 
this is balanced by an equal pres of the surrounding air upon the surface of the water, and which is 
transmitted (art 7, vert upwards against the immersed bottom of the floating body. 

f This buoyancy is made up of the parallel upward pressures of the 
innumerable vert filaments of the displaced water as shown by Fig 26, and 

the axis of flotation is their resultant, as in the case of parallel forces. 

I The shape of a hod v (as that of a sphere or cylinder TJ) may he such that the position of its cen of 
buov W relatively to that of its cen of gr G, is not changed bv the rotation of the body about, a given 
axis fas’auv axis of the sphere or the longitudinal axis of the cyl), but remains constantly in the 
same vert line with G, so that the body, in rotating, remains in equilib. Such a body is said to be 

in indifferent equilibrium about said axis. But if a cyl U be made to 

rotat- about, its transverse axis x x, it plainly comes under the remarks on Figs « and S, and may 
(before rotating) he in either stable or unstable equilib about that axis according to the way in which 

its wt is distributed. . .......... ..... 

II This metacenter shifts its position on the line t t according to the inclination of the latter. 

a Uneven loading, instead of a third force, may cause a vessel at rest to 
lean as at P • and yet the vessel so leaning may be in equilib: for its axis e. e of equilib may be vert, 
although not coinciding with the axis of symmetry of the vessel, as it does at 

( V In floatinq bodies, this nwv sometimes (as in Figs R and S) be the case even when the cen of 
W (not the metacenter ) is below the cen of gr G; because, when the body is forced to lean, W 
moves to another point in it, and this point may be such as to bring M above G. V, is always below 
G in bodies of uniform density, floating at rest if any part of the body is above water. \\ hen such 
bodies are entirely submerged, W and G coincide. 

16 









































236 


HYDRAULICS. 





Art. 19. A body lighter than water. If placed at 
the bottom of a vessel containing- water, will not 
rise unless the water can get under it, to buoy it, 
or press it upward, as the air presses a balloon oi 
smoke upward. Thus, if one side of a block of light wood, 

perfectly flat and smooth, be placed upon the similarly flat and smooth bottom of f 
vessel, and held there until the vessel is filled with water, the downward pres will 
keep it in its place, until water insinuates itself beneath through the pores of the 
wood. But if the wood be smoothly varnished, to exclude water from its pores, it * 
will remain at the bottom. 


Fi 3 27 


On the other hand, a piece of metal may be pre¬ 
vented from sinking in water, by subjecting it to a suffi¬ 
cient upward, pres only, while the downward pres is excluded. Thus, if the bottom 
of an open glass tube, t, Pig 27, and a plate of iron to, be made smooth enough to be 
water tight when placed as in the fig; and if in this position they be placed in a 
vessel of water to a depth greater than about 8 times the thickness'of the irou, the 
upward pres of the water will hold the iron in its place, and prevent its sinking,- 

QQPfi linwit rH hr o ool n n nf tvo tor h O o! nr tknn < Un .. .. J I . _ 


because it is pressed upward by a column of water heavier than both the column of air. and its own 
weight, which press it downward. On this principle irou ships float. 



Rem. 1. A retaining-wall, as in Fig 28, 
founded on piles, may be strong enough to re¬ 
sist the pres of the earth e behind it, in case water does not find , 
its way uuderneath; and yet may be overthrown if it does; or 
even if the earth ss around the heads of the piles becomes satu¬ 
rated with water so as to form a fluid mud. In either case, the ] 
upward pres of the water agamst the bottom of the wall will vir¬ 
tually reduce the wt of all such parts as are below the water surf, I 
to the extent of 6'2% lbs per cub ft; or nearly one-half of the or- I 
dinary wt of rubble masonry in mortar. 

Rem. 2. Although the piles under a wall, as in Fig 28, may be 
abundantly sufficient to sustain the wt of the wall; and the wall 
equally strong in itself to resist the pres of the backing e ■ yet if 


. ■ ^9 . J # * V o J * V M v J' • VO V X VU V wi*V U A U g v | J V V 11 

tne soil 8 8 around the piles be soft, both they and the wall may be pushed outward, and the latter 
overthrown by the pres of the backing e. From this cause the wing-walls of bridges, when built 
on piles in very soft soil, are frequently bulged outward and disfigured. In such cases, the piling 
and the wooden platform on top or it, should extend over the whole space between the walls; or else 
some other remedy be applied. 




Art. 20. Draught of vessels. Since a floating bodv displaces a wt of liquid 

equal to the wt of the body, we may determine the wt of a vessel and its cargo, by ascertaining how 
many cub ft of water they displace. The cub ft, mult by 62}^, will give the reqd wt in lbs. Suppose, 
for instance, a fiat-boat, with vert sides, 60 ft long. 15 ft wide, and drawing unloaded 6 ins, or .5 of 
a rt. In this case it displaces 60 X 15 X .5 — 450 cub ft of water ; which weighs 450 X 62H = 2812.' 
Ibs; which consequently is the wt of the boat also. If the eargo then be put in. and found to sin>- 
ft "? r i' we have for the wt of water ^P^ced by the cargo alone, 60 X 15 X 2 X 6214 - 
112o0O lbs ; which is also the wt of the cargo. So also, knowing beforehand the wt of the boat and 
•argo, and the dimensions of the boat, we can find what the draught will be. Thus, if the wt as before 

140625 

be 140625 Ibs. and the boat 60 X 15, we have 60 X 15 X 62>$ = 56250; and-1 - 2 5 ft the required 

. . . T , 56250 

draught. In vessels of more complex shapes, as in ordinary sailing vessels, the calculation of the 


amount of displacement becomes more tedious; but the principle remains the same. 

Art. 21. Compressibility of liquids. Liquids are not entirely in¬ 
compressible ; but for most engineering purposes they may be so considered. The bulk of water L 
diminished but about one-thousandth part by a pres of 324 Ibs per sq inch, or 22 atmospheres • vary 
mg very slightly with Us temperature. It is perfectly elastic ; regaining its original bulk when the 
pres is removed. 


HYDRAULICS. 


Art. 1. Hydraulics treats of the flow or motion of water through 

pipes, aqueducts, rivers, and other channels; also through orifices or openings of various kinds'- nr 
machmery for raising water; as well as that in which water furnishes the moving power Thl science 
of hydraulics, in many of its departments, is but imperfectly understood; therefore some oftherule, 
given on the subject are to be regarded merely as furnishing close approximaTons toTe Uu^f 


On the flow of water throug-h pipes. 

Inasmuch as the experiments on which the following rules are h-isod » 

fully laid in straight lines; and perfectly free from al? obltnmtlonfto’ the fl“w of The wt Tone 
a' low an ce must in practice be made for this circnmstance. Workmen do not lav „ some 

’ bo * h vert a " d hf)r; tbe ll so ! 1 I,self - ,n which the pipes are imbedded, especiallv^when^n enibkt 
W lit Tl" 7 5 especially in streets liable to heavy traffic, which not only frequently dertTges ’ 
but occasion ally breaks water pipes whose tops are 3 or 4 ft below the surf. The material use/fni 

w a it k er u h f 6 J °' DtS ,'i raay aa care, ® 8sl .y ,eft Projecting into the interior, and thus cause obstructions - the 
water is frequently muddy, or is impregnated with certain salts, or gases, which form denosit’s nr 
Incrustations, which materially impede the flow. Moreover the nines ‘ ? 

oast perfectly straight, or smooth, or of uniform diam; and irregular swellings, by producing eddi*es, 
























HYDRAULICS, 


237 


retard, the flow as well as contractions; and accumulations of air do the same. Under the most favor- 
able circumstances, therefore, it is expedient to make the diurns of pipes, even lor temporary pur¬ 
poses, sufficiently large to discharge at least 20 per ct more thau the quantity actually needed; and 
if there is occasiou to anticipate deposit, or incrustation, a still larger allowance should be made in 

I permanent pipes, especially in those of small diam; because in them the same thickness of incrusta¬ 
tion occupies a greater comparative portion of the area. Perhaps it would be best to allow an equal 
increase, of say from ^ to 1 % inch, to each diam, whether great or small; inasmuch as the thickness 
of incrustation will be the same for all diams, or nearly so. The cost of pipes does not increase as 
Rapidly as their discharging capacities; thus, if the diam be increased only part, the disch will 
be increased about 25 per cent; if % part, nearly 50 per cent; if ^ part, the disch will be doubled. 

Within these limits, the increase of thickness for the larger diams, and the increased 
expense of laying, will add but little to the cost; which will therefore augment only a little more 
rapidly than the diams. 

The increased diam involves no waste of water; since the disch may be regulated by stopcockB. 



The term HEAD or TOTAL IIEAI> of water, as applied to the flowage of 

rater through canals, pipes, or openings in reservoirs, &c, means the vert diet i v or p o, Fig 1, from 
the level surf, mi, of the water in the reservoir, or source of supply, to the center (or more properly to 
the cen of grav) o, of the orifice (whether the end of a pipe, r o,to, v o.za. I o; or any other kind of 
opening) through which the disch takes place freely, into the air; or the vert dist a u, or fg, from 
the same surf, m i, to the level surf, g u, of the water in the lower reservoir; when the disch takes 
place under water. Thus, in the case of disch into the air, the vert dist i v or po, is the total head 
for either of the pipes ro, t o, v o, z o, or l o ; and ik is the head for the orifice, k. in the side of the 
reservoir. And for disch under water, au. or / g. is the head for either the pipe j, or the opeuing n ; 
without any regard whatever to their depths below the surf of the lower water; which, according to 
the older authorities, do not at all affect their disch. 

A portion of a pipe may have a head greater than the total head of the entire pipe. Thus the 
point 6 in the pipe l o, has a head 6 1; while the entire pipe has only the head p o. 

Both in theory and in practice it is immaterial as regards 
the vel, ami the quantity of water discharged, whether the 
pipe is inclined downward, as ro. Fig;- 1; or hor, as ro; or in¬ 
clined upward, as lo; provided the total head po, ami also 
the length of the pipe, remain nnehanged. If one pipe is longer 

than another, its sides will evidently present more friction against the water, and thus diminish the 
vel and the quantity of disch. The inclined pipes, r o, l o, being of course a little longer than the 
hor one t>o, will therefore each disch a trifle less water: but if the hor one were extended slightly 
beyond o, so as to give it the same length as the others, then each of the three would disch the same 
tuantity in the same time. 


Art. 1 a . divisions of the Total Head. In any pipe, as so.ro, 

t o. v o, z o, or l o, Fig 1, the total head has three distinct duties to perform : 1st, to overcome the 
?sistauce to entry at s, r, t. v, z. or l ; 2d, to overcome the resistances within the pipe; and, 3d, to 
give to the water, entering the pipe, the uniform velocity with which it actually flows. 

For convenience, we regard the total head as divided into three portions, corresponding to these 
duties; namely, 1st, the entry head; 2d, the resistance, or friction, head; and, 3d. the velocity head. 

Art. 1 6. The velocity head is the height through which a body must 
fall, In vacuo, to acquire the vel witli which the water actually flows into the pipe. It is therefoie — 


, in which v is the vel in ft per sec; and g is the acceleration of gravity, or 32.2 


This head w ill be found in the table, p 258, opposite to the actual vel. 

Art. 1 e. Experiment shows that, with the usual sharp-edged entry, the en¬ 
try head is, near enough for practice, — half the vel head. If the entry is shaped 
like Fig 7, soaroelv anv entry head will he required. But, in pipes longer than about 1000 

diameters, the entry head bears so slight a proportion to the total head, that this advantage is of but 
little importance. It becomes more apparent in shorter pipes. 

Art. 1 <1. In Fig 1 we will assume that for any of the pipes, i s represents 

the sum of the vel and entry heads. Then the remainder sii, or w o, of the total head, is the 

frict ion liend; or the head winch is just sufficient to balance the friction and 

other resistances within the pipe ; and, since the entry head balances the resistance at the entrance 
to the pipe, the velocity head has only to give velocity to the water in the vessel, causing it to enter 





































238 


HYDRAULICS. 


the pipe as rapidly as it flows through it, and thus keeping the pipe supplied. If, by shortening the 
pipe, or by smoothing its inner surf, we diminish the total friction, then a less friction head will be j 
required ; but the vel will, at the same time, be increased, and this will require a greater vel he ad, I) 
and entry head, so that the three together make up Lhe total head, as before. Since the friction is 
equal to the force or head reqd to overcome it, it also is represented by too. 

Art. 1 e. The friction head may as in eo,zo,and lo, Fig 1, be all above the entrance ) 
to the pipe, aud therefore outside of the pipe : or, as in a pipe laid from s to o, it may be all below ; 
the entrauce, and within the pipe: or. as in ro and to.it may be partly above, and partly below, ihe 
entrance: and therefore partly within, and partly without, the (ipe. 1 he vel and disch, after rlie,| 
pipe is filled, are not affected by this difference in position of the entry end ; but the pressures in the'] 
pipe, and the vels while the water is filling an empty pipe, are affected by it, as explained in Arts 1 i 
aud I o 

Art. 1 f. Blit it i«t necessary that the entry end of the pipe 
should be placed so far below the surf m t, that there shall be left, 

above the cen of grav of the entry end, at least a head, i s, sufficient to perform the duties of the entry 
and vel heads. If the eutry end of any of the pipes be raised above s, a portion of the vel head will | 
be in the pipe. In other words, the head m the pipe wi 11 be more thau sufficient to overcome the 
resistances in the pipe : and the surplus will act as vel head, and will give greater vel to the water 
in the pipe. The reduced head thus left above the entry end will plainly be insufficient to maintain 
the supply for the greater vel. and the pipe will run only partly lull. 

In ordinary cases of pipes of considerable length, the sum of (he entry and vel heads theoretically 
required, is but a small portion of the total head, and tarely exceeds a foot. Indeed, in a pipe of 
considerable diameter, the upper half of its cross section at the entry eud may often he more than 
enough to provide sufficient eutry and vei heads above the cen of grav of said cross section : so that 
the top of the entry end might, so far as these considerations alone are concerned, project above the 
surf of the water in the reservoir. But the end of the pipe should in practice always be entirely be¬ 
low the surf; otherwise air and floating impurities will be drawn into it, and cause obstructions. 
Moreover, the water surf of reservoirs is always liable to considerable changes of height; and the 
entry eud of the pipe must he placed at such a depth that the water can flow into it with sufficient 
vel w hen at its lowest stages. As before stated, this will cause no diminution or increase of disch. 

Art. 1 fj. To find the frietion head reqd for any part of 
a pipe: knowing the fric bead reqd for the whole pipe. Sii.ce the*friction, in a 
pipe of uniform diam. is (other things being equal) in proportion to its length; and siuce w o. Fig 1, 
represents the total friction, or reqd friction head, we have 


w o 


The friction head reqd 
for that portion. 


A dist. ns s c, to be laid 
off from * on * o. 


Total length , Length of the 
of the pipe ♦ given portion 
Or, having drawn wo by scale, s w hor, and s o ; 

Total length . Length of the , 
of the pipe • given portion < 

O r • • »w ' A dist, as s b, to be laid off 

• • • from « ou t w. 

Then a vert line, as 6 c, drawn from b or c, and joining s w and s o, gives by scale the friction head 
reqd. 

Art. 1 Ji. If the pipe is straight,, as r o, v o, l o, the friction in any part begin¬ 
ning at the reservoir, as i 6 in the pipe l o, may be found at once by drawing a line 6 ‘l vert upward 

from the axis of the pipe at 6. The line_2J» will then give the friction in l 6. It td^Tgives the fnc- 
tiou in r 4, or in that part of v o which lies between v and the dotted line 1 6. It must be rtmera- 


cb~ 'a-—p 


n- 




V 


0 


—c>+ 

I 

I 

I 

I 

\ I 
\ I 


- --W 


VI ^ 

V v 

M 


-Tc 


bored that all the pipes in Fig 1 are supposed to he of the 
same actual length. They would thus end at different points 
o. and strictly, a separate diagram must be drawn for ea- h 
pipe. In a part of the pipe not beginning at the reservoir, 
as in r o, v o, or l o, between points vertically under c anti 
i. the amount of friction is given by the line d x, for it ia 
plainly — y x — be. 

Art. 1 j. If the pipe is vert, as v o. 

Fig 1 A; let is (on its axis to) represent, as before, the sum 
of the vel and entry heads. From s, v, and o, respectively, 
draw hor lines s w. v k. and o y. making o y~ v o Draw 
the oblique line s y. Then, to find the friction in any part 
as v g, beginning at the reservoir , from q lay off q d hor, amj 
equal to r q, and draw the vert line a d, crossing s y at a. 
Then b g will give the friction in v q. v 

Art. 1 A’. If the pipe is curved,and 

it the curvature is uniformly distributed along its length, or 
so slight that it may he neglected; the friction heads reqd 
Tor the several portions of the pipe, may he found in thi 
same way us for straight pipes, as in Art 1 H. Otherwise 
they must he found by proportion, as in Art 1 G. 

Art. I f. While water is tilling 
an empty pipe, the excess of the total head 
above the requirements of friction. &c, gives to the water a 
greater vel than it lias after tlie pipy is filled ; 
tprs the WMinn ,, „ . , , h ' lt l ^’ s gradually decreases as the advancing water eneoun' 

fills the Whote length »nd n h rei ^ ed , en « pipe mied: a " d fina,l - v l,ec ' ,mes leasl when the watei 

h i !ii 8 ' . to . t1o ' v from ,he d,sch end. »• if only the vel and entry 

oftntaV hita abo . ve the e,lt, y f ,ld ’ “® >" a pipe hi id from s to o, there will plainly be no such exces' 
f Wh»n h o d ’ d f C0u8e< l ue,lt| y. no KUch change of vel during the filling of the pipe. 

““ f i n " d, i ln i e ' er is flowing full, and is entirely open at its discharge end 

the vel throughout the pipe is equal to that at the outflow. 


Fig.l A 


:j y 


















HYDRAULICS 


239 


Rut if the opening o he contracted, but so shaped (see Fig 7, p 260) that the resistance of its edges 
may be neglected; then 

Area of . Area of . . vel of . vel throughout 

cross-section of pipe • cross-section of outflow • • outflow • the pipe. 

Art. 1 ill. Of the outward, or bursting 1 , pressure of water 
in pipes. When any pipe is full of water at rest, the attire head acts as pressure 
head, and the pres is greater than when the water is flowing through it. Thus, at the point 4 in the 
i aipe r o. Fig 1, it is that due to the head 4 1; at the point 6 in the pipe ( o, it is that due to the head 

> 1; at the point o, in any of the pipes, that due to o p. Therefore its amount in lbs per sq inch is 
; = Total head in ft X .434, as per rule p 224. 

.But if an opening he made in nny part of the pipe, or of the reservoir, the water will of course be 
;et in motion, and although the level in the reservoir be maintained, the pres in all parts of the pipe 
and of the reservoir will he reduced. The greater the area, and number, of such openings, the 
greater will be the vel, and the greater will be the reduction of pres. A part of this reduction is, in 
til cases, due to the vel of the water in the pipe, which requires the consumption of a part of the 
head (the vel head) for its maintenance. Another part is due to resistance to entry, and the remain' 
ler is due to friction within the pipe. 


Art. 1 «. The foregoing is true of all other vessels, as 
well as of pipes. Thus, if an opening o be made anywhere in 
a vessel V, Fig 1 B, the pres throughout will be reduced, and 
the water in the pipes p and < 7 , will no longer stand at the 
same level i as that in the vessel, although the vessel be kept 
full ; hut will fall to some lower one, /, which will depend for 
its height upon the relative areas of cross section of the orifice 
o and of the vessel V. 

Art. 1 o. The pressure head of running water upon 
any point in a pipe between the orifice and the reservoir, is 

. the bead consumed 

the head the in overcoming re- 

<lue to tlie -F entry + sistances in the pipe 
vel at head betiveen the reservoir 

that point 


(the total 
ss 1 head on 




(that point) 


> minus 


( : 


\ 

) 



head -- 

and the point. 

Thus, at the point 6 in the pipe ?o Fig 1, the pres head is = 3_6 = 1 6m inus (1 2 + 2 31 
= i s. 


2 being 

s, or = the sum of the vel aud entry heads. At 4 in the pipe ro, the pres head is only 3 4=1_4 

minus (1 2 + 2 3.) In a straight inclined or hor pil>e, the pres head at 

any pointTsYbus^given hy the length of a vert line drawn from the point to the 
line s o. 


Fi'r.l C 


4 rit. In. If the pipe has gentle curves, or if the cur¬ 

vature is uniformly distributed along the length of the pipe, the pres head may be 

fouud in the same wav as for a straight pipe in Art I o. But if the curves are of 

considerable extent, and unevenly distributed aloug the pipe, first find, by Art It/, 
the friction head reqd for that part of the pipe between the reservoir aud the point in 
auestion. Then find the pres head by the above formula. 

Art. 1 a. For a vert pipe v o Fig 1 A. draw the diagram as 
directed in Art 1 J. Then g d (= i q or a d — (a b + b g) ) gives the pres head at q. 

At the point o in any of the pipes, Fig 1 or Fig 1 A, the pres head is zero, sup- 
Dosing the pipe to be entirely open at that point. 

^ In a pipe laid along the line s o, Fig 1, the pres head will be zero at all points. 

Art 1 V. In a pipe r m or r m\ Fig 1 U, closed at its end, rn or 

m' but liavinc an orifice o between Its end and the reservoir, 

the pres at any point,*, between the orifice and the closed end, is equal to the pres in the tube 

opposite the orifice, plus that arising from the head o' x 
between the orifice aud the point. If the point, as x , is 
higher than o. this head Is negative, and the pres at* 
will be less than that opposite o. If the area of the ori¬ 
fice o is such as to pass the entire flow or the pipe with¬ 
out obstruction, the pres opposite o is zero. Otherwise 
there will be a pres at o varying with the amount of ob¬ 
struction to outflow at that point. . 

Art. 1 h. If a vert or oblique pipe be in¬ 

serted iuto one containing water under pres, the water 

I"' rVSftKft? «•» - etached. Such l.l*. .1. CM 

»i esEOi lie te es. or pressure-measurers. In oilier tint! Hie lit of the watei in tlieni 
!!-,r * know n thev are nl.de of glass, at least in that part of their length where the surface of the 
• i;i* P |tr to he o** else they are provided with a floating index. _ 

The »le*ome«e? is ii»«l for .let,dins the positions „r oh- 

ntpii(>t!oii« in a line of pipes. If the water in tlie piezometer is found at any time 

»f searching for it, avoided. „. 























240 


HYDRAULICS. 


Art. 1 f. If we imagine any pipe, full of water, to be supplied with a number 
of piezometers, then a line, joining the tops of the columns of water in the several 
piezometers, is called the hydraulic grade line. 

Art. 1 it. In a straight tube of uniform diam throughout, as ro, v o, or l o, Fig 
1. running full and discharging freely into the air, the hyd grade line is a straight iiue dtawn 

from its disch end o to a point s immediately over the entry eud of the pipe, aud at a depth below 
the surf equal to the sum of the vel and entry heads. 

If the orifice at o he contracted, the hyd grade line must be drawn 

from s to some point, as e, immediately over o, aud depending, for its height, upon the amount of 

contraction at o. Hut in this case 


4 


Fiff.l e 




Art. 1 v 


Fig.l F 


the point s will also he higher than 
before, because the vel in the pipe is 
reduced by the contraction ; aud the 
sum i s of the vel und entry beads 
will be less. 

If the disch at o is 
under water, the effect 

upou the position of the grade line 
will be the same as that of a con¬ 
traction of the orifice at o. The 
poiut e will be on the surf of the 
lower water, and immediately over o. 

- - - - If the pipe, of uniform 

diam, (whether discharging freely or through a cou- 
tracitd openiug at o, whether iuto the air or under 
Water), is bent or curved, the hyd grade 
line will still be straight, provided the 

resistances are equal in each equal division of the hor 
length of the pipe, as in Fig 1 K, where equal divisions 
o to, tc x, Ac, of the total length, correspond w ith equal 
divisions v a, a b, Ac, of the hor leugth. 

But in Fig 1 F, the hyd grade line will take the 
shapes a o. For ;f. in accordance with Art. 1 G, we 
div ide s o iuto two equal parts, s m, m o, correspond- 

si with t ! le tVT0 e< b‘ al l ,arts v r o. of length or the 
O pipe, we obtain rii c, ~ a e for the head consumed in the 

resistances in v r, leaving only r a for the pies head at r. 

. 

ex 


I 






. r_ __ . tesisiauces tun r, leaving only r a Tor the pres heat 

1 "• In a 'ery large vessel, the total head upon any point at the 1 
of the entrance I to a pipe loo Fig 1 G, is represented byas alreadv ex¬ 
plained p 2^7); but of this total head a portion, as is, is required to act as 
velocity bead and entry head for the entrance at l , leaving only si as the nres- 

su,e head upon a point in the pipe, immediately to 

lor right ot l \ T bu ? tlle pressure, in pounds 
per square inch, in the vessel at l , is 

.... , . P ~ i l X 0.434 . (see p 224) 

that in the pipe at / is \ v ) 

„ . ,P = s IX 0.434. 

But now a portion, as sv, of si, is expended in 
/o in balancing or “overcoming” the resistances 

th[ s G, work U rliv 00 !-° f . the > pe: and. in doing 
work, it gradually diminishes from sv (at /) to 

Thus n t"; /b at 0) - as f f dicated by the dotted IhieV* 
rims, at the point 6, a portion — be has alreadv been 

bet P wee^ 1 an°d V r r i° mi - ng ^ resistances in the pipe 
between / and 6, leaving c 6 as the pressure head at 

ances 1 n C fhp W 'w n r t • ti " ^ ex P ended against redst- 
! oo d .t p,pe between 6 and o, leaving 
o lm IvfTnf J* the pressure head for a point just 
to t he left of the contraction at o. The pressure in 

e ° e iV/ h (at ?) radually diminished from si (at 1) to 
eSl L re 9 Hired to act as ve- 


-l 




i \ 


ei 


BE 


'IV 

,|\ 
6 I \ 




0 

Fitr.l a 


Iocity and entry head for the entrance n t7, tL « 0 . 0 18 required to act as ve- 
because we need at o not only an addition!) 1 * narr o w er portion oo’ of the pipe; 
ance due to the square shoulder fortoed bl he * d *?• ov ercorne the resist- 
tionaL velocity head to give the inereasp nf col ? , r ? i ctlon i hut also an addi- 

water passes from the wide pine / n to th<T e J^ Clty which must take place as the 
pi|ie runs fi.ll and the dSr£%em ’'comaTth^.." / ^ 80 , to “* as » 
the pipe must be inversely as the aver of the in each part of 

each second the same quinitvof wafi ™i n of - t l ,at part ' bec &se in 
quantity is = area X S3? HeTiee thf each point 1. ‘l nd this constant 

increases. lty " Hence > as the area diminishes, the velocity 



















HYDRAULICS 


241 


the end o' of the pipe, as indicated by the dotted line so', being all expended in 
overcoming the resistances in o o'. We thus have, for the hydraulic gradient in 
Fig 1 G, the broken line ises'o'. 

W hen the pressure is thus diminished by overcoming resistances, or by ac¬ 
celerating velocity, the diminution is called loss of head. Thus we say that 
i s is lost at the entrance /, sv as Iriction head in to, ea' at the contraction o, and 
s' o as friction head in o o', 

When the increase of velocity at any point, as o, is very great, 

the velocity head required for such increase may (with the entry head) be as 
great as the entire available pressure head at the point. In such a case the 
pressure of course ceases entirely, and the hydraulic grade line drops to the 
level ot the axis on' oi the pipe. Indeed, the velocity head required may, and 
often does, exceed the available pressure head, causing a negative or inward pres¬ 
sure, or “suction ” (tendency to vacuum), so that if a piezometer be let downward 
from the pipe into an open vessel containing water, the pressure of the air upon 
the surface ot the latter will sustain, in the piezometer, a column of w T ater ex¬ 
tending upward from the vessel toward the pipe; and the hydraulic grade line 
lor the point where the piezometer joins the pipe, will be found below the axis 
o o'ot the pipe by a vertical distance equal to the height of said column. In 
such a case, the head is of course to be measured upward from this lower 
point and ot from the axis of the pipe. 

The syphon, or siphon. If one leg a 6 of a bent tube or pipe abc. 
Fig M, of any diam, filled with water, and with both its ends stopped, 
be placed in a reservoir of water, as in the fig; and if the stoppers be 

b then removed, the water in the reservoir will begin to flow out at c, and 
will continue to do so until its level is reduced to t, which is the same a3 
that of the highest end c of the pipe or syphon. The flow will then stop. 
The parts a b and b c are called the legs of the syphon, b being its high¬ 
est point; and this is correct so far as relates to it merely as a piece of 
tube; but considering it purely with regard to its character as a hydrau¬ 
lic machine, the part t a below the level of the highest end c, may be en¬ 
tirely neglected; for the water in the reservoir will not be drawn down 
below the level of the highest end, whether that be the inner or the outer 
one. Therefore, if the disch end be above the water in the reservoir, as, 
for instance, at w, no flow will take place. The vert height 6 o, from the 
highest part of the syphon, to the lowest level (, to which the reservoir 
is to be drawn down, must not, theoretically, exceed about 33 or 34 ft; 
or that at which the pres of the air will sustain a column of water. 
Practically it must be less, to allow for the friction of the flowing water, 
and for air which forces its way in. And still less at places far above sea 
level; for at such the reduced weight of the atmospheric column will not 
balance so great a height ot water. In order readily to understand, or 
at any time to recall the principle on which the syphon acts, bear in 
mind that we may theoretically consider the end of the inner leg to be 
not actually immersed below the water surf, but only to be kept precisely 
at it, as the surf descends while the water is flowing out; but may re¬ 
gard the vert dist b o as the length of the outer leg; and a varying dist, which at first is b s, and finally 
b o (as the surf of the reservoir descends) as the length of the inner leg; and that the flow continues 
only while this outer leg is longer than this inner one. The books are wrong in saying that the outer 
leg b c must be longer than the inner one b a, in order that the water may run at all. The principle 
then is simply this : that both these legs b c, and bi, being first filled with water, (the part ia being 
considered at first as a portion of the reservoir , and not of the syphon,) it follows that when the stop¬ 
pers are removed from the ends c aud a, the air presses equally against these ends; but the great vert 
head of water 6 o in the outer leg 5 c, presses against the air at c, with more force than the small head 
of water is in the inner leg bi, does against the air at a or i* Consequently, the water in 6 c will 
tend to fall out more rapidly than that in bi; and as it commences to fall, would produce a vacuum at 
b, were it not that the pres of the air against the other end a or i. forces the water up i b, to supply 
the place of that which flows out at c. In this manner the flow continues until the surf of the water 
in the reservoir descends to t. on the same level as c. The pressures of the vert heads bo, bo, in the 
two legs be, bt, being then equal, it ceases. 

The syphon principle may be employed for draining ponds into lower ground at a considerable dist, 
eveu though an elevation of several feet (in practice perhaps not exceeding about 28 ft above the level 
to which the pond is to be reduced) may intervene. In such a case an escape must be provided 
at the summit (or Rummits, if there are more than one) of the bends, for the disch of free air, which 
will inevitably enter, and soon stop the flow, unless this precaution be taken. The air-valve p.297, 
will not answer for this, because as soon as the valve v opens, the syphon becomes 
in effect two separate tubes open at top; and the water will fall in both. An ori¬ 
fice at the escape will be needed for filling the syphon at the start; and to pre¬ 
vent the water thus introduced, from running out. stopcocks must be provided at 
the ends, and kept closed until the filling is completed. 

The greatest pains must be taken to make all the joints perfectly air-tight. 
The motive power or head which causes the flow in a syphon, is the 
vert dist s o, from the surf of the reservoir, to the disch end c; or in other words, 
it is the ditf, s o, between the theoretical lengths b s and b o, of the two legs. Con- 



# Said pressure of the air is of course not exerted directly at a or i ; but is transmitted to a tbrougb 
the water in the vessel; and thence upward to i through the water iu the siphon. 













242 


HYDRAULICS. 



sequently, the farther c is below s the more rapid will be the flow; and it is plain 
that as the surf gradually sinks below s, the less rapid will the flow become. Hav¬ 
ing this head, the entire length ubcoi the syphon, and its diam, all in ft, the 
disch be lound approximately by eii her of the rules given in Art 2 for straight i 
I Hp , e . s \ J hes( r rule ® gi' e55/4galls per min, instead of the 43U galls actually dischd 
bj < ol C rozet s sj’phon, with a head of 20 ft, ns statcil on |> 1242. which sec. 
.i? n . R . lr . U u syphon, agnyo Fig ]£, free from air inside, and running full, ' 
* , toi.rl licnil po is measured verticallj - from the surface mi in the reser- ( i 
\oir to the center of gravity ol the outlet o, as in Fig ]; the hydraulic 
.ulit'ii] (with the restriction named in Art 1 v) is, as before, a straight line 

sro drawn from the foot s i 
of the combined entrj r and 


velocity heads to the end 
o; and the velocity and' 
discharge are the same as 
they would be if all parts 
of the pipe were brought 
below sro. But see cau¬ 
tions 1 and 2, below. 

The pressure at 
any point, g, n or y , is 
then given by a vertical 

,_,, ... .. line, gv. n r or yv, drawn 

lrom the point in question to sro: but for points, as «, situated above sro this 
pressure is negative or inward; while at points where sro and the pipe are at the 
same level, as at / and e, there is neither pressure nor vacuum. 

(ant ion 1. But if the water be admitted to the empty pipe at a, while the 
end o is open, the pipe will not form a true syphon. The part agn will then run 
lull, and will have sen as its hydraulic gradient; but upon reaching, at « a 
portion no of the pipe with a much steeper grade, the water will run off, in n o, 
AMth a velocity' greater than that with which it arrives from an. Hence the 
stream in no will have a less area of cross section than in an, and therefore can¬ 
not^ till no, but will run off in it as in an open gutter. 

( antion 2. r l he tendency to vacuum at points above sro causes an accu¬ 
mulation, at n, of particles of air that have been carried into the syphon by the 
water or have found their way in through imperfect joints, etc.; and these 
bring about a condition approaching that described in Caution 1; for their 
b L r( *n llCUlg . the nega J. i ' e P^ssure or vacuum n r at n, diminishes 
firm nf thl srnV,! °f thepart agn, while, by practically reducing the cross-sec- 
nled^d , at ? 'i tbey require that a portion of the remaining head be 

med at n, as entry head to overcome the resistance caused by the contraction 

r irt Sn y . head I? R1V ® the increase of velocity needed for passing the nar¬ 
rowed section at. n. Now since the friction head required for t he part a a n re- 

bdipd 8 ab !i U tv, the s . ame ’ tdie ve,ocit y head in the reservoir is considerably dimin- 
ished, and he water arrives at n too slowly to keep n o filled. The accumulation 

* tlms reta yds the flow and disturbs the distribution of the pressures 

so that the^e are no longer correctly indicated by vertical lines drawn t oTro 

svoho?f 7 q*f?i idse r Tu t "” e, J Vi r g inia, C°l. C. Crozet constructed a drainage 
syphon 1792 ft long of cast rron faucet pipes 3 ins bore, 9 ft long. Its summit was 
9 ft above the surface of the water to be drained ; and its discharge end was 20 ft 
below said surface, thus giving it a head of 20 ft. At the summit^TO ft fron ?he 
inlet, was an ordinary cast iron air-vessel with a chamber 3 ft high and 15 ins 
V h * e Stem connectin g it with the syphon wL a cut-off ston! 
a nd at its top was an openings ins diam, closed by an air tight screw' lid 
At each end of the syphon w'as a stopcock. To start the (low these end* 
cocks are closed, and the entire syphon and air-vessel are filled with water through 
“S °fr g a 11 t0p °J air - ves f i- This opening is then closed aHUght and thiTwo 
WJn« C a S f^F war(is ;. the cut-off cock remaining open. The flow then 

“ * aI ? * V l f ' should continue without diminution Scent so 
in nriS diminishes by the lowering of the surface level of the pond But 
in practice with very long syphons this is not the case, for air begins -it once 

wh'fre 6ltSe !i fr °- U tbe Y ater f and to travel up the syphon to the summit 
Jh vc!, t e A\ ters * he ai T r ; v cs«el, and rising to the top of the chamber gradually 
drives out the water. It this is allowed to continue the air would first fin the en¬ 
tire chamber, and then the summit of the syphon itself where it would a, » 
wad completely stopping the flow. The water-level intheair^»»!^«» a 
can be detected by the sound madeby tapping against the outside with a hammer! 






















HYDRAULICS. 


243 


To prevent this stoppage, the cut-off at the foot of the chamber is 
closed oelore the water is all driven out; and the lid on top being removed the 
chamber is refilled with water, the lid replaced, and the cut-off again opened. 
The flow in the meantime continues uninterrupted, but still gradually diminish¬ 
ing notwithstanding the refilling of the chamber; and after a number of refill¬ 
ing it will cease altogether, and the whole operation must then be repeated by 
filling the whole syphon and air chamber with water as at the start. 

At, Col. Crozet’s syphon at first owing to the porosity of the joint-caulking, 
which was nothing but oakum and pitch, air entered the pipes so rapidly as to 
drive all the water from the chamber and thus require it to be refilled every 5 or 
10 minutes; but still in two hours the syphon would run dry. The joints were 
then thoroughly recaulked with lead, and protected by a covering of white and 
red lead made into a putty with Japan varnish and boiled linseed oil. But even 
then the chamber had to be refilled with water about every two hours; and after 
six hours the syphon ran dry, and the whole had to be refilled. In this way it 
continued to work. 

In the writer’s opinion an inside, and probably an outside coating of the pipes 
and air-vessel with the coal pitch varnish, Art 33, p 291, would effect a great im¬ 
provement. 

Art. 2. Approximate formulae for the velocity of water in 
straight, smooth, cylindrical iron pipes, as ro, vo, lo, Fig 1, p 237. Having the 
total head po, and the length and diameter of the pipe. 


Approx "I 
mean vel > 

in ft per sec J 


coefficient 

m 

as below 



diam in ft X total head in ft 
total length in ft -f54 diams in ft 


Table of coefficients “ m 


/ diam X head 



diameter* of pipe, in feet 


\ length -)- 54 diams 

.05 

.10 

.50 

1 

| 1.5 

2 

3 

4 


in 

III 

m 

ill 

m 

ill 

ill 

m 

.005 

29 

31 

33 

35 

37 

40 

44 

47 

.010 

34 

35 

37 

39 

42 

45 

49 

53 

.020 

39 

40 

42 

45 

49 

52 

56 

59 

.030 

41 

43 

47 

50 

54 

57 

60 

63 

.050 

44 

47 

52 

54 

56 

60 

64 

67 

.100 

47 

50 

54 

56 

58 

62 

66 

70 

.200 l 

and over j 

48 

51 

55 

58 

60 

64 

67 

70 


The above coefficients are approximate averages deduced from a large number 
of experiments. In most cases of pipes in fair condition, carefully laid, and 
straight or nearly so, they should give results within say from 5 to 10 per cent 
of the truth. But slight differences as to roughness etc, may cause much 
greater variations, especially in small pipes, for in such a given roughness of 
surface bears a greater proportion to the whole area of cross section than in a 
pipe of large diameter. Extreme accuracy is not to be expected in such matters. 

As in a river the velocity half way across it, and at the surface, is usually 
greater than at the bottom and sides, so in a pipe the velocity is greater at the 
center of its cross section than at its circuraf. The mean velocity 
referred to in our rules is an assumed uniform one which would give the same 
discharge that the actual ununiform one does. 

Hence 

Discharge Mean velocity y Area of cross section 

in cub ft per sec ^ in It per sec of pipe in sq ft. 

See tables pp 125 to 140, 157, 247. 

1 cubic foot = 7.48052 U.S. gallons 
1 U. S. gallon = .13368 cubic foot = 231 cubic inches. 

For Kutter’s formula, as applied to pipes, see p 244. 

* For intermediate diameters, etc, take intermediate coefficients from the table by simple pro¬ 
portion. 































244 


HYDRAULICS. 


In the case of long pipes with low heads, the sum of the velocity and entry 
heads (see pp237, 238) is frequently so small that it may he neglected. Where 
this is the case, or where their amount can he approximately ascertained, Rut¬ 
ter’s formula, although designed for open channels, may be used. This 
formula is the joint production of two eminent Swiss engineers, E. Ganguillet 
and W. R. Kutter, hut for convenience it is usually called by the name of the 
latter. 

It is, properly speaking, a formula for finding the coefficient c in the well 
known formula, 


Mean velocity = c j/inean radunTx slope 



According to Kutter, 

For English measure. 


For metric measure. 




j/meau rad in feet 


1/mean rad in metres 


See also tables of c, pp 275 to 278. 

The mean radius is the quotient, in feet or in metres, obtained by divid¬ 
ing the area of wet cross section, in square feet or in square metres, by the wet > 

f ieri me ter (see below) in feet or in metres. In pipes running full, or exactly half i 
nil, and in semicircular open channels running full, it is equal to one fourth! 
of the inner diameter. 

1 , i j ► I 

The wet perimeter is the sum, a b co Figs 28,29,30, p 271, of the lengths, I 
ab, be, co, in feet or in metres, found by measuring (at right angles to the length] 
of the channel) such parts of its sides and bottom as are in contact with the 
water. In pipes running full, it is of course equal to the inner circumference. 


friction head wo Fig 1, p 237 

length of pipe measured in a straight line from end to end. 


The slope is = 


= sine of angle wso, Fig 1. 
Iu open channels, this becomes 


fail of water surface in any portion of the length of the channel 


slope = 


length of that portion 


= fall of water surface per unit of length of channel 
= sine of the angle formed between the sloping surface and the horizon. 


The number indicating the slope in any given case is plainly the same for 
English, metric and all other measures. 

“n ” is a “coefficient of roughness ” of wet perimeter, a>>d of course* 
depends chiefly upon the character of the inner surface of the pipe. For iron 
pipes in good order and from 1 inch to 4 feet diameter, n may be taken at from 
.010 to .012; the lower figures being used where the pipe is in exceptionally good 
condition. 

. If the diameter, or the mean radius, is in feet, metres etc, the velocity will be 
in feet, metres etc, per second. 






























HYDRAULICS 


245 


The diameter or the slope, required for a given velocity, 

inav be found by trial as follows: assume a diameter, or a slope, as the case may 
be; take the corresponding c from tables, pp 275, etc. Then say 

' 

Approx IMam required 

for the given vel 

) Approx Slope required _ 

for the given vel 

With the approximate diameter 

v' = c \/'mean radius X slope. If v' is near enough to the given velocity, the 
assumed diameter (or slope) is the proper one. If not, try again, assuming a 
greater diameter or slope than before ifV is less than the required velocity, and 
vice versa. 


mean ^ . 
radius * 4 


= ( 

/ velocity 

V c 


velocity \ 2 
cj/slope/ 


*)-(- 


velocity \ 2 




\/mean radius/ \cp| diam 
(or slope) and c, thus obtained, say 




To reduce cub ft to U. S. gallons, mult by 7.48. Since, therefore, 8 cub ft are equal to 60 gals, (very 
nearly,) if we divide the cub ft per 24 hours, by 8, we get the number of persons that may be 
daily supplied with 60 gals each, by a pipe constantly running full, and at the vel given In the third 
col. This condition does not exist In city water-pipes; the water in them being comparatively stag¬ 
nant. Therefore, the results of the rule and table do not at all apply to them. 

Rem. If the pipe, instead of being' straight, has easy curves. 

(say with radii not less than 5 diams of the pipe,) either hnr or vert, the disch will not be materially 
diminished, so long as the total heads, and total actual lengths of pipes remain the same: provided 
the tops of all the curves are kept below the hydraulic grade line; and provision be made for the 
escape of air accumulating at the tops of the curves. See Fig 44 A, p 297. 

Notwithstanding what is said about bonds on pages ‘266, 266, we 

advise to make the radius as much more than 5 diams as oan conveniently be done. 

To find either the area of pipe, opening, or channehway s 
or the mean % T eI; or the quantity discharged, when the other two 

are given. This applies to openings in the sides of vessels, to rivers, and to all other channels as 
well as to pipes. 

Disch in cub ft Disoh In cub ft 

Area in _ P er 8econ d Mean vel _. per second 

B( l feet ' mean vel in In ft per sec area in 

feet per sec. sq feet. 

Disoh in cub ft _ area In y mean vel In 
per second sq feet A ft per second. 

Or all the terms may be in inches instead of feet; and minutes or hours instead of seconds. 













246 


HYDRAULICS 




TABLE 2. Weight of water (at 62 l X lbs per cnb foot) con¬ 
tained in one foot length of pipes of different bores. (Original.) 


Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

X 

.00531 

2. 

1 3581 

3X 

5.0980 

’X 

19.098 

13* 

61.877 

22 

164.33 

x 

.02122 

X 

1.5.331 

4. 

5.4323 

X 

20.392 

14. 

66.545 

23 

179.60 . 

% 

.04775 

X 

1.7188 

X 

6.1325 

8. 

21.729 

X 

71.384 

24 

195.56 

X 

.08488 

% 

1.9150 

X 

6.8750 

X 

23.109 

15. 

76.392 

25 

212.20 

% 

.13263 

X 

2.1220 

X 

7.6601 

X 

24.530 

X 

81.568 

26 

229.51 

X 

.19098 

% 

2.3395 

5. 

8.4880 

X 

25.993 

16. 

86.916 

27 

247.51 

v. 

.25994 

X 

2.5676 

X 

9.3580 

9. 

27.501 

X 

92.434 

28 

266 18 

i. 

.33952 

X 

2 8063 

X 

10.270 

X 

30.641 

17. 

98.121 

29 

285.53 

X 

.42969 

3. 

3.0557 

X 

11.225 

10. 

33.952 

X 

103.97 

30 

305.57 

X 

.53050 

X 

3.3156 

6. 

12.223 

X 

37.432 

18. 

110.00 

31 

326.27 

X 

.64190 

X 

3.5862 

X 

13.262 

11. 

41.082 

X 

116.20 

32 

347.66 

X 

.76392 

X 

3.8673 

X 

14.345 

X 

44.901 

19. 

122.56 

33 

369.74 

y» 

.89654 

X 

4.1591 

X 

15.469 

12. 

48.891 

* 

129.10 

34 

392.48 

X 

1.0398 

% 

4.4615 

7. 

16.636 

X 

53.049 

20. 

135.81 

35 

415.90 

X 

1.1936 

X 

4.7745 

K 

17.846 

13. 

57.379 

21. 

149.73 

36 

440.00 . 


And in larger pipes, as the squares of their bores. Thus a pipe of 40 or 

SO ins bore, will contain 4 times as much as one of 20 or 30 ins bore; and one of -JL, * as much as 
•ne of y t iueh. At 62* lbs per cub ft, a sq iuch of water 1 ft high weighs .482292 of a B>. 




































HYDRAULICS, 


247 


TABLE 3. Areas and Contents of Pipes; and square roots 

of Biams. (Original ) Correct. 


Diaru. 

iu 

IllS. 

Diam. 

in 

Feet. 

Area in 
sq ft, also 
cub ft, 
in 1 foot 
length of 
Pipe. 

Sq. rt. 
of 

Diam. 
iu Ft. 

x 

.0208 

.0003 

.145 

516 

.0260 

.0005 

.161 

y» 

.0313 

.0008 

.177 

7-16 

.0365 

.0010 

.191 

X 

.0117 

.0014 

.204 

y -16 

.0169 

.0017 

.217 

% 

.0521 

.0021 

.228 

11-16 

.0573 

.0026 

.239 

X 

.0625 

.0031 

.250 

13 16 

.0677 

.0036 

.260 

Vs 

.0729 

.0042 

.270 

15-16 

.0781 

.0048 

.280 

1. 

.0833 

.0055 

.289 

1-16 

.0885 

.0062 

.297 

% 

.C938 

.0069 

.305 

3 16 

.0990 

.0077 

.314 

X 

.1042 

.0085 

.322 

5 16 

.1094 

.0094 

.330 

% 

.1146 

.0103 

.338 

7-16 

.1198 

.0113 

.346 

X 

.1250 

.0123 

.354 

9 16 

.1302 

.0133 

.361 

% 

.1354 

.0144 

.368 

11-16 

.1406 

.0155 

.375 

X 

. 1458 

.0167 

.382 

13-16 

.1510 

.0179 

.389 

v» 

.1563 

.0192 

.395 

1*16 

.1G15 

.0205 

.402 

2. 

.1667 

.0218 

.408 

1-16 

.1719 

.0232 

.414 

X 

.1771 

.0246 

.420 

3-16 

.1823 

.0260 

.427 

x 

.1875 

.0276 

.433 

5^6 

.1927 

.0291 

.440 

% 

.1979 

.0308 

.445 

7-16 

.2031 

.0324 

.451 

x 

.2083 

.0341 

.457 

9 16 

.2135 

.0358 

.462 

% 

.2188 

.0375 

.467 

11-16 

.2240 

.0394 

.473 

X 

.2292 

.0412 

.478 

13 16 

.2344 

.6432 

.484 

y» 

.2396 

.0451 

.489 

15 16 

.2448 

.0471 

.495 

S. 

.2500 

.0491 

.500 

H 

.2604 

.0532 

.510 

x 

.2708 

.0576 

.520 

y» 

.2813 

.0621 

.530 

x 

.2917 

.0668 

.540 

% 

.3021 

.0716 

.550 

y* 

.3125 

.0767 

.560 

X 

.3229 

.0819 

.570 


Oiam. 

iu 

Ius. 

Diam. 

iu 

Feet. 

Area in 
sq ft, also 
cub ft, 
in 1 foot 
length of 
Fipe. 

Sq. rt. 
of 

Diam. 
in Ft. 

4. 

.3333 

.0873 

.579 

X 

.3438 

.0928 

.588 

h 

.3542 

.0985 

.596 

% 

.3646 

.1040 

.604 

x 

.3750 

.1104 

.612 

% 

.3854 

.1167 

.621 

X 

.3958 

.1231 

.629 

y» 

.4063 

.1296 

.637 

5. 

.4167 

.1363 

.645 

H 

.4271 

.1433 

.653 

X 

.4375 

.1503 

.660 

X 

.4479 

.1576 

.669 

X 

.4583 

.1650 

.677 

% 

.4688 

.1725 

.685 

X 

.4792 

.1803 

.693 

y» 

.4896 

.1878 

.700 

6. 

.5 

.1964 

.707 

X 

.5208 

.2131 

.722 

y. 

.5417 

.2304 

.736 

% 

.5625 

.2485 

.750 

7. 

.5833 

.2673 

.764 

X 

.6042 

.2867 

.777 

X 

.6250 

.3068 

.791 

% 

.6458 

.3276 

.803 

8. 

.6667 

.3491 

.817 

X 

.6875 

.3712 

.829 

X 

.7083 

.3941 

.841 

X 

.7292 

.4176 

.854 

9. 

.75 

.4418 

.866 

X 

.7708 

.4667 

.879 

X 

.7917 

.4922 

.890 

% 

.8125 

.5185 

.902 

10. 

.8333 

.5454 

.913 

X 

.8542 

.5730 

.924 

X 

.8750 

.6013 

.935 

X 

.8958 

.6303 

.946 

11. 

.9167 

.6600 

.957 

X 

.9375 

.6903 

.968 

X 

.9583 

.7213 

.979 

X 

.9792 

.7530 

.990 

12. 

1 . 

.7854 

1.000 

54 

1.021 

.8184 

1.010 

X 

1.042 

.8522 

1.020 

X 

1 063 

.8866 

1.031 

13. 

1.083 

.9218 

1.041 

X 

1.104 

.9576 

1.051 

X 

1.125 

.9940 

1.060 

X 

1.146 

1.031 

1.070 

14. 

1.167 

1.069 

1.080 

x 

1.187 

1.108 

1.090 

X 

1.208 

1.147 

1.099 

X 

1.229 

1.187 

1.110 


liam- 

in 

Ins. 

Diam. 

in 

Feet. 

Area in 
sq ft, also 
cub ft, 
iu 1 foot 
length of 
Pipe. 

Sq. rt. 
of 

Diam. 
in Ft. 

15. 

1.250 

1.227 

1.118 

x 

1.271 

1.268 

1.127 

X 

1.292 

1.310 

1.136 

X 

1.313 

1 353 

1.146 

16. 

1.333 

1.396 

1.155 

X 

1.354 

1.440 

1.163 

X 

1.375 

1.485 

1.172 

X 

1.396 

1.530 

1.181 

17. 

1.417 

1.576 

1.190 

X 

1.437 

1.623 

1.199 

X 

1.458 

1.670 

1.207 

X 

1.479 

1.718 

1.216 

18. 

1.5 

1.767 

1.224 

X 

1.542 

1.867 

1.241 

19. 

1.583 

1.969 

1.258 

X 

1.625 

2.074 

1.274 

20. 

1 667 

2.182 

1.291 

X 

1.708 

2.292 

1.307 

21. 

1.750 

2.405 

1.323 

X 

1.791 

2.521 

1.339 

22. 

1.833 

2.640 

1.354 

X 

1.875 

2.761 

1.369 

23. 

1.917 

2.885 

1.384 

X 

1.958 

3.012 

1.399 

24. 

2.000 

3.142 

1.414 

‘25. 

2.083 

3.409 

1.443 

26. 

2.166 

3.687 

1.472 

27. 

2.250 

3.976 

1.500 

28. 

2 333 

4.276 

1.528 

29. 

2 416 

4.587 

1.555 

30. 

2.500 

4.909 

1.581 

31. 

2.584 

5.241 

1.607 

32. 

2.666 

5.585 

1.633 

33. 

2.750 

5.940 

1.658 

34. 

2.834 

6.305 

1.683 

35. 

2 916 

6 681 

1.708 

36. 

3.000 

7.069 

1.732 

38. 

3.166 

7.876 

1.779 

40. 

3.333 

8.727 

1.825 

42. 

3.500 

9.621 

1.871 

44. 

3.666 

10.56 

1.914 

48. 

4.000 

12.57 

2.000 

54. 

4.500 

15.90 

2.121 

60. 

5.000 

19 63 

2.236 

66. 

5.500 

23.76 

2.345 

72. 

6.000 

28.27 

2.449 

78. 

6.500 

33.18 

2.550 

84. 

7.000 

38.48 

2.646 

90. 

7.500 

44.18 

2.739 

96. 

8.000 

50.27 

2.828 


For contents in gallons, see p 157. 

V 



















































248 


HYDRAULICS. 


Art. .3. To find the total head required for a given velocity, or 
given discharge, through a straight, smooth, cylindrical iron pipe of 
known diam and length. 


If the pipe is carved, see Rem p 245. 

If the discharge is given, first find 

mean velocity _ discha rge in cubic feet per second 


Then 


in feet per second area ol cross section of pipe in square feet 


- 


I diam X head _ mean velocity in feet per second 

1 mt rrt li I o A i . ■ ■ > t o * 1* .. ^ « .. .— .. .O i* ii 


approx \ , ---——^-=--—i-^__ 

\ length -f 54 diams the proper divisor as follows 


diam of pipe in ft .05 .10 
divisor 40 43 


.50 

46 


1 

48 


1.5 

51 


2 

54 


3 

58 


4 

61 


(for intermediate diams, take intermediate divisors by guess.) 

From t able Art 2, p 243, take the coefficient m corresponding to this value of 
I diam X bead 


\length+54 diams' aDd to the 8 " e " diam ' Tllen 


Total tDer C sM ?x < len 8 th >" ft + 51 hian.s in ft) 

head =--- ; ——- ; - 

in feet in 2 X diam in teet 


To find the Friction head. Weisbach’s formula. 

.01716 \ Length 

— —t—-— K, in teet 
y/ vel in ft I 
• sec / 


Friction head 
in feet 


{ 


,0144 + 


per sec 


Diam 
in feet 



total head, we have only to add together, the friction head 
so found the velocity head, taken from the next table, or from Table 10 p 258 
oppos.te the given velocity, and the entry head (= say half the velocity head! 
Hie sum of the velocity head and entry head rarely amounts to a foot. * * 


0f , the vel, and discharge of water through straight smooth 
cylindrical cast-iron pipes; with the friction head required for eaclf 100 feet iii 

S ; T?„ alSO th f Z' OCit y h r h Ca,c " lated by means of Weisbach’s formula iy 
James Thompson, A M; and George Fuller, C E Belfast Ireland The vel i.e.,i 

water^n theffiM. ^ ° f P ‘ Pe5 beiDg de P® ndent only on the velocity of the 


5 


The entry head is equal to about half the vel head. 



























HYDRAULICS 


249 


TABLE 434. 


Diam. in Iuches. 


Yel. in 
Feet 
per Sec. 

Vel- 
head iii 
Feet. 

3 

3^ 

4 

4^ 

5 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

1 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
10O ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

f 

2.0 

2.2 

2.4 
2.6 
2.8 
3.0 

3.2 

3.4 

3.6 

3.8 
4.0 

4.2 

4.4 

4.6 

4.8 
5.0 

5.2 

5.1 

5.6 

5 8 
6.0 

6.2 

6.4 

6.6 

6.8 
7.0 

.062 

.075 

.000 

.105 

.122 

.140 

.160 

.180 

.202 

.225 

.250 

.275 

.302 

.330 

.360 

.390 

.422 

.455 

.490 

.525 

.562 

.600 

.640 

.680 

.722 

.765 

.659 

.780 

.911 

1.05 

1.20 

1.35 
1.52 

1.70 
1.89 
2.08 
2.28 

2.49 

2.71 

2.94 
3.18 
3.43 
3.68 

3.94 
4.22 

4.50 

4 78 
5.08 
5.39 
5.70 
6.02 

6.35 

5.89 

6.48 

7.07 

7.65 

8.24 

8.83 

9.42 

10.0 

10.6 

11.2 

11.8 

123 

12.9 

13.5 

14.1 

14.7 

15.3 

15.9 

16.5 

17.1 

17.7 

18.2 
1H.8 

19.4 
20.0 

20.6 

.565 

.669 

.781 

.901 

1.03 

1.16 

1.31 

1.46 

1.62 

1.78 

1.96 

2.14 
2.33 
2.52 
2.72 
2.94 

3.15 
3.38 

3.61 
3.85 
4.10 
4.36 

4.62 
4.89 

5.16 
5.45 

8.02 

8.82 

9.62 

10.4 
11.2 
12.0 
12.8 

13.6 
■ 14.4 

15.2 
16.0 
16.8 
17 6 

18.4 

19.2 
20.0 
20.8 

21.6 

22.4 

23.2 
24.0 
24.8 
25.6 

26.4 

27.3 
28.0 

.494 

.585 

.683 

.788 

.900 

1.02 

1.14 

1.27 
1.41 

1.56 
1.71 
1.87 
2.03 
2.21 
2.38 

2.57 

2.76 
2.96 
3.16 
3.37 
3.59 
3.81 
4.04 

4.28 
4.52 

4.77 

10.4 

11.5 

12.5 

13.6 

14.6 

15.7 

16.7 

17.8 

18.8 

19.9 

20.9 
22.0 
23.0 
24.0 

25.1 

26.2 

27.2 

28.2 
, 29.3 

30.3 

31.4 

32.4 
335 

34.5 

35.6 

36.6 

.439 

.520 

.607 

.701 

.800 

.905 

1.02 

1.13 

1.26 

1.39 
1.52 
1.66 
1.81 
1.96 
2.12 
2.28 
2.45 
2.63 
2.81 
3.00 
3.19 

3.39 
3.59 
3.80 
4.01 
4.24 

13.2 

14.6 
15.9 

17.2 

18.5 

19.8 

21.2 

22.5 

23.8 
25.2 

26.5 

27.8 

29.1 
30 4 

31.8 

33.1 

34.4 

35.8 

37.1 

38.4 

39.7 
41.0 

42.4 

43.7 
45.0 
46 4 

.395 

.468 

.547 

.631 

.720 

.815 

.915 

1.02 

1.13 

1.25 

1.37 
1.50 
1.63 
1.76 
1.91 
2.05 
2.21 

2.37 
2.53 
2.70 
2.87 
3.05 
3.23 
3.42 

3 61 
3.81 

163 

18.0 

19.6 

21 3 
22.9 

24.5 
26.2 

27.8 

29.4 
31.0 

32.7 

34.3 
36.0 

37.6 

39.2 

40.9 

42.5 

44.2 

45.8 

47.4 

49.1 

50.7 

52.3 
54.0 

55.6 

57.2 

Vel. in 
Keet 
per Sec. 

Yel- 
head in 
Feet. 

Diam. in Inches. 

6 

7 

8 

9 

10 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Kft per 
TOO ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

2 0 

.062 

.329 

23 5 

.282 

32 0 

.247 

41.9 

.220 

53.0 

.198 

65.4 

2.2 

.075 

.390 

25.9 

.334 

35.3 

.293 

46.1 

.260 

58.3 

.234 

72.0 

24 

.090 

.456 

28.2 

.390 

38.5 

.342 

50.2 

.304 

63.6 

.273 

78.5 

2 6 

.105 

.526 

30.6 

.450 

41.7 

.394 

54.4 

.350 

68.9 

.315 

85.1 

2 8 

.122 

.600 

32.9 

.514 

449 

.450 

58.6 

.400 

74.2 

.360 

91.6 

30 

.140 

.679 

35.3 

.582 

48.1 

.509 

62.8 

.453 

79.5 

.407 

98.2 

3.2 

.160 

.763 

37.7 

.654 

51.3 

.572 

67.0 

.508 

84.8 

.458 

105 

3 4 

.180 

.851 

40 0 

.729 

54.5 

.638 

71.2 

.567 

90.1 

.510 

111 

3.6 

.202 

.943 

42.4 

.808 

57.7 

.707 

75.4 

.629 

95.4 

.566 

118 

3.8 

.225 

1.04 

44.7 

.892 

60.9 

.780 

79.6 

.693 

101 

.624 

124 

4 0 

.250 

1.14 

47.1 

.979 

64.1 

.856 

83.7 

.761 

106 

.685 

131 

4.2 

.275 

1.25 

49.5 

1.07 

67.3 

.935 

87.9 

.832 

111 

.748 

137 

4.4 

.302 

1.35 

51.8 

1.16 

70.5 

1.02 

92.1 

.905 

116 

.814 

144 

4.6 

.330 

1.47 

54.1 

1.26 

73.7 

1.10 

96.3 

.981 

122 

.883 

150 

4 8 

.360 

1.59 

56.5 

1.36 

76.9 

1.19 

100 

1.06 

127 

.954 

157 

5.0 

.390 

1.71 

58.9 

1.47 

80.2 

1.28 

105 

1.14 

132 

1.03 

163 

5 2 

.422 

184 

61.2 

1.58 

83.3 

1.38 

. 109 

1.23 

138 

1.10 

170 

54 

.455 

1.97 

63.6 

1.69 

86.6 

1.48 

113 

1.31 

143 

1.18 

177 

5 6 

.490 

2.11 

65.9 

1.81 

89.8 

1.58 

117 

1.40 

148 

1.26 

183 

5 8 

.525 

2 25 

68.3 

1.93 

93.0 

1.68 

121 

1 50 

154 

1.35 

190 

60 

.562 

2.39 

70.7 

2 05 

96.2 

1.79 

125 

1.59 

159 

1.43 

196 

6 2 

.600 

2.54 

73.0 

2.18 

99.4 

1.90 

130 

1.69 

164 

1.52 

203 

64 

.640 

2.69 

75.4 

2.31 

102 

2.02 

134 

1.79 

169 

1.61 

209 

6 6 

680 

2.85 

77.7 

2.44 

106 

2.14 

138 

1.90 

175 

1.71 

216 

6 8 

.722 

3.01 

80.1 

2 58 

109 

2.26 

142 

2.01 

180 

1.81 

222 

7.0 

.765 

3.18 

82.4 

2.72 

112 

2.38 

146 

2.12 

185 

1.90 

229 
























































































250 


HYDRAULICS, 


TABLE 4^ —(Continued.) 


Diam. in Inches. 


Vel. in 
Feet 
per Sec. 

Vel- 
head in 
Feet. 

11 

12 

13 

14 

15 

Fr head 
Ft per 
100 ft. 

Cub ft 
per Min 

Fr head 
Ft per 
100 ft. 

Cub ft 
per Min 

Fr head 
Ft per 
100 ft. 1 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Fr head 
Ft per 
100 ft. 

Cub ft 
per Aim 

2.0 

.062 

.180 

79.2 

.165 

94.2 

.152 

110 

.141 

128 

.132 

147 

2.2 

.075 

.213 

87.1 

.195 

1 3 

.180 

121 

.167 

141 

.156 

162 

2.4 

.090 

.248 

95.0 

.228 

113 

.210 

133 

.195 

154 

.182 

176 

2.6 

.105 

.287 

103 

.263 

122 

.212 

141 

.225 

167 

.210 

191 

2.8 

.122 

.327 

111 

.300 

132 

.277 

156 

.257 

179 

.240 

206 

3.0 

.140 

.370 

119 

.339 

141 

.313 

166 

.291 

192 

.271 

221 

32 

.160 

.416 

127 

.381 

151 

.352 

177 

.327 

205 

.305 

235 

3.4 

.180 

.464 

134 

.425 

160 

.393 

188 

.365 

218 

.340 

250 

3.6 

.202 

.514 

142 

.472 

169 

.435 

• 199 

.404 

231 

.377 

265 

3.8 

.225 

.567 

150 

.520 

179 

.480 

210 

.446 

243 

.416 

280 

4.0 

.250 

.623 

158 

.571 

188 

.527 

221 

.4S9 

256 

.457 

294 

4.2 

.275 

.680 

166 

.62 4 

198 

.576 

v32 

.534 

269 

.499 

309 

4.4 

.302 

.740 

174 

.679 

207 

.626 

243 

.5S2 

2S2 

.543 

324 

4.6 

.330 

.803 

182 

.736 

217 

.679 

254 

.631 

295 

.5S9 

339 

4.8 

.360 

.867 

190 

.795 

226 

.734 

265 

.6S2 

308 

.636 

353 

5.0 

.390 

.935 

198 

.857 

235 

.791 

276 

.734 

321 

.6S5 

368 

5.2 

.422 

1.00 

206 

.920 

245 

.850 

2S7 

.789 

3:13 

.736 

383 

54 

.455 

1.07 

214 

.986 

254 

.910 

298 

.845 

346 

.789 

397 

5.6 

.490 

1.15 

222 

1.05 

264 

.973 

309 

.903 

359 

.843 

412 

58 

.525 

1.22 

229 

1.12 

273 

1.04 

321 

.964 

372 

.899 

427 

6.0 

.562 

1.30 

237 

1 19 

283 

1.10 

332 

1.02 

385 

.957 

442 

6.2 

.600 

1.38 

215 

1.27 

292 

1.17 

343 

1.09 

397 

1.01 

456 

6.4 

.640 

1.47 

253 

1.35 

301 

1.24 

354 

1.15 

410 

1.08 

471 

6.0 

.680 

1.55 

261 

1.42 

311 

1.31 

365 

1.22 

423 

1.14 

4S6 

6.S 

.722 

1.64 

269 

1.50 

320 

1 39 

376 

1.29 

436 

1.20 

500 

7.0 

.765 

1.73 

277 

1.59 

330 

1.46 

387 

1.36 

449 

1.27 

515 

— 1 1 


Hi am. in Inches. 


Vel. in 
Feet 
perSec. 

Vel- 
head in 
Feet. 

16 

17 

18 

19 

20 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Fr head 
Ft per I 
100 rt. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 1 

Cub ft 
per Alin 

Fr head 
Ft per 
100 ft. 

Cub ft 
per Mm 

2.0 

.062 

.123 

167 

.116 

189 

.110 

212 

.104 

236 

.099 

262 

2.2 

.075 

.146 

184 

.138 

208 

.130 

233 

.123 

260 

.117 

288 

2.4 

.090 

.171 

201 

.161 

227 

.152 

254 

.144 

283 

.137 

314 

2.6 

.105 

.197 

218 

.185 

246 

.175 

275 

.166 

307 

.158 

340 

2.8 

.122 

.225 

234 

.212 

265 

.200 

297 

.189 

331 

.180 

366 

3.0 

.1 40- 

.255 

251 

.240 

284 

.226 

318 

.214 

354 

.204 

393 

3.2 

.160 

.286 

268 

.269 

302 

.254 

339 

.241 

378 

.229 

419 

3.4 

.180 

.319 

284 

.300 

321 

.283 

360 

.269 

401 

.255 

445 

3.6 

.202 

.354 

301 

.333 

340 

.314 

382 

(298 

425 

.283 

471 

3.8 

.225 

.390 

318 

.367 

359 

.347 

403 

.1528 

449 

.312 

497 

40 

.250 

.428 

335 

.403 

378 

.380 

424 

.360 

472 

.342 

523 

4.2 

.275 

.468 

352 

.440 

397 

.416 

445 

.194 

496 

.374 

550 

4.4 

.302 

.509 

368 

.479 

416 

.452 

466 

29 

519 

.407 

576 

4.6 

.330 

.552 

385 

.519 

435 

.490 

488 

65 

543 

.441 

602 

4.8 

.360 

.596 

402 

.561 

454 

.530 

509 

.502 

567 

.477 

628 

5.0 

.390 

.642 

419 

.605 

473 

.571 

530 

.541 

590 

.514 

654 

5.2 

.422 

.690 

435 

.650 

492 

.614 

551 

.581 

614 

.552 

680 

5.4 

.455 

.740 

452 

.696 

511 

.657 

572 

.623 

638 

.692 

707 

5.6 

.490 

.791 

469 

.744 

529 

.703 

594 

.666 

661 

.632 

733 

5.8 

.525 

.843 

486 

.793 

548 

.749 

615 

.710 

685 

.674 

759 

6.0 

.562 

.897 

502 

.844 

567 

.798 

636 

.755 

709 

.718 

785 

6.2 

.600 

.953 

519 

.897 

586 

.847 

657 

.802 

732 

.762 

811 

6.4 

.640 

1.01 

536 

.951 

605 

.898 

678 

.851 

756 

.808 

838 

6.6 

.680 

1.07 

553 

1.01 

624 

.950 

700 

.900 

780 

.855 

864 

6.8 

.722 

1.13 

569 

1.06 

643 

1.00 

721 

.951 

803 

.904 

89G 

7.0 

.765 

1.19 

586 

1.12 

662 

1.06 

742 

1.00 

827 

.953 

916 






















































































HYDRAULICS. 


251 


TAISLE 4%. — (Continued.) 


Diam. in luches. 


Vel. In 
Feet 
perSec 

Vel- 
head ii 
Feet. 

22 

Frhear 
Ft per 
100 ft. 

24 

26 

28 

1 

30 

rr neat 
Ft per 
100 ft. 

Cub ft 
per Mil 

-j- 

Cub ft 
per Mit 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Mil 

Frhead 
Ft per 
100 ft. 

il~ 

Cub ft 
per Miu 

Frhead 
Ft per 

100 ft. 

Cub ft 
per Mia 

2.0 

2.2 

2.4 
2.6 
2.8 
3.0 

3.2 

3.4 

3.6 

3.8 
4.0 

4.2 

4.4 

4.6 

4.8 
5.0 

5.2 

5.4 

5.6 

5.8 

6.0 

6.2 

6.4 

6.6 

6.8 

7.0 

.002 

.075 

.090 

.105 

.122 

.140 

.100 

.180 

.202 

.225 

.250 

.275 

.302 

.330 

.360 

.390 

.422 

.455 

.490 

.525 

.562 

.600 

.610 

.680 

.722 

.765 

.090 

.106 

.124 

.143 

.164 

.185 

.208 

.232 

.257 

.284 

.311 

.340 

.370 

.401 

.434 

.467 

.502 

.538 

.575 

.613 

.652 

.693 

.735 

.778 

.821 

.867 

316 

348 

380 

412 

443 

475 

507 

538 

570 

601 

633 

665 

697 

728 

760 

792 

823 

855 

887 

918 

950 

982 

1013 

1045 

1077 

1109 

.082 

.097 

.114 

.131 

.150 

.170 

.191 

.213 

.236 

.260 

.285 

.312 

.339 

.368 

.397 

.428 

.460 

.493 

.527 

.562 

.598 

.635 

.673 

.713 

.753 

.794 

377 

414 

452 

490 

528 

565 

603 

641 

678 

716 

754 

791 

829 

867 

905 

942 

980 

1018 

1055 

1093 

1131 

1168 

1206 

1244 

1282 

1319 

.076 

.090 

.105 

.121 

.138 

.157 

.176 

.196 

.218 

.240 

.263 

.288 

.313 

.339 

.367 

.395 

.425 

.455 

.486 

.519 

.552 

.586 

.622 

.658 

.695 

.733 

442 

486 

531 

575 

619 

663 

708 

752 

796 

840 

885 

929 

973 

1017 

1062 

1106 

1150 

1194 

1239 

1283 

1327 

1371 

1416 

1460 

1504 

1548 

.070 

.083 

.097 

.112 

.128 

.145 

.163 

.182 

.202 

.223 

.244 

.267 

.290 

.315 

.341 

.3G7 

.394 

.423 

.452 

.482 

.513 

.544 

.577 

.611 

.645 

.681 

513 

564 

616 

667 

718 

770 

821 

872 

923 

974 

1026 

1077 

1129 

1180 

1231 

1283 

1334 

1385 

1437 

1488 

1539 

1590 

1642 

1693 

1741 

1796 

.066 
.078 
.091 
.105 
.120 
.136 
.152 
.170 
.189 
.208 
.228 
.249 
.271 
.294 
.318 
.343 
.368 
.394 
.422 
.450 
.478 
.508 
.539 | 
.570 
.602 ! 
.635 

589 

648 

707 

766 

824 

883 

942 

1001 

1060 

1119 

1178 

1237 

1296 

1355 

1414 

1472 

1531 

1590 

1649 

1708 

1767 

1826 

1885 

19 43 
2003 
2061 


) 


Power. 


0000100 

.0000110 

.0000122 

.0000131 

.0000147 
.0000161 
.0000176 
. .0000103 
.0000210 
1 .0000220 
t .0000240 
.0000270 
.0000203 
.0000318 
.0000341 
.0000371 
.0000*01 
.0000432 
.000046.3 
.0000500 
.0000538 
.0000577 
.0000619 
.0000663 
.0000710 
.0000754 
.0000895 
.000105 
.000122 


TABLE o. Of fifth roots nnd fifth powers. 


No. o 
Root. 

r Power. 

No. o 
Root. 

r Power. 

No. o 
Root. 

r Power. 

No. o 
Root 

Power. 

No. ot 
Root. 

Power. 

No. or 
Root. 

.1 

.0001*2 

.170 

.00*219 

.335 

.077760 

.60 

.695688 

.93 

8.11368 

1.51 


.00016* 

.175 

.0015** 

.3*0 

.08**60 

.61 

.733904 

.94 

8.66171 

1.54 

.102 

.000189 

.180 

.00*888 

.3*5 

.091613 

.62 

.773781 

.95 

9.2385)6 

1.56 


.000217 

1 .185 

.005252 

.350 

.0992*4 

.63 

.815373 

.96 

9.84658 

1.58 

.104 

.0002*8 

| .190 

.005638 

.355 

.10737* 

.64 

.858734 

.97 

10.4858 

1.60 


.000282 

.195 

.0060*7 

.360 

.116029 

.65 

.903921 

.98 

11.1577 

1.62 

• 10b 

.000320 

.200 

.000478 

.365 

.125233 

.66 

.950990 

.99 

11.8637 

1.64 


000362 

.205 

.00693* 

.370 

.135012 

.67 

1. 

1. 

12.60*9 

I 1.66 

.108 

000*08 

.210 

.007416 

.375 

.1*5393 

.68 

1.10*08 

1.02 

13.3828 

1.68 

.110 

.000*59 

.215 

.00792* 

.380 

,156403 

.69 

1.21665 

1.04 

14.1986 

1.70 

. 112 

.000515 

.220 

.008*59 

.385 

.168070 

.70 

1.33823 

1.06 

15.0537 

1.72 

.114 

.000577 

.225 

.009022 

.390 

.180*23 

.71 

1.46933 

1.08 

15.9495 

1.74 

.11b 

.0006** 

.230 

.009616 

.395 

.193492 

.72 

1.61051 

1.10 

16.8874 

1.76 

.118 

.000717 

.235 

.0102*0 

.400 

.207307 

.73 

1.76234 

1.12 

17.8690 

1.78 

.120 

0)0706 

.2*0 

.011586 

.4* 

.221901 

.74 

1.92541 

1.14 

18.8957 

1.80 

.122 

.000883 

.2*5 

.013069 

.*2 

.237305 

.75 

2.100.34 

1.16 

19.965)0 

1.82 

. 124 

.000977 

.250 

.01*701 

.43 

.253553 

.76 

2.28775 

1.18 

21.05)06 

1.84 

. j 2b 

.001078 

.255 

.010*92 

.** 

.270678 

.77 

2.48832 

1.20 

22.2620 

1.86 

.128 

.001188 

.260 

.018453 

.45 

.288717 

.78 

2.70271 

1.22 

23.48*9 

1.88 

.130 

.001307 

.265 

.020596 

.46 

.307706 

.79 

2.93163 

1.24 

24.7610 

1.90 

.132 

.001*35 

.270 

.022935 

.47 

.327680 

.80 

3.17580 

1.26 

26.0919 

X. 92 

.134 

.001573 

.275 

•025*80 

.48 

.3*8678 

.81 

3.43597 

1.28 

27.4795 

1.94 

. l $6 

.001721 

.280 

.0282*8 

.49 

.370740 

.82 

3.71293 

1.30 

28.9255 

1.96 

«np? 

.001880 

.285 

.0.31250 

.50 

.39390* 

.83 

4.00746 

1.32 

30.4317 

1.98 

.1*0 

.002051 

.290 

.03+503 

.51 

.418212 

.84 

4.32040 

1.34 

32.0000 

2.00 

.142 

.00223* 

.295 

.038020 

.52 

.443705 

.85 

4.65259 

1.36 

36.2051 

2.05 

.144 

.002*30 

.300 

.0*1820 

.53 

.470427 

.86 

5.00490 

1.38 

*0.8*10 

2.10 

. 14b 

.002639 

.305 

.0*5917 

.5* 

.498421 

.87 

5.37824 

1.40 

*5.9401 

2.15 

.1*8 

.002863 

.310 

.050328 

.55 

.527732 

.88 

5.77353 

1.42 

51.5363 

2.20 

.150 

.003101 

.315 

.055073 

.56 

.558406 

.89 

6.19174 

1.4* 

57.6650 

2.25 

.155 

.003355 

.320 

.060169 

.57 

.590490 

.90 

6.63.383 

1.46 

64.3634 

2.30 

.100 

.00.3626 

.325 

.065636 

.58 

.624032 

.91 

7.10082 

1.48 

71.6703 

2.35 

.165 

.003914 

.330 J 

.071492 

.59 

.659082 

.92 

7.59375 

1.50 

79.6262 

2.40 


17 

















































































































252 


HYDRAULICS 


TABLE 5. Of fifth roots anti fifth powers — (Continued.) 


Power. 

No. or 
Root. 

Power. 

No. or 
Root. 

Power. 

Mo. or 
Root. 

Power. 

Mo. or 
Root. 

Power. 

No. or 
Root. 

Power. 

No. or 
Root. 

KM *2735 

2.45 

2824.75 

4.90 

85873 

9.70 

2609193 

19.2 

20511149 

29.0 

459165024 

54. 


2.50 

2971.84 

4.95 

90392 

-9.80 

2747949 

19.4 

21228253 

29.2 

503284375 

55. 

]07 MVO 


3125.00 

5 AH) 

95099 

9.90 

2892547 

19.6 

21965275 

29.4 

550731776 

56. 

1 IK MI4 

2.60 

3450.25 

5.10 

100000 

10.0 

3043168 

19.8 

22722628 

29.6 

601692057 

57. 

];:() 1WK 


3802.04 

5-20 

110408 

10.2 

3200000 

20.0 

23500728 

29.8 

656356t6b 

58. 

1 [A 1M9 

2.70 

4181.95 

5.30 

121665 

10.4 

3363232 

20.2 

21300000 

30.0 

714924299 

59. 

7 57 *27 fi 

2.75 


5.40 

133823 

10.6 

3533059 

20.4 

20393634 

30.5 

777600000 

60. 

]7*2 104 

2 .h 0 

5032.84 

5.50 

146933 

10.8 

3709677 

20.6 

28629151 

31.0 

844596301 

61. 

],KH 029 

2.85 

5507.32 

5-60 

161051 

11.0 

3893289 

20.8 

31013642 

31.5 

916132832 

62. 

205 111 

2.90 

6016.92 

5-70 

176234 

11.2 

4084101 

21.0 

33554432 

32.0 

992436543 

63. 

22iL414 

2.95 


5.80 

192541 

11.4 

4282322 

21.2 

36259082 

32.5 

1073741824 

64. 

213.000 

3.00 

7149.24 

5.90 

210034 

11.6 

4488166 

21.4 

39135393 

33.0 

1160290625 

65. 

26:1.936 

3.05 

7776.00 

6-00 

228776 

11.8 

4701850 

21.6 

42191410 

33.5 

1252332576 

66. 

286.292 

3.10 

8445.96 

6.10 

248832 

12.0 

4923597 

21.8 

45435424 

34.0 

1350125107 

67. 

310 136 

3.15 

9161.33 

6-20 

270271 

12.2 

5153632 

22.0 

48875980 

34.5 

1453933568 

68. 


3.20 

9924.37 

6.30 

293163 

12.4 

5392186 

22.2 

52^21875 

35.0 

1564031349 

69. 

362.591 

3.25 

10737 

6-40 

317580 

12.6 

5639493 

22.4 

56382167 

35.5 

1680700000 

70. 

391.354 

3.30 

11603 

6.50 

343597 

12.8 

5895793 

22.6 

60466176 

36 0 

1804229351 

71. 

421.419 

3.35 

12523 

6.60 

371293 

13.0 

6161327 

22.8 

64783487 

36 5 

1934917682 

72. 

454.354 

3.40 

13501 

6.70 

400746 

13.2 

6436.343 

23.0 

69343957 

37.0 

2073071593 

73. 

488.760 

3.45 

14539 

6.80 

432040 

13.4 

6721093 

23.2 

74157715 

37.5 

2219006624 

74. 

525.219 

3.50 

15640 

6.90 

465259 

13 6 

7015834 

23.4 

79235168 

38.0 

2373046875 

75. 

563.822 

3.55 

16807 

7.00 

500490 

13.8 

7320825 

23.6 

84587005 

38.5 

2535525376 

76. 

604.662 

3.60 

18042 

7-10 

537824 

14.0 

7636332 

23.8 

90224199 

39.0 

2706784157 

77. 

647.835 

3.65 

19349 

7.20 

577353 

14.2 

7962624 

24.0 

96158012 

39.5 

2887174368 

7B. 

693.440 

3.70 

20731 

7.30 

619174 

14.4 

8299976 

24.2 

102400000 

40.0 

3077056399 

79. 

741.577 

3.75 

22190 

7.40 

663383 

14.6 

8648666 

24.4 

108962013 

40.5 

3276800000 

80. 

792.352 

3.80 

23730 

7.50 

710082 

14.8 

9008978 

24.6 

115856201 

41.0 

3486784401 

81 • 

845.870 

3.85 

25355 

7.60 

759375 

15.0 

9381200 

24.8 

123095020 

41.5 

3707398432 

82. 

902.242 

3.90 

27068 

7.70 

811368 

15.2 

9765625 

25.0 

130691232 

42.0 

3939040643 

83. 

961.580 

3.95 

28872 

7.80 

866171 

15.4 

10162550 

25.2 

138657910 

42.5 

4182119424 

■84. 

1024.00 

4.00 

30771 

7.90 

923896 

15.6 

10572278 

25.4 

[47008443 

43.0 

443705312;> 

85. 

1089.62 

4.05 

32768 

8.00 

984658 

15.8 

10995116 

25.6 

155756538 

43.5 

4704270176 

86. 

1158.56 

4.10 

34868 

8.10 

1048576 

16.0 

11431377 

25.8 

164916224 

44 0 

4984209207 

87. 

1230.95 

4.15 

37074 

8.20 

1115771 

16.2 

11881376 

26.0 

174501858 

44.5 

5277319168 

88. 

1306.91 

4.20 

39390 

8.30 

1186367 

16.4 

12345437 

26.2 

184528125 

45.0 

5584059449 

89. 

1386.58 

4.25 

41821 

8.40 

1260493 

16.6 

12823886 

26.4 

195010045 

45.5 

5904900000 

90. 

1470.08 

4.30 

44371 

8.50 

1338278 

16.8 

13317055 

26.6 

105962976 

46.0 

6240321451 

91. 

1557.57 

4.35 

47043 

8.60 

1419857 

17.0 

13825281 

26.8 

>17402615 

46 5 

6590815232 

92. 

1649.16 

4.40 

49842 

8.70 

1505366 

17.2 

14348907 

27.0 

229345007 

47 0 

6956883693 

93. 

1745.02 

4.45 

52773 

8.80 

1594947 

17.4 

14888280 

27.2 

241806543 

47.5 

7339040224 

94. 

1845.28 

4.50 

55841 

8.90 

1688742 

17.6 

15443752 

27.4 

254803968 

48.0 

7737809375 

95. 

1950.10 

4.55 

59049 

9.00 

1786899 

17.8 

16015681 

27.6 

268354383 

48.5 

8153726971 

96. 

2059.63 

4.60 

62403 

9.10 

1889568 

18.0 

16604430 

27.8 

>82475249 

49.0 

8587340257 

97. 

2174.03 

4.65 

65908 

9.20 

1996903 

18.2 

17210368 

88.0 

297184391 

49.5 

9039207968 

98. 

2293.45 

4.70 

69569 

9.30 

2109061 

18.4 

178:33868 

28.2 

112500000 

50.0 

950990049! 

99. 

2418.07 

4.75 

73390 

9.40 

2226203 

18.6 

18475.309 

28.4 

145025251 

51. 



2548.04 

4.80 

77378 

9.50 

2348493 

18.8 

19135075 

28.6 

180204032 

52. 



2683.54 

4.85 

81537 

9.60 

2476099 

19.0 

19813557 

28.8 

118195493 

53. 





















































HYDRAULICS, 


253 


TABLE 6. Of the square roots of the fifth powers of num¬ 
bers. In this table the numbers and the roots are supposed to be in the same di¬ 
mensions ; that is, both in inches, or both in feet, &c. See the next table. 


No. 

Sq. Rt. 
of 5th 
Power. 

No. 

Sq. Rt. 
of 5th 
Power. 

No. 

Sq. Rt. 
of 5th 
Power. 

No. 

Sq. Rt. 
of 5th 
Power. 

No. 

Sq. Rt. 
of 5th 
Power. 

No. 

Sq. Rt. 
of 5th 
Power. 

.25 

.031 

7. 

129.64 

17.5 

1281.1 

31. 

53ol 

49 

16807 

76 

50354 

.5 

.177 

7.25 

141.53 

18. 

1374 6 

31.5 

5569 

50 

17678 

77 

52027 

.75 

.485 

7.5 

154.05 

18.5 

1472.1 

32. 

5793 

51 

18575 

78 

.53732 

i. 

1 . 

7.75 

167.21 

19. 

1573.6 

32.5 

6022 

52 

19499 

79 

55471 

1.25 

1.747 

8 . 

181.02 

19.5 

1679.1 

33. 

6256 

53 

20450 

80 

57243 

1.5 

2.756 

8.25 

195.50 

20 . 

1788.9 

33.5 

6496 

54 

21428 

81 

59049 

1.75 

4.051 

8.5 

210.64 

20.5 

1902.8 

34. 

6741 

55 

22434 

82 

60888 

2 . 

5.657 

8.75 

226.48 

21 . 

2020.9 

34.5 

6991 

58 

23468 

83 

62762 

2.25 

7.594 

9. 

243. 

21.5 

2143.4 

35. 

7247 

57 

24529 

84 

64669 

‘2.5 

9.882 

9.25 

260.23 

22 . 

2270.2 

35.5 

7509 

58 

25620 

85 

66611 

2.75 

12.541 

9.5 

278.17 

22.5 

2401.4 

36. 

7776 

59 

26738 

86 

68588 

3. 

15.588 

9.75 

296.83 

23. 

2537. 

36.5 

8049 

60 

27886 

87 

70599 

3.25 

19.042 

10 . 

316.23 

23.5 

2677.1 

37. 

8327 

61 

29062 

88 

72646 

35 

22 918 

10.5 

357.2 

24. 

2821.8 

37.5 

8611 

62 

30268 

89 

74727 

3.75 

27 232 

11 . 

401.3 

24.5 

2971.1 

38. 

8901 

63 

31503 

90 

76843 

4. 

32. 

11.5 

448.5 

25. 

3125. 

38.5 

9197 

64 

32768 

91 

78996 

4.25 

37.24 

12 . 

498.8 

25.5 

3283.6 

39. 

9498 

65 

34063 

92 

81184 

4.5 

42.96 

12.5 

552.4 

26. 

3446.9 

39.5 

9806 

66 

35388 

93 

83408 

4.75 

49.17 

13. 

609.3 

26.5 

3615.1 

40. 

10119 

67 

36744 

94 

85668 

5. 

55.90 

13.5 

669.6 

27. 

3788. 

41. 

10764 

68 

38131 

95 

87965 

5.25 

63.15 

14. 

733.4 

27.5 

3965.8 

42. 

11432 

69 

39518 

96 

90298 

5.5 

70.94 

14.5 

800.6 

28. 

4148.5 

43. 

12125 

70 

40996 

97 

92668 

5.75 

79.28 

15. 

871.4 

28.5 

4336.2 

44. 

12842 

71 

42476 

98 

95075 

6 . 

88.18 

15.5 

945.9 

29. 

4528.9 

45. 

13584 

72 

43988 

99 

97519 

6.25 

97.66 

16. 

1024. 

29.5 

4726.7 

46. 

14351 

73 

45531 

100 

100000 

6.5 

107.72 

16.5 

1105.9 

30. 

4929.5 

47. 

15144 

74 

47106 



6.75 i 

118.38 

17. 

1191.6 

30.5 | 

5138. 

48. 

15963 

75 

48714 




TABLE 6%. Numbers, in inches. Square roots of fifth powers, in feet.. 



Sq. Rt. of 
5th Pow. 


Sq. Rt. of 
5th Pow. 


Sq. Rt. of 
5th Pow. 


Sq. Rt. of 
5th Pow. 


Sq. Rt. of 
5th Pow. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

H 

.00006 

Wi, 

.0547 

12 . 

1.000 

22X 

4.813 

42 

22.92 

X 

.00017 

4. 

.0641 

X 

1.108 

23 

5.086 

43 

24.31 

X 

.00035 

X 

.0731 

13. 

1.221 

X 

5.365 

44 

25.74 

% 

.00062 

X 

.0827 

X 

1.342 

24 

5.657 

45 

27.23 

X 

.00098 

X 

.0971 

14. 

1.470 

25 

6.264 

46 

28.77 

X 

.00144 

5. 

.1120 


1.605 

26 

6.909 

47 

30.36 

1. 

.0020 

X 

.1271 

15. 

1.747 

27 

7.593 

48 

32.00 

X 

.0027 

X 

.1428 

X 

1.896 

28 

8.316 

49 

33.69 

X 

.0035 

X 

.1590 

16. 

2.053 

29 

9.079 

50 

35.44 

y» 

.0044 

6 . 

.1768 

X 

2.217 

30 

9.882 

51 

37.25 

x 

.0055 

X 

.2160 

17. 

2.389 

31 

10.73 

52 

39.13 

% 

.0067 

7. 

.2599 

X 

2.567 

32 

11.61 

53 

41.02 

X 

.0081 


.3088 

18. 

2.756 

3.3 

12.54 

54 

42.96 

X 

.0096 

8 . 

.3628 

X 

2.950 

34 

13.51 

55 

44.97 

2 . 

.0113 

X 

.4228 

19. 

3.155 

35 

14.53 

56 

47.05 

X 

.0152 

9. 

.4871 

X 

3.365 

36 

15.59 

57 

49.17 

X 

.0198 

X 

.5577 

20 . 

3.586 

37 

16.69 

58 

51.35 

X 

.0252 

10 . 

.6339 


3.813 

38 

17.84 

59 

53.60 

3. 

.0312 

X 

.7162 

21 . 

4.051 

39 

19.04 

60 

55.90 

X 

.0383 

11 . 

.8043 

X 

4.297 

40 

20.29 

61 

58.27 

X 

1 

.0459 

X 

.8990 

22 . 

4.551 

41 

21.58 


































































254 


HYDRAULICS 




Art. 4 a. To find the discharge through a compound pipe 

v n, Fig 1 H, composed of any number of pipes, a, b, e, z , of different diuill- 
eters, which decrease from the reservoir toward the outflow o. 


a 


--v- 


- T 

H 


Fiff.lII 


First find what part of the total head H is employed in forcing the water through 
the last jripe z alone, thus : 

Let L a, L b, he, and L z, be the lengths in ft of the pipes a, b, c, and z, respectively; 
D a, D b, D c, and D z their diameters in ft; and A a, A b, A c, and A z the areas of 
their cross sections in sq ft. Then 

The head in \ 
ft employed f 

in forcing the ( __ The total head H in feet _ 

walkthrough T - f A z2 x D z /La+(5*XD«) , L6+(54XD&) , Lc+(5*XDc)\n 
the last P>pe ^ 1+ * \ A a* X D a + ^Tb^lTb + A c2 X l)c 


z alone " J i_L zq-(54XL> z) Ua’XDa A 62 x D 6 A c« X l)c 

However many divisions the pipe may have, proceed in the same way as above, using f 


Area 2 X Liam 


Length + 54 Diams 


.-- for the last or narrowest division; and 

i o m a 


Length + 54 Diams 


Area 2 X Diam 


for each of the others. 

Then, by the formulae, Art 2, find the velocity in ft per second, and discharge in 
cub ft per second, of the last pipe z, using its actual diameter, length, and cross £ 
sectional area, and the head just found. Said discharge is evidently the discharge 
for the compound pipe. 

For the velocity in any portion, as b, say 

area of cross section , area of cross . , velocity . velocity in the 

of the given portion • section of z • • in z • given portion. 

For the above rule and formula, we are indebted to Mr. Howard Murphy, C E, 
of Phila; and for the opportunity of testing it experimentally, to Messrs Morris! 
Tasker & Co, Limited, Pascal Iron Works, Philadelphia. 





























HYDRAULICS, 


255 


Art. 5. On the resistance which curved hen<ls oppose to the 
flow of water through rouml pipes. Well-rouuded bonds of large rad, 

whether ven or hor, produce but little resistance; except so g. 

far as the first ma_v cause accumulations of sediment, or of 
air. According to Weisbach, the rules of DuBuat, Navier, 
and other authorities, are erroneous; aud he gives the follow¬ 
ing one for ascertaining the additional head reqd to overcome 
the resistance produced in a circular pipe, b.v a bend formed 
by an arc of a circle : Knowing the rad re of the pipe, (or. in 
other words, half its diam,) iu feet; the rad r«, of the axis 
moot the bend, in feet; the central augle r s o iu degrees ; 

(which is equal to the augle d bx, or c b a,) aud the reqd vel 
of the water in the pipe, in ft per sec. 

Rule. Div the central angler so in deg, by 180. 

Call the quot a. Next, square the reqd vel. Div this sq by 
the constant number 64.4. Call the quot 6. Div the inner rad 
re of the pipe, in ft, by the rad rs of the axis rno of the 
bend, in ft. Call the quot c. Take from the following Table 
7. the number in column d, which corresponds to c; (unless 

c be less than .1, in which case always take .13 as d.) Finally, mult together this number d. the 
quot a, and the quot 6. The prod will be the reqd head in feet; which must either be added to the 
head previously calculated for the straight pipe, if the original vel is required to be maintained ; or 
must be subtracted from it, in case the head does not admit of increase, and a new calculation made 
to ascertain the diminished vel under the head thus reduced. If there is more than oiie bend of the 
same dimensions, an equal alteration of head must be made for each ; or, if they are of diff radii, 
and with diff central angles r so. a separate calculation must be made for each. Rennie's experi¬ 
ments, at the end of this Art, seem to prove that this is by no means the case. So far as the writer 
is aware, we have no reliable data for calculating the effects of a succession of bends. 

In shape of a formula. Weishach's rule stands thus: R being 

the rad of the axis of the bend, and r the rad of the pipe; 



Additional 
head in feet 


ft\ ! square ot 
= .131 + 1.847 2 x in ft per 

vR in ft/ * - 


square of vel central angle 
sec x in degrees 

’ " 180. 


The expression ^ means the sq root of the 7th power. When the rad of the bend exceeds 5 diams 
of the pipe, then 1.847 becomes inappreciable in practice, and may be omitted from the for¬ 


mula, When the pipe is square, instead of circular, the formula becomes 

vel2 
v 


Additional head =.124+ 3.104 X 


central angle 

6474 X 180] 


TABLE 7. 


c. 

d. 

c. 

d. 

c. 

d. 

c. 

d. 

c. 

d. 

.1 

.131 

.325 

.17 

.5 

.29 

.675 

.60 

.85 

1.18 

.15 

.135 

.35 

.18 

.525 

.32 

.7 

.66 

.875 

1.29 

.2 

.138 

.375 

.195 

.55 

.35 

.725 

.73 

.9 

1.41 

.225 

.145 

.4 

.206 

.575 

.39 

.75 

.80 

.925 

1.54 

.25 

.15 

.425 

.225 

.6 

.44 

.775 

.88 

.95 

1.68 

.275 

.155 

.45 

.24 

.625 

.49 

.8 

.98 

.975 

1.83 

.3 

.16 

.475 

.264 

.65 

.54 

.825 

1.08 

1 . 

2. 


Ex. A straight pipe 1 mile long, and 18 ins diam, with a total head of 20 ft, will disch water with 
a vel of 4 ft per sec; but it has been found necessary to introduce a circular bend of 90°, with a rad 
sr. Fig 2, of 5 feet. What addition mu-t be made to the 20 ft head, to compensate for the additional 
resistance caused by the bend ; so that the reqd vel of 4 ft per sec may still be maintained? 

Here, 90° + 180 = ,5 = a. Next, the square of the reqd vel in ft per sec, is 4 X 4 = 16. Ajid 16 + 
61.4 = .2484 = 6. The rad r e of the pipe (.75 ft), div by the rad r s of the bend (5 ft), = .75 + 5= 15 
= c ; and opposite this .15 in the column c oT the foregoing table, we find d — .135. Finally, 
a X b X d = 5X -2484 X .135 = .0168 ft, or about one-fifth inch only, the additional head reqd. See 
next table. No. 8. 

I>u Boat’s rule for the additional head required to over- 
come the resistance of circular bends in water pipes. Having 

diam of pipe, in ft; rad of bend, in ft, central angle w s o. Fig 2; and vel in ft per sec. Div e r, Fig 
2, or half the diam of the pipe, by the rad s tv of the outer side of the bend. The quot will be the 
nat versed sine of Du Buat'3 angle of reflexion. Take this versed sine rrom unity, or 
1. The Rem will be the nat cosine of the same angle. From the Table of Nat Sin and Tang, take 
both the angle and the nat sine corresponding to this nat cosine. Call the angle R. Also, square the 
nat sine, aud call this square S. Take the angle w s o from 180°. Div the rem by twice the angle It 
of reflexion just fouud. Call the quot T. Finally, mult together the constant dec .00375, the square 
of the vel in ft per sec, the quot T, and the square S. The prod will be the reqd extra head in feet. 











































256 


HYDRAULICS. 


Ex. The same as the foregoing one for VTeisbach’s rule ; that is, a pipe of 18 ins, or 1.5 ft diam; 
rad s to of outer side of bend, 5 75 ft; vel 4 ft per sec. What extra head will the bend require, ia 
order that this vel may not be diminished? 


.75 


Here, ~ .13043 = nat versed sine of angle of reflexion. 
5.7o 


And 1 — .13043 = .86957 = nat cos of 


same angle. In the Table of Nat bin. &c, we find, opposite the nat cos .86957, the angle R — 

29° 35'; and its nat sine .4937. The square of .4937 - .2437 = S. Again, 180° — 90° = 90°. And 
90° 5400 min 

-:= — — -—:— = 1.52. or T. Finally, the square of the vel is 16; heuce, we have .00375 X 
ofi- 5 10 3oo0 nun 

16 X 1 52 X .2437 = .0222 ft. the reqd extra head; or about ^ of an inch. Weisbach's rule pave 
.0168 ft, or about one-fifth of an inch. Hence we see that the resistance produced by well-rounded 


bends is not great. 



Fig. 3. 


When the rad r s, Fig; 2, of the bend, is less 
than about two diains of the pipe, which w ill rarely hap¬ 
pen, the resistance to the flow of the water iucreases very rapidly; while, on 
the other hand, by' Weisbach’s rule, as we understand it, uu advantage ap¬ 
pears to be gained by using a rad greater than 5 diauis of the pipe.* Employ* 
ing Weisbach's formula, the writer has drawn up the following table of heads 
reqd to overcome the resistance of one bend of 90°, for diif vets in ft per sec; 
and for any diam whatever. This table exteuds from a rad of 5 diauis down 
to one of diam; which is the smallest possible, inasmuch as it leads to a 
bend like Fig 3. 

A vel of 12 ft per sec is equal to 8.18 miles per hour; one which will rarely 
occur, iuasmuch as it requires a head of about 330 feet per mile. 


TABLE 8. Heads required to overcome the resistance in 
circular bends of 90°. Original. 


Rad = 5 diams of 
the pipe. 

1 ft. 

.001 

.004 

.009 

Veloc 
4 ft. 

HE 

.016 

ity in 
5 ft. 

ADS 

.025 

feet per Se 
6 ft. j 7 ft. 

IN FEET. 

.036 | .050 

cond. 

8 ft. | 

.065 

9 ft. | 

.082 

10 ft. 

.101 

12 ft. 

.145 

Rad = 3 diams... 

.001 

.004 

.010 

.017 

.027 

.038 

.052 

.069 

.086 

.106 

.153 

Rad = 2 diams... 

.001 

.005 

.011 

.019 

.029 

.012 

.057 

.074 

.094 

.116 

.167 

Rad= 1% diams... 

.001 

.005 

.012 

.021 

.033 

.048 

.066 

.086 

.108 

.134 

.192 

Rad = 1% diam... 

.002 

.007 

.015 

.026 

.041 

.059 

.080 

.104 

.132 

.163 

.235 

Rad = l diam... 

.002 

.009 

.020 

.036 

.056 

.081 

.110 

.144 

.182 

.225 

.324 

Rad ~ % diam... 

.005 

.018 

.041 

.072 

.113 

.162 

.221 

.288 

.365 

.450 

.649 

Rad — % diam... 

.016 

.062 

.140 

.248 

.388 

.559 

.761 

.994 

1.26 

1.55 

2.24 


i ^ Ce Ilio a /i “ft 1 ®* s °> ^‘8’ should be either greater, or 
less than 90 , then the heads given in the table, must be increased, or dimin¬ 
ished directly in the same proportion. 


Experiments by Rennie, with a pipe 15 ft long; and % inch bore ; with 
4 ft head, gave the following disch in cub ft per sec : 


Straight.00699 cub ft. 

15 semicircular beads.00617 


One bend at right angles near end.00556 

24 bends at right angles.00253 


The mean of many careful experiments tried at Eiveri>ooI. 
England, with a leaden pipe, 75 ft long, % inch bore, under 8 ft head gave the 
following number of secs to discharge one gallon of water: * 


75 ft pipe, straight and horizontal.. 81.56 sec. 
2 hor bends near discharge end ... 83.33 
2 hor bends near supply end.81.80 

The rad of the bends is not stated. 


4 vertical bends near discharge end 85.00 sec, 
4 vertical bends near supply end... 84.00 “ 


Os 


s» 


See Minutes Trans Inst Civ Eng, vol 12, page 501. 

On knees, or angular bends in water 

\e ^ f 4 * . The bends in lines of water pipes 

. mmm U snonld always form circular arcs; because knees create a much 

greater resistance. According to Weisbach, the head in ft reqd to 
overcome the additional resistance caused by a knee in a round 
f pipe is as follows : 

Additional __ Sq of vel in ft per sec 

rig. 4. head in ft ■ X constant in following 

table opposite the angle of deflection, a e o, or d e f.t Fig 4 . 



•Notwithstanding this, we advise to use as many more than 5 as can conveniently be done 
angle of del*; X Malian'the 4th power V thTsam/stae^^ ^ square of nat sine of half the 























































































































HYDRAULICS. 


257 


TABLE 9. 


Ang. of 


Ang.of 


1 

Ang. of 


Def. 

Constant 

Def. 

Constant. 

Def. 

Constant. 

in Pegs. 


in Deg. 


in Degs. 


140° 

2.431 

70° 

.533 

25° 

.049 

130 

2.158 

60 

.364 

20 

.030 

120 

1.861 

50 

.234 

15 

.016 

110 

1.556 

40 

.139 

10 

.007 

100 

1.260 

35 

.102 

5 

.002 

00 

.984 

30 

.073 



80 

.740 






Constants intermediate of those in the table may be obtained near enough by simple proportion. 



n a m 


Fig-. 5. 


The disch is diminished by swellings, or enlargements in pipes, 

as well as by contractions, bends, and knees; ou account of the eddies which they produce, &c. 

Art. 6. Inasmuch as the pres of quiet water against, and perp to, any given 
surf, is (other things being equal) iu proportion to the vert height of the water above the cen ot grav 
of the pressed surf, (see Art 1, Hydrostatics,) it follows that in two pipes of the same dianis, as a b, 
and c h, Fig 5, the pres agaiust, and at right 
angles to, the equal bases, mn of the vert pipe, 
and op of the inclined one, are equal; because 
the vert heights, a b and h g, of the water above 
the cen of grav a and c. of the equal bases, are 
equal in the two pipes. If the base of the inclined 
pipe be cut so that ty becomes the base, then the 
base is no longer a circle, but an ellipse; the 
area of which will always be greater than that 
of the circular one; and since the vert height ft g 
remains unchanged, the pres against the base t y, 
and perp to it, will be greater than that against 
o p, in the same proportion as the two areas. 

The upright pipe may be but 1 ft long ; and the 
inclined one 1 mile, or 10 miles long, still the pres 
at the base mn will be the same as that at the 

base o v. so long as the vert height a 6 is equal to .... _, 

the vert height gh. The greater weight of the water in c ft. does not increase the pres at its lower end 
o n ■ said weight beiug sustained by the under part p w of the inclined pipe. 

therefore, two steam pumps, with plungers of equal diameter were 
employed • one to force the water up the one foot long vertical pipe, and the 
other to force it up the ten miles of inclined pipe; botli engines would have to 
exert the same force to balance their respective columns of water ; i e, to uphold 
them. In other words, the static pressure of the water is the same in both cases, 
beiug equal to the weight of a cylindrical column of water of the same diameter 
as the plunger* and as long as the vertical stretch, a box gh, of the pipes..... . 

But in order to move the water in either pipe at a given velocity, an additional 
force is required, equal to the weight of a cylindrical column of water, the 
diameter of which is equal to that of the plunger, and the length of which is 
equal to the total head (calculated by Art 3, p 248) required to force water at 
the given velocitv througli a pipe of the given dimensions. Ihe weight of this 
second column is’the pressure, or motive force, necessary to give the required veloc¬ 
ity to tiSe water, to overcome its friction in the pipe, and to put water into the 
nine at its lower end as fast as it passes out at the upper end. See Art 1 a, p 237. 
P The total force or total steam pressure required in the cylinder, is the sum ot 
these two pressures ; namely, of the static pressure and the motive force. 

In the foregoing we have assumed that the resistance to entry is only such as 
would be encountered by water when forced by any means from a reservoir into 
n.e men end of an ordinary pipe ; so that the “entry head” may be taken as 
the open °, V( ,i oc itv head In practice, a much greater resistance to 

2S* s SSSfti sudden bewls iu pipes leading to and from . r 

chambers etc. Still, in most eases the entry head, even as thus increased, is 
but trifling iu comparison with the total head. 

aw 7 The flow of water throngh openings, or apertures, 

tolf - rt - ” 

Because the plunger uow rorrns practically the bottom, nm or op, of the pipe. See Item 2 
p 223. 














































258 


HYDRAULICS 


64.4, and take the sq rt of the prod. In practice, we may use 8 and 64, as near enough. The theo¬ 
retical, as well as the actual disch, or the quantity in cub ft, which Hows out per sec, is evidently 
equal in all cases to the prod of the theoretical, or of the actual vel, (as the case may be,) in ft per 
sec, mult by the area of the opeuing in sq ft. 

These theoretical laws apply equally to all fluids, whatever may be their spgrav ; thus, theoretically, 
mercury, water, air, &c, will all flow with equal vels from opeuiugs of equal sizes, under equal heads. 

Practically, however, only the mean vel, aud the disch through the VClia COIltraCta, or 
Contracted vein, (see Fig 11,) which forms itself just outside of certain kinds of openings, 
(and which is smaller thau the opeuiugs themselves,) are actually very nearly equal to the theoret¬ 
ical ones; but through the very opening itself they are usually less. The discrepancy is greater in 
some cases than in others; depending chiefly on the shape of the opening. 

On this account, the theoretical vel and disch found by the foregoing rule, must usually be dimin¬ 
ished by mult them by certain decimal numbers corresponding to the various kinds of openings: aud 
called coefficients of discharge. These coeffs have in many cases been determined by experiment 
very approximately; and will be found in the following articles. It will be seen in Remark 5, p260, 
that by the use of a peculiarly formed adjutage, or attachment, to small openings, the actual disch 
may even be increased beyond the theoretical one. 

The following table will save the trouble of calculating the theoretical vel, previously to mult it by 
the corresponding coeif of disch, for obtaining the actual vel. The coeffs for diff kinds of openings 
will be found further on. 

TABLE 10. Of the theoretical velocities in feet per sec, 

with which water should flow out into the air. under diff heads, through opeuiugs in the bottom or 
sides of the containing reservoir; the surf level of which remains constantly at the same height. 

Weisbacb says (see third footnote to Art 9) that when water flows out of an opening under water, 
as at »», Fig 1, the vel and disch are about ^ part less than when it flows into the open air, under 
equal heads. When the disch is made under water, the vert dist a u. Fig 1. between the surf levels 
of the two reservoirs, must be taken as the head. These theoretical vels are very nearly the actual 
mean ones at the contracted vein; see Art 9. Calling the head, H, then 

per^sec ^ 2 9 B ~ \/ 64-4 B ~ 8.03 times the sq rt of the head in ft. 


Theoretical head _ 2 _ (square of thcoret vel\ x 01 -- 

in feet 2 g 64.4 ^ in ft per sec J 


Head 

Vel. 

Head 

Vel. 

Head 

Vel. 

Head 

Vel. 

Head 

Vel. 

Head 

Vel 

Head 

Vel. 

Feet. 

Ft per 

Feet. 

Ft per 

Feet. ; 

F t per 

Feet. 

Ft per 

Feet. 

Ft per 

Feet. 

Ft per 

Feet. 

Ft per 


sec. 


sec. 


sec. 


sec. 

sec. 

sec. 

sec. 

.005 

.57 

.29 

4.32 

.77 

7.04 

1.50 

9.83 

7. 

21.2 

28 

4*2.5 

76 

69.9 

.010 

.80 

.30 

4.39 

.78 

7.09 

1.52 

9.90 

.2 

21.5 

29 

43.2 

77 

70.4 

.015 

.98 

.31 

4.47 

.79 

7.13 

1.54 

9.96 

.4 

21.8 

30 

43.9 

78 

70.9 

.020 

1.13 

.32 

4.54 

.80 

7.18 

1.56 

10.0 

.6 

22.1 

31 

44.7 

79 

71.3 

.025 

1.27 

.33 

4.61 

.81 

7.22 

1.58 

10.1 

.8 

22.4 

32 

45.4 

80 

71.8 

.030 

1.39 

.34 

4.68 

.82 

7 26 

1.60 

10.2 

8. 

22.7 

33 

46.1 

81 

72.2 

.035 

1.50 

.35 

4.75 

.83 

7.31 

1.65 

10.3 

.2 

23.0 

34 

46.7 

82 

72.6 

.040 

1.60 

.36 

4.81 

.84 

7.35 

1.70 

10 5 

.4 

23.3 

35 

47.4 

83 

73.1 

.045 

1.70 

.37 

4.87 

.85 

7.40 

1.75 

10.6 

.6 

23.5 

36 

48.1 

84 

73.5 

.050 

1.79 

.38 

4.94 

.86 

7.44 

1.80 

10.8 

.8 

23.8 

37 

48.8 

85 

74.0 

.055 

1.88 

.39 

5.01 

.87 

7.48 

1.85 

10.9 

9. 

24.1 

38 

49.5 

86 

74.4 

.060 

1.97 

.40 

5.07 

.88 

7.53 

1.90 

11.1 

.2 

24.3 

39 

50.1 

87 

74.8 

.065 

2.04 

.41 

5.14 

.89 

7.57 

1.95 

11.2 

.4 

21.6 

40 

50.7 

88 

75.3 

.070 

2.12 

.42 

5.20 

.90 

7.61 

2. 

11.4 

.6 

21 8 

41 

51.3 

89 

75.7 

.075 

2.20 

.43 

5.26 

.91 

7.65 

2.1 

11.7 

.8 

25.1 

42 

52.0 

90 

76.1 

.oso 

2.27 

.44 

5.32 

.92 

7.70 

2.2 

11.9 

10. 

25.4 

43 

52.6 

91 

76.5 

.085 

2.34 

.45 

5 38 

.93 

7.74 

2.3 

12.2 

.5 

26.0 

44 

53.2 

92 

76.9 

.090 

2.41 

.46 

5.44 

.94 

7.78 

2.4 

12.4 

ii. 

26.6 

45 

53.8 

93 

77.4 

.095 

2.47 

.47 

5 50 

.95 

7.82 

2.5 

12.6 

.5 

27.2 

46 

54.4 

94 

77.8 

.100 

2.54 

.48 

5.56 

.96 

7.86 

2.6 

12.9 

12. 

27.8 

47 

55.0 

95 

78.2 

.105 

2.60 

.49 

5.62 

.97 

7.90 

2.7 

13.2 

.5 

28.4 

48 

55.6 

96 

78.6 

.110 

2.66 

.50 

5.67 

.98 

7.94 

2.8 

13.4 

13. 

28.9 

49 

56.2 

97 

79.0 

.115 

2.72 

.51 

5.73 

.99 

7.98 

2.9 

13.7 

.5 

29.5 

50 

56.7 

98 

79.4 

.120 

2.78 

.52 

5.79 

1 Ft. 

8.03 

3. 

13.9 

14. 

30.0 

51 

57.3 

99 

79 8 

.125 

2.84 

.53 

5.85 

1.02 

8.10 

3.1 

It.l 

.5 

30.5 

52 

57.8 

100 

80.3 

.130 

2.89 

.54 

5.90 

1.04 

8.18 

3.2 

14.3 

15. 

31.1 

53 

58.4 

125 

89.7 

.135 

2.95 

.55 

5.95 

1 06 

8.26 

3.3 

14.5 

.5 

31.6 

54 

59.0 

150 

98.3 

.140 

3.00 

.56 

6.00 

1.08 

8.34 

3.4 

14.8 

16. 

32.1 

55 

59.5 

175 

106 

.145 

3.05 

.57 

6 06 

1.10 

8.41 

3.5 

15. 

.5 

32.6 

56 

60.0 

200 

114 

.150 

3.11 

.58 

6.11 

1.12 

8.49 

3.6 

15.2 

17. 

33.1 

57 

60.6 

225 

120 

.155 

3.16 

.49 

6.17 

I 14 

8.57 

3.7 

15.4 

.5 

33.6 

58 

61.1 

250 

126 

.160 

3.21 

.60 

6.22 

1.16 

8.61 

3.8 

15.6 

18. 

34.0 

59 

61.6 

275 

133 

.165 

3.26 

.61 

6.28 

1.18 

8.72 

3.9 

15.8 

.5 

34.5 

60 

62.1 

300 

139 

.170 

3.31 

62 

6.32 

1 20 

8.79 

4. 

16.0 

19. 

35.0 

61 

62.7 

350 

150 

.175 

3.36 

.63 

6 37 

1.22 

8.87 

.2 

16.4 

.5 

35.4 

62 

63.2 

400 

160 

.180 

3.40 

.64 

6.42 

1.24 

8.94 

.4 

16.8 

20. 

35.9 

63 

63.7 

450 

170 

.185 

3.45 

.65 

6.47 

1.26 

9.01 

.6 

17.2 

.5 

36.3 

64 

61.2 

500 

179 

.190 

3.50 

.66 

6 52 

1.28 

9.08 

.8 

17.6 

21. 

36.8 

65 

64.7 

550 

188 

.195 

3.55 

.67 

6 57 

1.30 

9.15 

5. 

17.9 

.5 

37.2 

66 

65.2 

600 

197 

.200 

3.59 

.68 

6.61 

1.32 

9.21 

.2 

18.3 

22. 

37.6 

67 

65.7 

100 

212 

.21 

3.68 

.69 

6.66 

1.34 

9.29 

.4 

18.7 

.5 

38.1 

68 

66.2 

800 

227 

.22 

3.76 

.70 

6.71 

1.36 

9.36 

.6 

19. 

23. 

38.5 

69 

66.7 

900 

241 

.23 

3.85 

.71 

6.76 

1.38 

9.43 

.8 

193 

.5 

38.9 

70 

67.1 

1000 

254 

.24 

3.93 

.72 

6.81 

1.40 

9.49 

6. 

19.7 

24. 

39.3 

71 

67.6 


.23 

4.01 

.73 

6.86 

1.42 

9.57 

.2 

20.0 

.5 

39.7 

72 

68.1 



.26 

4.09 

.74 

6.91 

1.44 

9.63 

.4 

20.3 

25 

40.1 

73 

68.5 



.27 

4.17 

.75 

6.95 

1.46 

9.70 

.6 

20.6 

26 

40.9 

74 

69.0 



.28 

4.25 

.76 

6.99 

1.48 

9.77 

.8 

20.9 

27 

41.7 

75 

69.5 


















































HYDRAULICS. 


259 


Art. 8. On the flow of water 
through vertical opening fur¬ 
nished with short tubes. When water 

“T fro ® a reservoir, Pig 6, through a vert partition 
tn ma a, the thickuess a m of which is about 2K or 3 times 
the least transverse dimension of the opening', (whether 
that dimension be its breadth, or its height;) or when if 
the partition be very thin, as n n, the water flows through 
>a tube, as at (, the length of which is about 2 or 3 times its 
least trausverse dimension, then the effluent stream will 
eutirelv nil the opening, or the tube, as shown in Fig 6 • o r 
in technical lauguage, will run with a full flow; or a ’full 
feore • and will disch more water in a given time, than if 
the tube were either materially longer or shorter. For if 
longer than 3 times the least trausverse dimension, the 
flow will be impeded by the increased friction against the 
sides of the tube ; and if shorter than about twice the least 



-~ vu W , rtuu II snurier man about twice the least 

trausverse dlui ® nsl0 “; th ® w f ater n UI ® ot flow in a ful > stream, but in a contracted one, as shown by 
g ' Tbls W1 1 be the case whether the tube be circular, or rectilinear, in its cross-section. 

a PP ro xin*atcly the actual vel. and disch Into the 
* ,1 . ro “S , » a tube, or opening, either circular or recti¬ 
linear in its outline, or cross-section ; and whose length c i, 
or c e 9 in the direction of the flow, is about 2]4 or 3 times its 
least transverse dimension ; w hen the surface-level, #, Fisr 6. 

,,le sa,,,e height; and which height 
must not be below the upper edge of the tube, or opening. 

•o R m';h 0 1 ; o .T t ake , OUt lhC theoret j cal ' -el from Table 10, corresponding to the head measured vert 

, 1,1 Vht ~ ™ 0 r e ,P r oper]y, the cen of grav) c, of the opening, to the level water surf*. Mult 
it by the coeff of disch .81. The prod will be the reqd vel, in ft per sec. Mult this actual vel bv the 
; ESiT 86 area the °* )eD , i ,"f’ ft - Ir Circular, knowing its diam. this area will be found in 
iacne 3. The prod will be the quantity of water dischd, in cub ft per sec; within, probably, 3 

or « per c0d t. 

1 veffn ft 2 ‘er F sec d ^ B<1 Ft ° f the h ° ad in ft- M “ U thiS 8q rt by 6 ' 5 ’ The prod wil1 be the actual 

Ex. An opening c o ; or box-shaped tube c t, Fig 6, is 3 feet wide, by .25 of a ft high ; and its length 
in the direction c i or c e in which the water flows is about .62 of a ft, or about 2% times its least 
transverse dimension, or its height. The head from the cen of grav c, of the openiug, to the constant 
surt-level $, is 4 feet. \\ hat will be the vel of the water; and how much will be dischd per sec? 
a The theoretical vel (Table 10, ) corresponding to a head of 4 ft* is 16 ft per sec. 

Ana lb X .81 — 12.96 ft per sec, the actual vel reqd. Again, the transverse area of the opening, or of 
the tube, is 3 ft X -25 ft — .75 sq ft. And .75 X 12.96 = 9.72 cub ft,* the quantity’ dischd per sec. 

• fe 8C l rfc I s ^* 2 X 6.5 = 13 ft per sec. the reqd vel, as before; the very slight 

ain being owing to the omission of small decimals in the coeffs. 

Rem. 1. If the short tube t projects partly inside of the vert 

partition n n y the disch will be diminished about Vg part. In that case, use .71 

or 7 instead of the .81 of Rule 1 ; or 5.7 instead of the 6.5 of Rule 2. 

Rem. 2. W hen the thickness am of the vert partition m m a a; or the length c e of the tube t , Fig 
o. is increased to about 4 times the least transverse dimension of the opening ; or of the diam, when 
-'•nlar: then the additional friction against its sides begins appreciably to lessen the vel and disch 

so nr fnr ** ♦ 111 opooIo- ..- inn j: __‘ 1 _ i. _ ’ j _ 


V mar. men tne annitional rriction against its sides begins appreciably to lessen the vel and disch. 
R case, or for still greater lengths, up to 100 diams, they may be found approximately, by using 

j instead of the coefF of disch .81 in Rule 1, the following coeffs, by which to mult the theoretical vela 
of Table 10. ri r non Dnlo r» *>12 


Or use Rule, p 243. 


TABLE 11. 


t: 


Length of 
Pipe 

in Diams. 

Coeff. 

Length of 
Pipe 

in Diams. 

Coeff. 

4 

.80 

40 

.62 

6... 

... .76 

50... 

... .60 

10 

.74 

60 

.57 

15.. 

... .71 

70... 

... .55 

20 

.69 

80 

.52 

25... 

... .67 

90... 

... .50 

30 

.65 

100 

.48 


Rem. 3. When the length of the opening or tube, in the direction in which the water flows, becomes 
If* than about twice its least transverse dimension, the disch is diminished ; so that for lengths from 
times, down to openings in a very thin plate, we may use .61, instead of the .81 of Rulel. For 
*uch openings, however, see Arts 9 and 10. 

Rem. 4. But on the other hand, the disch through such short openings and tubes as are shown in 
Fig 6. may he increased to nearly the theoretical ones of Table 10, by merely rounding off neatly the 
edges of the entrance end or mouth, as in Fig 7; which is the shape, and half actual si*e of one with 
which Weisbach obtained .975 of the theoretical vel and discharge, when the head was 10 ft; and ,958 































260 


HYDRAULICS 






with a head of one foot; so that in similar cases, .975, and .958 may be used instead of the eoeff .81 
in Rule 1. 






As much as .92 to .94 mat be obtained by widening the opening, m n, toward its outer mouth, o s. 
Fig, 8, making the divergence, or angle a. about 5°; or by widening it toward its inner mouth, as at 
t c, Fig 9; but increasing the angle of divergence, at b, to from 11° to 18°. In all cases, we consider 
the. small end as being the opening whose area must be multiplied by the vel togeu.be discharge. 


In some experiments made with large pyramidal wooden 

troughs 9.5 ft long, with an inner mouth of 3.2 X 2.4 ft, and a discharging one 


troughs 

of .62 X -44 ft; and under a head of 9X feet, the discharge was .98 of the theoretical one, due to the » 
smaller end. Therefore, .98 mar be used in such cases, instead of the .81 of Rule l. 

Rem. 5. The discharge through a short opening of small 
transverse section may even be made 50 per cent greater 
than the theoretical one, by adopting the shape, Fig 10; where m n is sup -1 

posed to be the diameter of the opening 


DC -«i!£U:::E 



The best proportions appear to be about as 
follows: oy —9 inches; m n~ 1 inch ; b 
■=. 1.8 inch ; os~ X inch ; a d — 2 ins; the 
curves, am, and d n, being quadrants; 
the angle, x, of divergence, about 5° 6'; 
and the tube of polished metal. In this 
case use 1.55, or more safely, 1.5, instead 
of the .81 of Rule 1. The only experi¬ 
ments with this form have been on a very 
small scale. To what extent it may be 


applicable is unknown. 

So far as regards the ordinary operations of the engineer, this subject is perhaps more curious than 
useful; for he will rarely have any difficulty in making his openings large enough, without resorting to 
such aids; except, perhaps, that of rouudiug off the iuuer edges, as in Fig 7 : which is usually done. 

Art. 9. tin the disch of water through openings In thin 
vert partitions, with plane or flat faces, ee, or nn, Fig 11 * If thoj 

face e e, or n n, instead of beiug plane, aud vert, should be curved, 
or inclining in diff directions toward the opeuing, then the disch 
will be altered. When water flows from a reservoir. Fig 11, through 
a vert plaue plate or partition nn, which is not thicker than about i 
the least trausversedimension of the opening, whether thatdimension 
be its breadth, or its height o o; t or wheu, if the partition e e itself 
is much thicker, we give the opeuing the shape shown at b, (whicM 
evidently amounts to the same thing.) then the effluent stream will 
not pass out with a. full flow, as in Fig 6. but will assume the shape 
shown in Fig 11; formiug, just outside or the opening, what is 
called the vena contracta, or contracted vein. In order that this 
contraction may take place to its fullest extent, or become complete, 
the inner sharp edges of the opening must not approach either the 
surf of the water, or the bottom or sides of the '•eservoir, nearer! 
than about IX times the least transverse dimension of the opening. 
The contracted vejn occurs at a dist or about half the smallest di- 
mension of the orifice, from the orifice itself. In a circular orifice,! 
at about half the diam dist; and ordinarily its area is about .62. or nearly % that of the orifice itse' 



At this point the actual mean vel of the stream is very nearly (about .97)’the theoretical vel given u 
Table 10, and hence the actual dischs are but .62, or nearlv % of the theoretical ones. 

Case 1. 


lo find the actual disch into air.t through either a 
circular or rectilinear^ opening in a thin vert plane parti* 


•cl 


A 


Fig. 13. 


* We believe that these rules for thin plate are also sufficiently approximati 
for most practical purposes, if the opening be in the bottom of the reservoir 
or in an inclined, instead of a vert side. 

t When the side of a reservoir, or the edge of a plank, Ac. over which watei 
nows, has no greater thickness than this, the water is said to flow through 
or over, thin plate, or thin partition. 

7 Should the disch take place under water, as in Fig 12. both surf levels re 
maining constant, then the head to be used is the vert diff a o. of the tw 
levels. After making the calculation with this head, we should, according ft 
Weisbach, deduct the y 1 ^- part; inasmuch as he states that the disch is thai 
much less when under water, than when it fakes place freelv into the air 
Otherexperimenters, however, assert that it is precisely the same in bot* cases 
§ If the shape of the opening is oval, triangular,’ or irregular, the hea< 
must be measured vert front its cen of gray. 







































HYDRAULICS 


261 


JJ®": ^ hen th . e contraction is complete; and when the snrf- 
»!i* 1 *11 " H constantly at the same height; water being* 
* d to the r cscrvoir as fast as it runs out at the open- 

onenhiV'to ‘, h ? hC f d ’ “«K SUred Vert from the center (or rather from the cen ° r gray) c, of the 

feast t.l'n?vprLT - * <■ ft 0 reser . volr .- ,s 1,01 less than 1 ft. nor more thau 10 ft; and when the 

see due to the head rT a hl'e°m 0pC ?h g .v “ 0t ‘2? S . than au iach ' niult the theoretical vel in ft per 
sec aue to the head, (Table 10, ) by the coefficient of disch .62. The prod will be the actual 

DrodVin be Ihp ^ al er.through the opeuing. Mult this vel by the area of the opening in sq ft; th« 
prod will be the disch in cub ft per sec, approximately. ’ 

When the head is greater than 10 ft, use .6, instead of .62. 

Rltlb 2. Find the sq rt of the bead in ft. Mult this sq rt by 5; the prod will be the vel in ft per 
sec ; which inult by the area as before for the disch. v 

■ EX ‘ ?f haC Wil1 he the disch through an opening in complete contraction, whose dimensions are 6 
• feet? 1 ver * > aQ d -4 It hor ; the vert head above the cen of gray of the opening being constantly 


_ I*, Jhetheoretica 1 vel (Table 10, ) corresponding to 6 ft head, is 19.7 ft per sec. And 

,'oJ l ,: 214 ft ’ the reqd veI - A 8 aiu > tbe area of the opeuing = .5 X 4 r= 2 sq ft; and 12.214 X 
2 = 24.42a cub ft per sec; the disch. 

By Buie 2. The sq rt of 6 = 2.45; and 2.45 X 5 = 12.25 ft per sec, the reqd vel; and 12.25 X 2 = 
24.0 cub ft per sec, the disch. ~ 

Both very approx even if the orifice reaches to the surface of the issuing water. 

Rem. 1. The coef .62 is a mean of results of many old experimenters. 

In 1874 Genl. T. G. Ellis of Massachusetts conducted an elaborate series (Trans Am Soc C E, Feb 
1876) on a large scale, the general results of which, within less than 1 per ct, are given iu the follow¬ 
ing table. See also Rem 3. The sharp edged orifices were in iron plates .25 to .5 inch thick. 


Orifice. 

Head above Center. 

Coef. 

2 ft sq. 

2. to 3.5 ft. 

.60 to .61 

2 “ long, I ft high 

1.8 to 11.3 “ 

.60 to .61 

2 “ long, .5 high 

1.4 to 17.0 “ 

.61 to .60 

2 “ diam. 

1.8 to 9.6 “ 

.59 to .61 


Rem. 2. Extreme care is reqd to obtain correct results; but for many 

purposes of the engineer au error of 5 to 10 per ct is unimportant. 

It will rarely happen that greater accuracy is required than may be obtained by the foregoing 
rules; but when such does occur, aid may be derived from the following tabic deduced 
from the experiments of Lesbros and Poncelet. on opeuings 8 ins 

wide, of did heights, and with diff heads. Use that coeff iu the table which applies to the case, in¬ 
stead of the .62 of Rule 1. In some of the cases in this table, the upper edge of the opening is 
nearer the surf-level of the reservoir than 1 % times its least transverse dimension. 


TABLE 12. Coefficients for rectangular openings in thin 
vertical partitions in full contraction.* 


Head 
above cen. 
of grav. of 
opening 
in Feet. 

Head 
above cen. 
of grav. of 
opening 
in Inches. 

Ins. 

8 

The bre 

H 

Ins. 

6 

adth in all the ope 

EIGHT OF O: 

Ins. I Ins. 

4 | 3 

nings ~ 

PENIN 

Ins. 

2 

inches. 

G. 

Ins. 

1 

[ Ins. 

■ 4 

.033 

.4 







70 

.0666 

.8 







fiQ 

.0833 

1 






.64 

68 

.125 

1J4 





.61 

.64 

.68 

.1666 

2 




.60 

.62 

64 

68 

.‘2083 

214 



.59 

.61 

£2 

64 

67 

.250 

3 



.60 

.61 

.62 

64 

67 

.2917 

3h> 


.57 

.60 

.61 

.62 

.64 

^66 

.3333 

4 


.58 

.60 

.61 

.63 

.64 

.66 

.3750 

4M 

.56 

.59 

.60 

.61 

.63 

.64 

.66 

.4167 

5 

.57 

.59 

.61 

.62 

.63 

.64 

.66 

.6666 

8 

.59 

.60 

.61 

.62 

.63 

.64 

.65 

1 

12 

.60 

.60 

.61 

.62 

.63 

.63 

.64 

3 

36 

.60 

.60 

.61 

.62 

.62 

.63 

.63 

5 

60 

.60 

.60 

.61 

.61 

.62 

.62 

.62 

10 

120 

.60 

.60 

.60 

.60 

.60 

.61 

.61 


Rem. 3. Careful experiments on openings 4^ ft wide, and is 

ins high, under heads of from 6 to 15 ft, show that the coeff .62 will give results 
correct within Ay part, for openings of that size also, under large heads; although the thickness of 
Hie partition varied on its diff sides, from 12 to 20 ins. It must be recollected, however, that nothing 
more than close approximations are to be attained in such matters. 

Rem. 4. It has been asserted by some writers, that when two or more 
contiguous openings are discharging at the same time from the same reser¬ 
voir, they disch less in proportion than when only one of them is open. Other experiments, how¬ 
ever, seem to show that this is not the case; it is therefore probable, at least, that the diff, if any, 
is but trifling. See Art 1 N, p. 239. 


* See first footnote on preceding page. 









































262 


HYDRAULICS, 


m 


m 


Case 2. The discharge through thin vert partitions in com* 
plete contraction, when the stir lace-level, in, Fig 13 , descends 
as the water Hows out into the air. In this case, if the reservoir is 

prismatic, that is, if its hor sections are everywhere equal; aud if no water is (lowing into the reser¬ 
voir, to supply the plac • of that which Hows out, theu, to find the time reqd to discb the reservoir. 

Rule. Inasmuch as the time in which such a reservoir entirely discharges itself, is twice that in 
which the same quantity would How out uuder a constant head, as iu Case 1, therefore, cal¬ 

culate the disch iu cub ft per sec by Rule 1, Art SI: div the number of cub ft con¬ 
tained in the reservoir, above the level g of the bottom of the opening, Fig 13, by 
this disch; the quot will be the number of sec iu which a volume equal to that in 
the reservoir, to the depth g, would ruu out iu Case 1, of a constant head. Aud 
twice this number will be the seconds reqd to empty the reservoir iu Case 2, of a 
varying head. 

Rem. If it should be reqd to find the time in which such a prismatic reservoir 
would partly empty itself, as, for instance, from m to n. Fig 13, first calculate, by 
the above rule, the secs necessary to empty it if it had oulv been filled to n : aud 
afterward calculate as if it had been filled to to. The ditf betweeu trie two times 
will evidently be the time reqd to empty it from m to n. If the opening is not in 
complete contraction, see Arts 11, &c. 

If the disch is into a lower reservoir, whose 
suri-level remains constant, proceed in the same manner; 

only use the diff of level of the two surfs us the head, aud afterward (according 
to.Weisbach) iucrease the time part. 

Disch from a reservoir Et, Fig 14, the smrf-Ievel, #, 


- 9 - 


Fig. 13. 


Art. 10. 

of which remains constantly at the same height; through 
an opening, o, in thin vert partition; and in complete con¬ 
traction; but entirely under water; and into a prismatic 
reservoir, in. 



Seconds required 


... - . '** ./height a c 

to discharge a quantity — w in ft ^ 


C da, the 
constant. 


level c remaining = 


hor area of 
m in sq ft 


area of opening 


o in sq ft 


X -62 X $.03 


Seconds required _ 

to raise level iu m from c to a ~ 


^/ficightac v hor area of., „ 
in ft * m iu sq ft x 2 


"“.;;sr«" s x“x8.03 


Seconds required a h C t7- a r. d ) x 

to raise level in m from e to — \ DOtll IU it / 

Area of opening 


hor area of 


to raise level in m from c to — 
any other level, d. 


m in sq ft 


XJ 


X .62 X $.03 


Rem. 1. 

its bottom 


the first rule in Art 9, 


o in sq ft 

If it should he reqd to find the time of tilling m, from 
up 


to 


(l. we may do so very approximately by calculating by 

. , the time reqd from e to the center of the opening o, as if all that portion off* 

the disch took place into air; and afterward, from the center of the opening to d, by the rule just 
given. 1 his case is similar to that of filling a lock from the canal reach above, in which the surf- 
level may be considered constaut. 

Rem. 2. It the bottom of the opening o. should coincide with 
the bottom of the reservoir, then tbe coeff will become greater than .62. 
oee Art 11, for obtaining coetTs for imperfect contraction. 

Rem. 3. Df the opening, instead of being in complete eon- 

traction. is of any of the shapes Figs 6 to 9, then a reference to Art 8 will show 

what coeff must be substituted for .62. 

CUss 3. Disch from one prismatic reservoir. Fig 15, W, into 
another, X, of any comparative sizes whatever, through an 


opening o, in a plane thin vert partition, and in complete 
contraction; when the water rises iu X, while it falls in W. 



To find the time in which the water, flowing from W into X, through 
o, will fall through the diet as, so as to stand at the same level s c, in 
both reservoirs. 

In this case, the water reqd to fill X from « to d, (d being the bottom 
of the opening o,) flows out into the air; and the time necessary for it 
to do so, must be calculated separately from that reqd above d, which 
flows into water. 

Rule. First from e to d. Find the hor area of each reservoir, in 
sq ft. Mutt the hor area of X, by the vert depth de in ft, Tor the cob 
ft contained iu that portion. Div these cub ft by the hor area of W. 
The quot will be the dist am, in feet, through which the water in W 
must descend, in order to fill X to d. 


I 


Seconds re¬ 
quired to low¬ 
er from a to m , and 
raise from e to d. 


Twice the 
hor area 
W iu sq 


of v ( an _ s/ head m n \ 

ft V ,n ft in ft / 


Area of opening 


O iu sq ft X - 62 X 8-03 


Fig. 15. 













































HYDRAULICS. 


263 


Seconds required 

to lower from m to s, and raise 
from d to c. (Very approx) 


Hor area of y twice the hor area v . /head m n 
X tu sq ft * of W in sq rt x v in ft 

(i 


i sq 
Area of 
opening 
o in sq ft 


/hor area 
of W 
\in sq ft 


hor ares', 

+ of 

in 


r area-, 
of X ) 
sq ft/ 


X .62 X 8.03 


«** *- ■» 

m ' u ‘*«•«.* r~. -». sS uS “iiasrs? i 


Hence, 


4.X 100 X 60 X 2 


48000 


160 X 8.03 X 8 X .62 2389.73" 


m n — 4 ft; and the sum of the 2 areas = 100 + 60 = 160. 

20.1 sec; the additional time reqd, very approximately 

Note 1. If the opening, as d. Fig: 16, reaches 
to the very bottom of the reservoirs, we mav 

consider all the water flowing from K into T, as flowing into water. 

1 herefore, using the head am, we at once calculate the time necessary 
for the water iu the two reservoirs to arrive at the same level sc b» 
the last process of the preceding rule; or, in other words, by the pro- 
cess given iu the preceding example. But in this case it must be borne 
in mind that the opening o is no longer in complete contraction, inas. 
mucn as the contraction along its lower edge is suppressed. 

fhe disch will consequently be somewhat increased; and a coeff 
greater than .62 becomes necessary. The method of finding this is 
given in the following Case 4. A reference to Art 8 will give the coeff 
in case the opening is shaped as Figs 6 to 9. 

Art, 11. Cas«4 The discharge through openings in plane 
tions: (Hit in _ F e 



Fig - . 16. 


thin vert partitions; hut in incomplete co/.traction. 


The opening may be such that contraction will take place 
along one portiou of its perimeter, or at the top of the open¬ 
ing a, Fig 17 ; while it is suppressed on another portiou ; as 
at the bottom and two ends of the opening a ; where suppres¬ 
sion is caused by the addition of short side and bottom pieces 
c, c, c. Or it may be caused by the bottom, or ends, or botb 
coinciding wilh the bottom and sides of the reservoir. In 
such cases the disch will be greater than iu those of complete 
contraction; but less than iu those of full flow ; inasmuch as 
the opening now partakes somewhat of the character of the 
short tubes of Art 8; and the coeff will rise from .62, or that 
which usually pertains to openings in full coutraction ; and 
will approach .8, or that of full flow, in proportion to the ex¬ 
tent of perimeter along which contraction is suppressed ; or 
even to .9 or .98 by the use of such openings as are shown bv 
Figs 7, 8, 9. J 



Fig. 17. 


.- To 1 J”" <1 approximately a new coeff of disch; and tlie disch 
Itself, in cases ot incomplete contraction. 

Rule. First find by the foregoing rules, what would be the disch in the particular case that, mav 
be under cousideratmn, supposing the contraction to be complete. Then div that portion of the 
? e rTn eD,ng °“ contraction is suppressed, by the entire perimeter. Mult the quot 

by the dec .152 if the opening is rectangular, or by .128 if circular. To the prod add unity, or 1. Call 
« 9 v Tn h ?ho Say ’a aS unlt - v ’or 1.. is to 9, so is the coeff for complete contraction in ordinary cases 
coeff in y the place^f e^ F repeat the on S inal calculation, only substituting this new 

According to this rule, we have the following coeff of discharge Tor rectangular openings within pro- 
bahlv 3 or 4 per cent when contraction is not suppressed on more than % of the perimeter. The theo¬ 
retical discharge multiplied by the corresponding coeff will give the actual discharge. When the con¬ 
traction is carried farther, the coeff becomes extremely irregular, and is probably indeterminable. 

For complete contraction ( ordinarily ). g 2 

When contraction is suppressed on Ki the perimeter .. ca 

“ “ “ “ 14 “ •• e * 

“ “ “ “ h “ “ . 

“ “ “ “ entirely around the orifice . .80 

Intermediate ones can be estimated nearly enough, mentallv. 


Rem. 1. When, instead of a short spout, as in Fie- 17 the 
opening* is provided with an indefinitely Iona hor trough 

similarly attached and open at top, there will be no practically appreciable diminutibn of disch below 
that through the simple opening as at a, Fig 11; provided the head measured above the een of grav 
of the opening be at least as great as 2 or 2 l fi times the height of the opening itself. Therefore under 
such circumstances the disch may be calculated by the rules in Art 9. But with smaller heads the 
disch diminishes considerably ; so when the bead above the center becomes but as great as the height 
of the opening, it will be but about ^ of the calculated one. With still smaller heads, the flow 
becomes less much more rapidly; but has not been reduced to any rule. 

Rem. 2. If, instead of being hor, the trough is INCLINED 






































264 


HYDRAULICS, 


- 


as mncli as 1 in 10, the disch will be increased very slightly, (some 3 or 4 per 

cent) over that calculated by the rules in Art 9, for the plain opening. These results were obtained 
by experiments on a very small scale; and should be considered as mere approximations. 

Art. 12. In a case like Fig 18, where contraction is supposed 
to be suppressed at the bottom, and at both vert sides oi' the 

opening o, in consequence of their coinciding 
with the bottom and sides of the reservoir; but where the 
front of the reservoir, instead of being vert, is sloped as at/; 
and when the water, after leaving the opeuiug. flows away j 
over a slightly slopiug apron, g, then the disch in cub ft per ]» 
sec may be approximately found by Rule 1, Case l, Art 9, 
only substituting .8 in place of .62, when / slopes back 46°, 
or 1 to 1; or .74 when / slopes back 63°, or with a base of 1 ) 

to a rise of 2. In such cases of iuclined fronts, the height of 
the openiug must be measured vert, or rather at right angles 
to the floor of the reservoir; and not in a line with the 
sloping front. 

Rem. When the front, /*, of the reservoir is vert, and a sloping- 
apron or trough, </, is used, having its upper edge level with the bottom 

of the opening, the disch is not appreciably diminished below that which takes place freely into the 
air, provided the head above the cen of grav of the opening is not less than from 

18 to 24 ins, for an opening 6 to 9 ins high. 

12 to 16 “ “ •• “ 4 ins high. 

9 “ “ “ 2 ins or less, high. 

Art. 13. To find, approximately, the time reqd for the emp¬ 
tying of a pond, or any other reservoir, as Fig- 19, which is 
not of a prismatic shape; through an openiug, a, near the 
bottom. 




Rule. First ascertain the exact shape and dimensions 
of the reservoir. If large, and irregular, it must be care¬ 
fully surveyed; aud souudiugs taken, and figured upon a 
correct plau aud cross-sections. Next, consider the entire 
body of water to be divided into a series of thin hor strata, 
A, B, C, D ; the top line of the lower one being at least a 
few ins above the top of the opening n. It is not necessary 
that these strata should be of equai thickness; although 
the thinner they are, the more correct will the result he. 
The depth of the lower one, D, will vary to some extent 
with the height of the opeuiug ; those next above it should 


not exceed about a foot in thickness, until a depth of 6 or 8 feet is reached; then they may conve¬ 
niently, and with sufficient accuracy, be increased to about 2 ft, for 6 or 8 ft more ; and so on; be¬ 
coming thicker as they approach the surf. By aid of the drawings, calculate the content of each 
stratum in cub ft. Now, since the strata are thin, we may, without serious error, assume each of 
them to he prismatic, as shown by the dotted lines; and may assume that the head under which each 
stratum (except the lowest) empties itself through n, is equal to the vert height from the center of 
the opening to the center of the stratum. Thus, m n will be the head of A ; w n. the head of B ; in, 
the head of C. Theu, for the stratum A. by Rule 1, Art 9, (only using tom ns the head instead of on,) 
and instead of the coeff .62 of that rule (which can only be used if n is in complete contraction) using 
.64, or whatever other coeff near the end of Art 11 applies to the case, calculate the disch in cub 
per sec. Div the content of the stratum A by this disch, and the quot will be the number of sec reqd’ 
> for discharging A. Using the head w n, proceed in precisely the same way with the stratum B ; and 
using the head xn, do the same with C. Finally, for the lower stratum 1),’find by Rule 1, Art9, (with 
the same caution as before respecting the proper coefT.) in what time it would empty itself under a 
constant head equal to y n, measured from its surf to the center of the opening. Double this time will 
be that reqd to empty itself in the case hefore us, under its raryitig head. Finally, add together all 
these separate times; aud their sum will be the entire time reqd to empty the pond, or reservoir, ap¬ 
proximately enough for practical purposes. 


I' 


Art. 14. On the discharge of water over weirs, or overfalls. 

Experiments and observations on a grand scale, were made on this subject at Lowell, Mass, by 
i^Ir Janies B. 1 rancis, C E, one of the most accomplished hvdraulicians of the age. (See 
his “ Lowell Hydraulic Experiments.”) To apply the rule arrived at by Mr Francis, the following 
conditions must exist. 

The crest a, Fig 20, or top of the weir over which the water discharges, must be a hor. sharp cor¬ 
ner, in thin plate, or thin partition. (See p 260; first footnote.) Its inner side must form a vertical 
straight line, a h, witli the inner face of the dam, to a depth a b, not less than twice the depth, or 
head a m. measured vert from a to the level o m of the hor portion of the water surf; and not to c. the 
curved surr of the falling sheet of water. The head a m mnv vnrv from 6 to 24 ins in height. The 
ends ah, ah, of the weir, Figs 21 and 22, must be vert; and its length, a, a, not less than 3 times the 
head a to. 

These conditions being observed, we may distinguish 2 cases; nnmelv. Case 1st. that in which, ns 
in Fig 21, the weir extends entirely acroSs the reservoir; so that its ends ah. ah, coincide with, or 
form portions of, the sides s, s. of the reservoir: in which case contraction takes place only along the 
upper edge a, a, of the weir, Fig 21, as shown at a in Fig 20; but is suppressed entirely at the ends 
so that the water flows out as shown in plan by Fig 23. And Case 2d. in which, as in Fig 22, the 
vert ends ah, ah, as well as the crest a, a. are formed with a sharp corner in thin plate; and 'are 
moreover, removed from the sides v. it, of the reservoir, a dist equal at least to the head am : no that 
contraction takes place at the ends of the weir, as well as along its crest; and less water flows out 
as shown in plan at a, a, Fig 24. 


If 


I 


! 
















HYDRAULICS. 


265 


(V A 






Fw20 

o 


B<22. 

O 






11 


! 


Ed 23 

%3 

To find the discharge over a weir in thin plate. 

Case 1. When there is no contraction at either end of the 
Weir, Figs 21 and 23 : 

Discharge __ 3 33 x the length aaof v . / cube of the head 

in cab ft per sec the overfall, in ft A / V a to in ft. 

Ex. How many cab ft per sec will flow over such a weir in thin plate, 200 ft long; having a head. 
am, of 1.5 ft, measured to the level surf o to, of the reservoir ; and with no contraction at either end ? 

Here, the cube of l.o — 3-375. And the sq rt of 3.375 — 1.837. And 3.33 X 200 X 1 837 — 1223 4 
cub ft per sec, the reqd disch. 

This rule will also he very approximate even when there 
is contraction at both ends, provided the length of the weir 
is at least 10 times as great as the head am; and provided the head 

is not less than 2 or 3 ins in depth. Iudeed, it will be within about 6 per ceut of the truth for weirs 

with contraction at the ends, and whose lengths are but 4 times the head ; and for the many cases in 
which no closer approximation is reqd, the disch may be taken at once from table 13. 

«*?ea. When contraction takes place at both ends, as in Figs 
2-and 24, or at one end only, use the same formula as in Case 1; except that 
when there is contraction at both ends, one-fifth of the head a m is to be taken from the length a a 
before using it as a multiplier ; and when there is contraction at one end only, one-tenth of a to must 
first be taken from the length a a. 

EX ' . How many cub ft per sec will flow over such a weir in thin plate, 200 ft long; having a head 
a m, of l.o ft; with contraction at both ends ? 

, H . e f.^ ,lle 3.375. And the sq rt of 3.375 — 1.837. Again, one-fifth of a m = .3 of a ft. 

And 200 — .3-199.7. Therefore we have 3.33 X 199.7 X 1.837 = 1221.6 cub ft per sec, the reqd disch ; 
or (in this cas e\ practically the same as when there is no contraction. 

Rkm. If, instead of 3.33, we use 3.41. the two foregoing rules will apply to heads a to, as small as 
>5 an inch; and coeffs between 3.33 and 3.41 may be used for heads between about 5 inches and « 
an inch, where more than common accuracy is aimed at. We may also use 3.3 instead of 3 33 for 
“ads greater than 2 feet. ’ 

-Eytelwein’s formula for weirs, over a thin edge, and having no contrac¬ 
tion at either end, is identical with the above. His coefficient is 3.4. ' 

From the Lowell experiments, by Mr Francis, it appears that 

when the depth, a to, is 1 foot, and the entire sheet of w ater, after passing over the weir strikes a 
" or floor placed only about 6 iDS below the crest a of the weir, the disch is thereby diminished 
but about the one-thousandth part; and that when the head a m is about 10 ins, and falls into water 
of considerable depth, no diff whatever is perceptible in the di.-ch, w hether the surf of that water be 
about 4, or about 13 ins below the crest a; and that a fall below the crest a, equal to one-half of the 
head « to, is quite sufficient. 

Art. 14 A. If the water in the reservoir, or In the feeding 
canal, instead of being stagnant, has a slight current toward 
w ®* p s the disch will be but very little increased thereby when the head am 
is several ins. Mr rrancis observed that a current of 1 foot per sec, or nearly .7 of a mile per hour, 

, reused the disch but about 2 per cent, when the head was 13 ins ; and one of 6 ins per sec, about 
per ct, when the head was 8 ins. Whenever the effect of the current, how'ever, is so great as to re- 
q ire notice, proceed as follows: First find the approx disch as directed in Art 14, under Case 1 or 
Case 2, according to circumstances. Then, in Case 2. find the area in sq ft of 
t™f. S » ectlon °f channel of approach at o b’, Fig 20. measured between its two sides, and from bottom 
. ~ r ' ° ,^ a ttns area length of the weir in ft. corrected for contraction as directed 

ju Case 2 above. Call the quot D. In ( ase 1, I) is = the depth oh'. In either 

case, divide the head a to or H. in ft, by D. Find this last quot In the column in table 13 A, p 266. 
Mult the approx disch by the corresponding coeff in column K. 

C 7r» m, /°n V, i 0f apr, :? ch ' Rnd U ‘ r tah,e IS A, we are indebted to Messrs A. 
I, ' n a f d f Har ‘. Oiv Engs of bowel), Mass. They are, in fact, simply a convenient 

Fins - T°K 1 a £ p yiDg ? ,r * f sands’ rule* for similar cases, which is as follows: 
thii a the ,h ,‘i 0r T, t C '?.i heHd * corresponding to the observed vel of approach. Add 

thla head to the head a to, or H, Fig 20. Then, in the formula, Art 14, instead of 

V cube of head am, use p' cubeof CM- It)\ — p'^ube of h. 
















































266 


HYDRAULICS. 


TABLE 13. Of actual discharges in cub ft per sec, for ear 
font ill lenath of weir in thin plate: and without contractio 

firt .‘Ke'^Sd; «*>. « rt -,“" Ue ?r.l?e Vc« 

the head am. Very approximate also, whe.1 there is eo 

traction at both ends, provided the length be at least j 
times the head. And but about 6 per cent in excess ot tl 
truth, if the length be but 4 times the head. (Original.) 

The decimal .01 of a foot, is precisely .12 of an inch; or scant % inch._ 


Head, 
am, in 
Feet. 

Cub. ft. 
per 

Second. 

Head, 
am, in 
Feet. 

Cub. ft. 
per 

Second. 

Head, 
am, in 
Feet. 

Cub. ft. 
per 

Second. 

Head, 
am, in 
Feet. 

Cub. ft. 
per 

Second. 

Head, 
am, in 
Feet. 

Cub. ft. 
per 

Second. 

03 

0 1 7 

.22 

.351 

.58 

1.47 

.94 

3.04 

2.4 

12.2 

04 

027 

.24 

.401 

.60 

1.54 

.96 

3.14 

2.5 

13.0 


.038 

.26 

.452 

.62 

1.62 

.98 

3.24 

2.6 

13.8 

06 

.050 

.28 

.505 

.64 

1.70 

1. 

3.:t3 

2.7 

14.6 

07 

.063 

.30 

.560 

.66 

1.78 

1.1 

3.85 

2.8 

15.4 

08 

077 

.32 

.603 

.68 

1.86 

1.2 

4.38 

2.9 

16.2 

.09 

.092 

.34 

.659 

.70 

1.95 

1.3 

4.94 

3. 

17.1 

.10 

.108 

.36 

.719 

.72 

2.03 

1.4 

5.51 

3.1 

18.0 

.11 

.124 

.38 

.780 

.74 

2.12 

1.6 

6.11 

3.2 

18.9 

.12 

.142 

.40 

.842 

.76 

2.21 

1.6 

6.73 

3.3 

19.8 

.13 

.160 

42 

.907 

.78 

2.30 

1.7 

7.37 

3.4 

20.7 

.14 

.178 

.44 

.972 

.80 

2 38 

1.8 

8.04 

3.5 

21.6 

.15 

.198 

.46 

1.04 

.82 

2.47 

1.9 

8.72 

3.6 

22.5 

.16 

.218 

.48 

1.1 L 

.84 

2.56 

2. 

9.42 

3.7 

23.5 

.17 

.239 

.50 

1.18 

.86 

2.65 

2.1 

10.0 

3.8 

24.4 

.18 

.260 

.52 

1.25 

.88 

2.74 

2.2 

10.8 

3.9 

25.4 

.19 

.282 

.54 

1.32 

.90 

2.84 

2.3 

11.6 

4. 

26.4 

.2 

.305 

.50 

1.40 

.92 

2.94 




1 f 


In calculating this table, the coeff 3.41 was used for heads from .03 ft to .3 rt; then 3.33 to 
then 3.3 to the end. 


TABLE 13 A. 


Coefficients K for velocity of approach. 

the use of this table, see Art 14 A, p‘265. 


JH 

D 

K 

H 

D 

K 

H 

D 

.01 

1.0000 

.09 

1.0020 

.17 

.0-2 

1.0001 

.10 

1.0025 

.18 

.03 

1.0002 

.11 

1.0030 

.19 

.01 

1.0004 

.12 

1.0036 

.20 

.05 

1 0006 

.13. 

1.0042 

.21 

.06 

1.0009 

.14 

1.0049 

.22 

.07 

1.0012 

.15 

1.0056 

.23 

.08 

1.0016 

| .16 

1 1.0064 



1.0072 

1.0081 

1.0090 

1.0100 

1.0110 

1.0121 

1.0132 


H_ 

D 


.24 

.25 

.26 

.27 

.28 

.29 

.30 


K 

H 

D 

K 

1.0143 

.31 

1.0239 

1.0155 

.32 

1.0254 

1.0168 

.33 

1 0271 

1.0181 

.34 

1.0287 

1.0195 

.35 

1.0305 

1.0209 

.36 

1.0322 

1.0224 

.37 

1.0341 


'llie following empirical rule by the author gives the discha 
over a weir nearly enough for ordinary purposes; and probably quite as cl< 
as possible without actual measurement in each case that presents itself. 


Having 

the length of the weir in feet, a a , Figs. 21 and 22, p. 265 ; 
the head in feet, measured to the level surface in the reservoir, a m, Fig. 5 

the theoretical velocity in feet per second ( = v/2 g h ) due to that head ] l 
table, p. 258; and 


that coefficient from the following table which agrees most nearly wit! 
case: then 


Discharge, in cubic feet per second = length X head X velocity X coeffi 


Example. How much water will be discharged over a weir 60 fp.*t long; the 
of which is level, smooth and 3 feet wide, or thick; and over which the head, 
Fig. 20. is 8 inches, or .6666 feet thick? Here the theoretical velocity for a h< 
8 inches, (Table 10, p. 258), is 6.55 feet per second. The coefficient for a weir \ 
crest is level, and 3 feet wide, with a head of 8 inches, is by the following tabl 
Consequently, 60 X -6666 X 6.55 X -31 = 81.21 cubic feet per second; the req, 
discharge, approximately. 




































































HYDRAULICS 


267 


n 


TABLE 14. Of coefficients of approximate discharge over 
weirs of different thicknesses, varying from a sharp edge to 
5 feet. —(Original.) 


Head 
a m 

in Feet. 

Head 
a m 

in Inches. 

Sharp 

Edge. 

2 Inches 
Thick. 

3 Ft Thick; 

smooth; 
sloping out¬ 
ward: and 
downward, 
from 1 in 12 
to 1 in 18. 

3 Ft Thick ; 
smooth, 
and level. 

.0833 

1 

.41 

.37 

.32 

.27 

.1666 

2 

.40 

.38 

.34 

.30 

.25 

3 

.40 

.39 

.34 

.31 

.3333 

4 

.40 

.41 

.35 

.31 

.4166 

5 

.40 

.41 

.35 

.32 

.5 

6 

.39 

.41 

.35 

.33 

.5833 

7 

.39 

.41 

.35 

.32 

.6666 

8 

.39 

.41 

.34 

.31 

.8333 

10 

.38 

.40 

.34 

.31 

1 . 

12 

.38 

.40 

.33 

.31 

2. 

24 

.37 

.39 

.32 

.30 

3. 

36 

.37 

.39 

.32 

.30 























267 a 


HYDRAULICS 



M. Bazin* experimented with weirs shaped as in Fig 24 
A, from about 1}/ to &y 2 feet long and from about 9 inches to 
4^ feet high. After comparing his experiments with others 
by Messrs. Fteley and Stearns,! he gives the following values 
for the coefficient m in the formula: 


Q = m 1 hp/2 g h, 


or 


Dischargee, in cubic feet per second, = m X length in 
X head in feet X 1/2 g X head in feet. 


feet 


Fig 24 A 
measurements 
in metres 

Table 15. 


M. Bazin gives these values as varying not more than 1 per 
cent from the truth, provided 

(1) that there is no lateral contraction ; 

(2) that the air nas free access to the back of the sheet of 

water falling over the weir. 

(3) that the weir is shaped as in Fig 24 A. 


Note. 


head, a m, Pig 20. 


metres. 


.05 

.06 

.07 

.08 

.09 


.10 

.12 

.14 

.16 

.18 


.20 

.22 

.24 

.26 

.28 


approximate 


feet. inches. 


.164 

.197 

.2H0 

.262 

.295 


.328 

.394 

.459 

.525 

.591 


.30 

.32 

.34 

.36 

.38 


.40 

.42 

.44 

.46 

.48 


.50 

.52 

.54 

.56 

.58 

.60 


.656 

.722 

.787 

.853 

.919 


.984 
1.050 
1.116 
1.181 
1.247 


1.312 

1.378 

1.444 

1.509 

1.575 


height, a b, Fig 20, of crest of weir above bed of up-stream channel. 


9 ( feet 


0.80 1.00 1.50 2.00 


metres 0.20 0.30 0.40 0.50 0.60 

0.656 0.984 1.312 1.640 1.969 2.624 3.280 4.920 6.560 j. 2 

inches 7.87 11.81 15.75 19.69 23.62 31.50 39.38 59.07 78.76 ’"o 


1.97 

2.36 

2.76 

3.15 

3/54 


Ill 

.458 


111 

.453 
.456 .450 

.455 .448 

.447 


.456 

.457 


111 

.451 

.447 

.445 

.443 

.442 


Ill 111 

.450 .449 

.445 


.445 
.443 .442 


.441 
.440 .438 


3.94 

4.72 

5.51 

6.30 

7.09 


.459 

.462 

.466 

.471 

.475 


.448 

.450 

.453 

.456 


.443 

.444 

.445 


.439 

.438 


.437 


.439 


7.87 

8.66 

9.45 

10.24 

11.02 


11.81 

12.60 

13.39 

14.17 

14.96 


1.640 

1.706 

1.772 

1.837 

1.903 

1.969 


15.75 

16.54 

17.32 

18.21 

18.90 


19.69 

20.47 

21.26 

22.05 

22.83 

23.62 


.480 

.484 

.488 

.492 

.496 


.459 

.462 

.465 

.468 

.472 


.447 

.449 

.452 

.455 

.457 


.440 

.442 

.444 

.446 

.448 


.436 

.437 

.438 

.440 

.441 


.500 


.475 

.478 

.481 

.483 

.486 


.460 

.462 

.464 

.467 

.469 


.450 

.452 

.454 

.456 

.458 


.443 

.444 

.446 

.448 

.449 


.489 

.491 

.494 

.496 


.472 

.474 

.476 

.478 

.480 


.459 

.461 

.463 

.465 

.467 


.451 

.452 

.454 

.456 

.457 


.482 

.483 

.485 

.487 

.489 

.490 


.468 

.470 

.472 

.473 

.475 

.476 


.459 

.460 

.461 

.463 

.464 

.466 


Ill 

Ill 

Ill 

Ill 

.449 

.449 

.448 

.448 

.444 

.443 

.443 

.443 

.441 

.*40 

.440 

.439 

.438 

.438 

.437 

.437 

.436 

.436 

.435 

.434 

.435 

.434 

.433 

.433 

.433 

.432 

.430 

.430 

.432 

.430 

.428 

.428 

.431 

.429 

.427 

.426 

44 

.428 

.426 

.425 

<4 

<1 

.425 

.423 

1 4 

II 

.424 

44 

.432 

II 

44 

.422 

* 4 

.429 

41 

44 

.433 

44 

44 

44 

.434 

.430 

It 

.421 

.436 

«< 

Cl 

44 

.437 

.431 

it 

4 4 

.438 

.432 

li 

44 

.439 

44 

It 

44 

.440 

.433 

41 

41 

.441 

.434 

.425 

44 

.442 

.435 

4 4 

44 

.443 

1* 

it 

44 

.444 

.436 

44 

44 

.445 

.437 

.426 

It 

.446 

.438 

14 

II 

.447 

It 

44 

II 

.448 

.439 

.427 

it 

.449 

.440 

44 

II 

.451 

.441 

44 

IS 


Ill 


.4203 


.4187 

.4181 


.4174 

.4168 

.4162 

.4156 

.4150 


.4144 

.4139 

.4134 

.4128 

.4122 


.4118 

.4112 

.4107 

.4101 

.4096 

.4092 


For heads from 4 inches to 1 foot, M. Bazin gives, 
the formula: 6 


as sufficiently approximate, 

111 = 0.425 + 0.21 

where p is the height aft of the weir, Fig 20. P 


1888!' EXp6lienCeS nouveIles sur I’&oulement en dfivenwir." Annales des Pontes et Chafes! Oct. 
t Transactions, American Society of Civil Engineers, Vol XII, 1533, 




-The coefficient in, for any given case, remains the same 

for English, metric or other measure, provided the length and the head are meas¬ 
ured in the same unit, and the discharge in the cube of that unit. 


I 


1 







































































HYDRAULICS. 


267 6 



Article 15. If tl»e inner face of tl»e weir and dam, instead of 
betas vertical, as a b, Fig. 30, is sloped, as a or b, Fig. 25; the contrac¬ 
tion on the crest will be diminished; and consequently the 
discharge will be increased. This will also be the case if the 
inner corner or edge of the crest be rounded off, instead of 
being left sharp; or if the sides of the reservoir eonve-ge 
more or less as they approach the weir, so as to form wings 
tor guiding the water more directly to it; or if a b, Fig. 20, 
be h ss than twice a m. Indeed, so many modifying circum¬ 
stances exist to embarrass experiments on this and similar 
subjects, that some of those which have been made with great 
care are rendered inapplicable as other than tolerable approximations, in conse¬ 
quence of the neglect to tike into consideration some local peculiarity, which was 
not at the time regarded as exertiug au appreciable effect. Unless, therefore, cir¬ 
cumstances admit of our combining all the conditions mentioned in the first part of 
Article 14. p. 264, thereby securing very approximate results, we must either resort to 
an actual measurement of the discharge in a vessel of known capacity; or else be 
contented with rules which may lead to errors of 5, 10, or more per cent in propor¬ 
tion as we deviate from these conditions. Frequently even 10 per cent, of error may 
be of little real importance. 

Remark 1. When the water, after passing over 
a weir, Fig. 26, instead of falling freely into the 
air, is carried away by a slightly incliued apron or 
trough, T, the floor of which coincides with the 
crest” a, of the weir, then the discharge is not ap¬ 
preciably diminished thereby when the head a m, 
is 15 inches or more. But if the head a m is but 
1 fnni then the calculated discharge must be , , 

reduced about one-tenth; if 6 inches, two-tenths; if 2% inches, three-tenths; and 
if 1 inch, five-tenths, or one-half, as approximations. 

Remark 2. Professor Thomson, of Dublin proposed the use of triangular notches, 
or weirs, for measuring the discharge; inasmuch 
as then the periphery always bears the s me 
ratio to the area of the stream flowing over it; 
which is not the case with any other form. 

Experimenting with a right-angled triangular 

lb. bottom of the notch, ,0 <k 
level surface of the quiet water , he found discharge in cubic feet per second 
_ .0051 X \/ fifth power of head in inches. (For such roots, see tables, p. 253.) 


_ zA 



1 

% 

i 

' ' ' 'i ~ ' 'f/SsJ*.’— 




— 


Fiq 36 


90° 
































268 


HYDRAULICS. 


OX THE FLOW OF WATER IX OPEX CHAXXEES. 

Art. 16. The mean velocity of flow is an imaginary uniform one, 
which, if given to the water at every point in the cross section, would give the 
same discharge that the actual uuunilorm one does. Or 

. ... volume of discharge 

mean velocity =- - -- 

area of cross section 

In channels of uniform cross section, the maximum velocity is found 
about midway between the two banks, and generally at some dist below the sur¬ 
face. This dist varies in diff streams; but, as an average, it seems to be about 
one third of the total depth. Where the total depth is great in proportion to 
the width, (say £ the width or more), the max vel has been found as deep as 
midway between surf and bottom; while in small shallow streams it appears to 
approach the surf to within from .1 to .2 of the total depth. Many experiments 
upon shallow' streams have indeed indicated that the max vel was at the surf. 

The ratio between the velocities in different parts of the 
cross section varies greatly in diff streams; so that but little dependence 
can be placed upon rules for obtaining one from the other. With the same surf 
vel, wide and deep streams have greater mean and bottom vels than small shal¬ 
low ones. In order to approximate roughly to the mean vel when 
the greatest surf vel is given, it is frequently assumed that the former is = $ 
(or .8) of the latter. But Mr. Francis found, in his experiments at Lowell, that 
surface floats of wax, 2 ins diam, floating down the center of a rectangular flume 
10 ft wide, and 8 ft deep, actually moved about 6 per cent slower than a tin tube 

2 ins diam, reaching from a few ins above the surf, down to within ins of the 
bottom of the flume; and loaded at bottom with lead, to insure its maintaining 
a nearly vert position. While the wax surf float moved at the rate of 8.73 ft per 
sec, the rate of the tube (which was evidently very nearly the same as that of 
the center vert thread of water) was 3.98 ft per sec. Also, that in the same flume, 
with vels of the center tube varying from 1.55 to 4 ft per sec, the vel of the tube 
was less than that of the mean vel of the entire cross section of water in the 
flume, about as .96 to 1, for the lesser vel; and .93 to 1 for the greater vel. 
While, in another rectangular flume 20 ft wide and 8 ft deep, with vels varying 
from 1.16 to 1.84 ft per sec, that of the tubes was greater than that of the entire 
mass of water, about as 1.04 to 1. In a flume 29 It wide, by 8.1 ft deep, with vels 
of about 3 ft per sec, it was as 1 to .9; and in a flume 36£ ft wide, by 8.4 ft deep, 
with vels of about 3£ ft per sec, as 1 to .97. 

Hilaries Ellet. Jr, C E, found in the Mississippi “at diff points 

on the river, in depths varying from 54 to 100 ft; ami in currents varying from 

3 to 7 miles an hour that the speed of a float supporting a line 50 ft long, is al¬ 
most always greater than that of the surf float alone.” The same results w ere 
obtained with lines 25 and 75 ft long; the excess of the speed of the line floats 
being about 2 per cent over that of the simple floats: and Mr. Ellet concludes, 
therefore, that the mean vel of the entire cross section of the Mississippi, instead 
of being less, is absolutely greater by about 2 per cent, than the mean surf've 1. 
He, however, employed .8 of the greatest surf vel as representing approximately, 
in his opinion, the mean vel of the entire cross section of water. In shallow 
streams, he always found the surf float to travel more rapidly than a line float. 

European trials of the mean vel of separate single verticals , in tolerably deep 
livers, have resulted in from .85 to .96 of the surf vel at each vertical. The mean 
of all may be taken at .9. 

Bottom velocity. In streams of nearly uniform slope and cross section, 
there is a great reduction of vel near the bottom. As a very rough approxima¬ 
tion, the deepest measurable vel, in streams of uniform slope etc, appears to be 
from h to | of the mean vel. But see rems on “scour ”, p 279/. 

Art. 17. To measure the surface velocity, select a place where 

the stream is for some dist (the longer the better) of tolerably uniform cross 
section; and free from counter-currents, slack water, eddies, rapids, etc. Ob¬ 
serve, by a seconds-watch,or pendulum, how long a time afloat (such as a small 
block ot wood) placed in the swiftest, part of the current, occupies in passing 
through some previously measured dist. From 50 feet fur slow streams, to 150 ft 
for rapid ones, will answer very well. This dist in ft, or ins, div by the entire 
number of seconds reqd by the float to traverse it, will give the greatest surf vel 
in ft or ins per sec. 

The surf vel should l»e ineasd in perfectly calm weather, 

so that the float may not be disturbed by wind ; and, for the same reason, the 
float should not project much above the water. The measurement should be 



HYDRAULICS. 


269 


repeated several times to insure accuracy. In very small streams, the banks 
and bed may be trimmed for a short dist,*so as to present a uniform channel¬ 
way. The float should be placed in the water a little dist above the point for 
commencing the observation ; so that it may acquire the full vel of the water, 
! * before reaching that point. 


Art. 18. To gauge a stream 
toy means of Its velocity. Select 
a place where the cross-section remains, for 
a short distance, tolerably uniform, and 
free from counter-currents, eddies, still 
water, or other irregularities. Prepare a 
careful cross-section, as Fig. 27. By means 
of poles, or buoys, n, n, divide the stream into sections, «, 6, c, Ac. Plant two range- 
poles, R, R, at the upper end, and two others at the lower end, of the distance 
through which the floats are to pass; for observing by a seconds watch, or a pendu¬ 
lum, the time which they occupy in the passage. Then measure the mean velocity of 
each section a, b, c, Ac., separately, and directly, by means of long floats, as F L, 
reaching to near the bottom: and projecting a little above the surface. The floats may 
be tin tubes, or wooden rods; weighted in either case, at the lower end, until they 
will float pearly vertical. They must be of different lengths to suit the depths of 
the different sections. For this purpose the float may be made in pieces, with screw- 
joints. The area of each separate section of the stn am in square feet, being multi¬ 
plied by the observed mean velocity of its water in feet per second, will give the 
discharge of that section in cubic feet per second. And the discharges of all the 
separate sections thus obtained, when added together, will give the total discharge 
of the stream. And this total discharge, divided by the entire area of cross-section 
of the stream in square feet, gives the mean velocity of all the water of the stream, 
in feet per second. 



Rem. If the channel is in common earth, especially if sandy, 
the loss bysoakage into the soil, and by evaporation, will frequently abstract so 
much water that the disch will gradually become less and less, the farther down 
stream it is measured. Long canal feeders l bus generally deliver into the canal 
but a small proportion of the water that enters their upper ends. 

The double float is used for ascertaining vels at diff depths. It consists 
of a float resting upon the surface of the water, and of a heavier body, or “lower 
float”, which is suspended from the upper float by means of a cord.* The depth 
of the lower float of course depends upon the length of the suspending cord 
(which may be increased or diminished at pleasure until the lower float is be- 
ieved to be at that depth for which the vel is wanted), and upon its straight¬ 
ness, which is more or less affected by the current. Owing to this latter circum¬ 
stance, it is difficult to know whether the lower float is really at the proper 
depth. Moreover it is uncertain to what extent the two floats and the string 
interfere with one another’s motions. In deep water the string may oppose a 
greater area to the current than the lower float itself does. It thus becomes 
doubtful to what extent the vel of the upper float can be relied upon as indicat¬ 
ing that of the water at the depth of the lower one. 

Art. 19. Castelli’s quadrant, or hydrometric pendulum, 

consisted of a metallic ball suspended by a thread from the center of a graduated 
arc. The instrument was placed in the current, with the arc parallel to the 
direction of flow; and the vel was then calculated from the angle formed be¬ 
tween the thread and a vert line. 

Gauthev’s pressure plate was a sheet of metal suspended by one of its 
ends, about which it waS left free to swing. The plate was immersed in the 
stream, with its face at right angles to the current. The vel was estimated by 
means of the weight required to make the plate hang vert in opposition to the 
force of the current. 

Pitot’s tube is a tube bent, at right angles like the letter L. One leg is 
held hor under the water, with its open end facing the current. The vel is measd 
by the height to which the water rises in the vert leg above the general surf. 

These instruments could not well be used except for points near the suriacej 
and they gave^pnly the vel at the time of observation. 










270 


HYDRAULICS. 


Art. 20. Wheel meters. For accurate current measurements, espe¬ 
cially in streams of irregular cross section, where long floats cannot well he 
used; wheel meters, similar in principle to Woltinann’s tachometer, are now 
largely employed. 

Such a meter consists of a wheel which is turned by the current, and which j 
communicates its motion, by means of its axle and gearing, to indices which j 
record the number of revolutions. The inst may be clamped to anv part of a J 
long pole reaching to the bottom of the stream, and thus may be used at any ( 
depth. The observer, by means of a wire, rod or string, reaching down to the , 
inst, throws the registering apparatus first into, and then out of, gear with the | 
wheel (applying a brake to the former at the instant it is thrown out of gear), i 
and carefully noting the times when he does so. The inst is then raised, the 
number of revolutions in the measd time is read off from the indices, and from 
it the vel is calculated. 

Various plans have been tried for so arranging such wheel 
meters that their revolutions may be registered above the 
surface at the time. This is now generally accomplished by electricity ; 
the wheel, at each rev, automatically breaking and re-establishing a galvanic 
current generated by a battery. The wire carrying this current is thus made 
to operate Morse telegraphic registering apparatus placed in a boat or on shore. 

A number of meters, so arranged, can be attached at diff points on the same I 

{ ole at the same time, and thus simultaneous observations of veloe- i 
ties at different depths may be made and registered. 

Wheel meters are made by Messrs. Buff & Berger, No 9 Province 
Court, Boston. The prices range approximately from $125 to $225 each. Most 
of their meters are so arranged that they can freely swing lior about the long 
vert pole to which they are clamped, and are provided each with a vane or tail 
similar to that of a wind-mill, for keeping the wheel in the proper position as 
regards the current. The wheels are generally made like those of a wind-mill; 
i e with blades set at such an angle as to present a sloping surface to the cur¬ 
rent; and with the axis of the wheel parallel to the direction of flow. The axis 
runs in agate bearings. When desired, the rim of the wheel is furnished with 
an air-chamber, which just counter-balances the weight of the wheel and thus 
removes journal friction due to it. Meters are made both with and without 
electrical registering apparatus. In the latter case the gearing and indices etc 
are sometimes enclosed in a glass case, to prevent them from becoming clogged 
by weeds, sediment etc. 

In all of the above methods, except those with floats which move along with 
the current, it is necessary, in order to calculate the vel from an observation to ; 
first rate the meter; i e, to ascertain what effects different vels produce , 
upon the inst. This is best done by moving the inst at a known vel through still I 
water, and noting the effect, produced. In this way a coefficient is obtained for ' ‘ 
each meter, which, when multiplied by the number of revs etc recorded in an 
given case, gives the vel for that case. v 









HYDRAULICS. 


271 


21. Iilitter's formula for the mean vel of water flowing in open 
channels ol uniform cross section and slope throughout. 

^ a **H*>M. The use of all such formulae is liable to error arising from the 
, j - ascertaining the exact condition of the stream as regards roughness 
of bed, surface slope,* etc. 

Rem. 1. Care must be taken that the bottom vel Is not so 
great as to wear away the soil. If there is any such danger artificial 
means must be applied to protect the channel-way; or it may be advisable to 
reduce the rate of fall, and increase the cross section of the channel; so as to 
secure the same disch, but with less vel. A liberal increase should also be made 
in the dimensions of such channels, to compensate for obstructions to the flow, 
aiisiug from the growth of aquatic plants, or deposits of mud from rain- 
vvashes, etc; or even lrom very strong winds blowing against the current. See 
also Rem, p 269. 

Rem. 2 Water running in a channel with a horizontal bed, 
or bottom, cannot have a uniform vel, or depth, through¬ 
out its course; because the action of gravity due to the inclined plane of a 
sloping bottom, is wanting in this case; and the water can flow only by forming 
its surjace into an inclined plane; which evidently involves a diminution of 
depth at every successive dist from the reservoir. 



Fig. 28. Fig. 29. Fig. 30. 


Theory of flow. It is generally held that the resistances to the flow of 
water in a pipe or channel are directly proportional to the area of the bed sur¬ 
face with which the water comes in contact ( i e, to the product of the “wetted 
perimeter” as abco Figs 28, 29, 30 mult by the length of the channel, or of the 
portion of it under consideration); and to the square of the vel of the flowing 
water; and, inasmuch as the resistance at any given point in the cross section 
appears to be inversely as the dist of that point from the bottom or sides, we 
conclude that the total resistances are inversely as the area of the cross section ; 
because the greater that area, the greater would be the mean dist of all the par¬ 
ticles from the bottom and sides. (The resistance is independent of the pressure. 
See p 374c.) 

In short, the resistances are assumed to be in proportion to 

vel 2 X wet perimeter X length v 2 pl 

-- - -— or —- 

area of cross section a 


and the head h" in feet or in metres etc, required to overcome those resist¬ 
ances, is 


resistance 

head 


a coefficient 
C 


X 


vel 2 X wet perimeter X length 
area of wet cross section 


or 



a 


from which we have 



and 


v — 



1 a A" 
Cp T 


or 


vel = 



area of wet 
cross section 

wet perimeter 


X 


resistce 

head 

length 


* “In measuring the slope of a large river, the ordinary errors of the most careful leveling are a 

large proportion of the whole fall; the variation of level in the cross section of the surface is often as 
great as the slope for ten miles or more ; the exact point where the level should be taken is often 
uncertain : the rise and fall of the water makes it extremely difficult to decide when the levels should 
he taken at the upper and lower points ; waves of translation may affect the inclination to a great 
and uncertain degree, and may even make the surface slope the reverse wav.” Genl T. G. Ellis, 
Trans Am Soc Civ Engrs, Aug 1877. 

























272 


HYDRAULICS. 


But 


area of wet cross section 


or 


a 


— is the “hydraulic radius” or 
P 


wet perimeter r 

“ mean depth ” or “ mean radius,” R, of the cross section ; 

. resistance head h" 

ana ------ or —— is the inclination or slope, S, (fre- 


length ( 

quently denoted by “I”) of the hydraulic grade line, p 240, or the sine of the 
angle tv so Fig 4 , p 240. In open channels, it is = the fall of the surface per 
unit of length. r 


We therefore have velocity = ^~r X 1 /mean radius X slope 
or, by using a coeff(c) = 


velocity = coefficient c X l/mean radius X slope 
or v — c i/7tS 


T he earlier hydraulicians gave (each according to the results of his investiga¬ 
tions) fixed values for the ooelf c, (generally about 95 to 100 for channels 
in earth or gravel, as in our early editions), making it, in other words a con¬ 
stant, and independent of the shape, size, slope and roughness of the channel 
But according to Messrs. E. Ganguillet and W. R. Kutter, eminent Swiss 
engineers, the coeff is affected by diffs in any of these particulars. 

mtUa C ”) r<iing l ° theIr fortuula (generally called, for convenience, “Kutter’s for- 


For English measure. 


For metric measure. 


ai a . - 00281 , 1.811 

41.6 + : —)- 


c = 


slope 


n 


1 + 


. , .00281 \ 


23 + + i 

_ slope n 


l/inean rad in feet 


1 + 




1 /mean rad in metres 


Tables giving values of c for diff grades, mean radii and decree* of 

roughness, and for English and metric measures, are given on pp 275 etc 

TT n S r. A L.L. „ _ „ A* _ • . j « _ 


No general formula is applicable to cases of <leei«le<l bentU in tiit. ^ 

S »“ a, r a s,reai "' orAarUea irrejjamHti^ In <1.2 c^hhZT 

ion. Such cases would require still higher coefficients n than thnfce V 

each *™* 8 an 1 Cana 1 1 f l but the y would have to be ascertained bv experimenter 
each ease, and would be useless for other cases. For such streams we t / 

of r the e disch. UPOn aCtUal measurements of the velocity, either direct or by means 


1 


j T 

I I 


C 


| ' 


I L 




























HYDRAULICS. 


273 


There is much room for the exercise of judgment in the selection of the 
proper coefficient n for any given case, even where the condition of the 
channel is well known. It may frequently be necessary to use values of n inter¬ 
mediate between those given ; for careless brickwork may be rougher than well 
finished rubble; side slopes in “ very firm gravel ” may have very diff degrees 
of roughness; etc etc. The engineer should make lists of values of n from his 
own experience, fully noting the peculiarities of each case, and calculating n 
from the tables, pp 275 etc, as directed. 

A given diff in the deg n of roughness exerts a much greater effect upon the 
coefficient c, and thus upon the velocity, in small channels than in larger ones. 
It is therefore especially necessary in small channels that care be exercised in 
finding (by experiment if necessary) the proper value of ra ; and, where a large 
diseh is desired, the sides of small channels should be made particularly smooth. 


Table of n, or coefficient of roughness. 

In anv given case the value of n is the same whether the mean 
radius is given in English, metric or any other measure. 


Artificial channels of uniform cross section. 

Sides and bottom of channel lined with n = 

well planed timber...-.009 

neat cement* (applies also to glazed pipes and very smooth iron pipes). .010 
plaster of 1 measure of sand to 3 of cement;* (or smooth iron pipes), .oil 

unplaned timber (applies also to ordinary iron pipes)..012 

ashlar or brickwork...013 

rubble.-.-.^ 


Channels subject to irregularity of cross section. 


Canals in very firm gravel-...y • 

Canals and rivers of tolerably uniform cross section, slope and direction, 
in moderately good order and regimen, and free from stones and 

weeds.*.••• 

having stones and w’eeds occasionally. 

in bad order and regimen, overgrown with vegetation, and strewn 
with stones and detritus.. 


.020 

.025 

.030 

.035 


\rt 22. The following tables give values of the coefficient 

c as obtained by Rutter’s formula for dill' slopes (S) mean radii (R) and degrees 
of roughness (n).f 

Caution. Different values of c must be used with English and with metric 
measures. We give tables for both measures. 


1st Havino- the slope S, the mean rad R and the deg n of roughness; to 
find'the coeff e. Turn to the division of the table corresponding to the 
given slopeS. In the first column find the given mean rad, R. In the same 
line with this R, and under the given w, is the proper value of c. f 

•>«l- Having the slope S, the mean rad R and either c or the actual or reqd 
veT v; to find the actual, or the greatest permissible, deg n of 
roughness of channel. If the vel is given, and not c, first find 

c _ velocity __ Turn to the division of the table corresponding to 

1 /slope X mean radius . „ _ 

the given S and in the first col find the given R. In the same line find the 
value given, or just obtained, for e; over which will be found the reqd n.f 
3d Having the slope S, the deg n of roughness, and the actual or required 
vel v: to find the actual or necessary mean rad. It. Assume a 
mean rad; and from the division of the table corresponding to the given S take 
out the value of e corresponding to the given n and the assumed R. lhen say 


r/ = c so found X f/asstimed mean radius X elope 


* Fnr experiments on abrasion of cements, see p€78. ... .. . , .. . .. . ,. 

lit is often necessary to interpolate values of S, R, n and c intermediate of those in the tables. 

this may be done mentally by simple proportion. 
























274 


HYDRAULICS. 


If this / is the same as the given vel, or near enough to it, take the assumed R 
as the proper one. Otherwise, repeat the whole process, assuming a new B, 
greater than the former one if / is less than the given vel, and vice versa* 

4t.h. Having the dimensions of the wetted portion (abco Figs 28, 29, 30,) of 
the channel, the degw of roughness, and the actual or reqd vel: to find the 

actual or necessary slope, S : 


Find the mean rad, R = 


area of wet cross section 


length, abco, of wet perimeter 

Assume one of the four slopes of the tables to be the proper one. From the 
corresponding division of the table take out the value of c corresponding to the 
given R and n. 

It R is 3.28 feet, or 1 metre, the value of c thus found is the proper one (be¬ 
cause then c, for any given n ,remains the same for all slopes): and the slope, S. 
may be found at once, thus: r ^ 

Slope,S = ( _ gi yep velocity \2 
\c X l/mean radius/ 


\c X l/mean radius) 
But if R is greater or less than 3.28 feet, or 1 metre, say 


v ' — e thus found X l/mean radius X assumed slope 

If this t f is near enough to the given vel, take the assumed S as the proper one 
Otherwise, assume a new S, greater \\vak the former one if / is less than the given 
vel, and vice versa; and repeat the whole process.* 


* 11 is necessary to interpolate values or S, R, n and c intermediate of those in the tables. 
This may be done mentally by simple proportion, ^ laules - 




Table of coefficient c, for mean radii in feet. 



Mean 

—• 

rad K 

v—1 

)l 

ft et 

•cT 


"So 

.1 

c * 

a, 0) 

.2 

r *z 

.4 

o a 

.6 

.8 

1 

U O 

*5 50 

1.5 

2 

3 


3.28 

gll 

4 

© o 

6 

©.§ 

8 

•o 

11 ^ 

10 

n 

12 


16 

© 

20 

& 

30 

0 

50 

* 

75 


100 



Coefficients n of roughness 


.009 

.010 

.011 

.012 

.013 

.015 

.017 

1.020 

1.025 

.030 

.035 

.040 

c 

c 

e 

c 

e 

e 

e 

e 

c 

c 

c 

c 

65 

57 

50 

44 

40 

33 

28 

23 

17 

14 

12 

10 

87 

7,5 

67 

59 

53 

45 

38 

31 

24 

19 

16 

14 

111 

97 

87 

78 

70 

59 

51 

42 

32 

26 

22 

19 

127 

112 

100 

90 

81 

69 

60 

49 

38 

31 

26 

22 

1 

122 

109 

99 

90 

77 

66 

55 

43 

35 

30 

25 

148 

131 

118 

106 

97 

83 

72 

60 

47 

38 

32 

28 

16b 

148 

133 

121 

111 

95 

83 

69 

55 

45 

38 

33 

179 

160 

144 

131 

121 

104 

91 

77 

61 

50 

43 

37 

197 

177 

160 

147 

135 

117 

103 

88 

70 

59 

50 

44 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 


2uy 

188 

172 

158 

146 

127 

113 

96 

78 

65 

56 

49 

226 

206 

188 

174 

161 

142 

126 

108 

88 

74 

64 

57 

238 

216 

199 

184 

171 

151 

135 

117 

96 

82 

71 

63 

246 

225 

207 

192 

179 

159 

142 

124 

102 

87 

76 

68 

253 

231 

214 

198 

186 

165 

349 

129 

107 

92 

81 

72 

263 

242 

223 

208 

195 

174 

157 

138 

115 

100 

88 

7q 

271 

249 

231 

215 

202 

181 

164 

144 

121 

106 

94 

84 

283 

261 

243 

228 

215 

193 

176 

157 

133 

117 

104 

95 

29/ 

274 

257 

241 

228 

207 

190 

170 

147 

130 

117 

107 

306 

284 

267 

251 

238 

217 

200 

180 

157 

140 

127 

117 

332 

290 

273 

257 

244 

223 

207 

187 

163 

147 

134 

124 


Mean 
rad K 

feet 


.1 

.2 

.4 

.6 

.8 

1 

1.5 

2 

3 

3.28 

4 


I v 


» 

1 - 


\l 

« 


6 

8 

10 

12 

16 

20 

30 

50 

75 

100 








































































HYDRAULICS 


275 


Table of coefficient c, for mean ra<lii in feet. — Continued. 



Mean 



Coefficients n of roughness 




Mean 


rad K 













rad It 

c 

feet 

.009 

.010 

.011 

.012 

.013 

.015 

.017 

.020 

.025 

.030 

.035 

.040 

feet 

II 


c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 



.1 

78 

67 

59 

52 

47 

39 

33 

26 

20 

16 

13 

11 

.1 


.15 

91 

79 

69 

62 

56 

46 

39 

31 

23 

19 

16 

13 

.15 

w • 

c 

.2 

100 

87 

77 

68 

62 

51 

44 

35 

26 

21 

18 

15 

.2 

O ^ 

.3 

114 

99 

88 

79 

71 

59 

50 

41 

31 

25 

21 

18 

.3 

s 
c - 

.4 

124 

109 

97 

88 

79 

66 

57 

46 

35 

28 

24 

20 

.4 

-w o> 

.6 

139 

122 

109 

98 

90 

76 

65 

53 

41 

33 

28 

24 

.6 

c IT 

.8 

150 

133 

119 

107 

98 

83 

71 

59 

46 

37 

31 

27 

.8 

P o 

1 

158 

140 

126 

114 

104 

89 

77 

64 

49 

40 

34 

29 

1 

<1; 

1.5 

173 

154 

139 

126 

116 

99 

87 

72 

57 

47 

40 

34 

1.5 


2 

184 

164 

148 

1&5 

124 

107 

94 

79 

62 

51 

44 

38 

2 

*5 oj 

3 

198 

178 

161 

148 

136 

118 

104 

88 

71 

59 

50 

44 

3 

© II 

3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

© 

4 

207 

187 

170 

156 

145 

126 

111 

95 

77 

64 

56 

49 

4 

©.I 

6 

220 

199 

182 

168 

156 

137 

122 

105 

85 

72 

63 

56 

6 

• o 

li S 

8 

228 

206 

189 

175 

163 

144 

129 

111 

91 

78 

68 

61 

8 

II e** 

10 

234 

212 

195 

181 

169 

149 

134 

116 

96 

82 

72 

64 

10 

* 

12 

238 

217 

200 

185 

173 

153 

138 

120 

99 

86 

75 

68 

12 

« 

16 

245 

223 

206 

191 

180 

160 

144 

126 

106 

91 

81 

73 

16 

ft 

20 

250 

228 

211 

196 

184 

165 

149 

131 

110 

96 

85 

77 

20 

V 

30 

257 

236 

219 

204 

192 

172 

157 

139 

118 

103 

92 

84 

30 


50 

266 

245 

228 

213 

201 

181 

165 

148 

127 

112 

101 

93 

50 


75 

272 

250 

233 

218 

207 

187 

171 

153 

133 

119 

108 

99 

75 


l . 100 

275 

254 

237 

222 

210 

190 

175 

158 

137 

123 

112 

104 

100 

_T 

.1 

90 

78 

68 

60 

54 

44 

37 

30 

22 

17 

14 

12 

.1 


.2 

112 

98 

86 

76 

69 

57 

48 

39 

29 

23 

19 

16 

.2 

C,2i 

.3 

125 

109 

97 

87 

78 

65 

56 

45 

34 

27 

22 

19 

.3 

0> —« 

— * p 

.4 

136 

119 

106 

95 

86 

72 

62 

50 

38 

31 

25 

22 

.4 


.6 

149 

131 

118 

105 

96 

81 

70 

57 

44 

35 

30 

25 

.6 

^ 2 

.8 

158 

140 

126 

114 

103 

88 

76 

63 

48 

39 

33 

28 

.8 

*rH ^ 

P w 

1 

166 

147 

132 

120 

109 

93 

81 

67 

52 

42 

35 

31 

i 

3 O 

1 .=; 

178 

159 

144 

130 

120 

103 

89 

75 

59 

48 

41 

35 

1.5 

^ «2 
a> 

2 

187 

168 

151 

138 

127 

109 

96 

81 

64 

53 

45 

39 

2 

p,CO 

^CnJ 

3 

198 

178 

162 

149 

137 

119 

104 

89 

71 

59 

51 

45 

3 


3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

© H 

4 

206 

186 

169 

155 

143 

125 

111 

94 

76 

64 

55 

49 

4 

© o' 

6 

215 

195 

178 

164 

152 

134 

119 

102 

84 

71 

61 

54 

6 

• o 

8 

221 

201 

184 

170 

158 

139 

124 

107 

88 

75 

66 

59 

8 

II© 

10 

226 

205 

188 

174 

162 

143 

128 

111 

92 

78 

69 

62 

10 

* a 

15 

233 

212 

195 

181 

169 

150 

135 

118 

98 

85 

75 

68 

15 

©— 

20 

237 

216 

200 

185 

173 

154 

139 

122 

102 

89 

79 

71 

20 

Sr* 

30 

243 

222 

206 

191 

179 

160 

145 

128 

108 

95 

84 

77 

30 

0 II 

50 

249 

227 

211 

197 

185 

166 

151 

134 

114 

100 

91 

83 

50 

35 

100 

255 

234 

218 

204 

191 

172 

158 

140 

121 

108 

98 

91 

100 


• 

,i 

99 

85 

74 

65 

59 

48 

41 

32 

24 

18 

15 

12 

.i 

^ . 

.2 

121 

105 

93 

83 

74 

61 

52 

42 

31 

25 

21 

17 

•2 

tc V 

r— — 

.3 

133 

116 

103 

92 

83 

69 

59 

48 

36 

29 

24 

20 

.3 

± a 

.4 

143 

125 

112 

100 

91 

76 

65 

53 

40 

32 

27 

23 

.4 

v- s_ 

.6 

155 

138 

122 

111 

100 

85 

73 

60 

46 

37 

31 

26 

.6 

w 13 

Q. 

.8 

164 

145 

131 

118 

107 

91 

79 

65 

50 

41 

34 

29 

.8 

ZZ -*-« 

1 

170 

151 

136 

123 

113 

96 

83 

69 

54 

44 

37 

32 

1 

0) 

1.5 

181 

162 

146 

133 

122 

105 

91 

77 

60 

49 

42 

36 

1%5 


2 

3 

188 

200 

170 

179 

154 

163 

140 

149 

129 

137 

111 

119 

97 

105 

82 

89 

64 

72 

54 

59 

45 

51 

40 

45 

2 

3 

C! • ’ 

4 

205 

185 

168 

155 

143 

125 

111 

94 

76 

63 

55 

48 

4 

© II 

6 

213 

193 

176 

162 

150 

132 

117 

100 

82 

69 

60 

53 

6 

© , 

8 

218 

198 

181 

167 

155 

137 

122 

105 

87 

73 

64 

67 

8 

|) g 

10 

222 

201 

185 

170 

158 

140 

125 

108 

89 

76 

67 

60 

10 

II o 

Tg) ^ 

15 

228 

207 

190 

176 

164 

145 

131 

113 

95 

82 

72 

65 

15 


20 

231 

210 

194 

180 

168 

149 

134 

117 

98 

85 

76 

68 

20 

* '•*> 

30 

235 

215 

198 

184 

172 

154 

139 

122 

103 

89 

80 

73 

30 

0 :i 

50 

240 

220 

203 

189 

177 

158 

143 

126 

108 

94 

85 

78 

50 

* 

100 

245 

224 

208 

194 

182 

163 

148 

131 

113 

99 

90 

83 

100 
















































































276 


HYDRAULICS 


Table of* coefficient c, for mean radii in feel.— Continued. 



Mean 




Coefficients n of roughness. 




Mean 

x: 

rad 11 













rad It 

"Sd • 

c ^ 

feet 

.009 

.010 

.011 

.012 

.013 

.015 

.017 

.020 

.025 

.030 

.035 

.040 

leet 

w ■ 

»—i r* 


c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 


o 

.1 

104 

89 

78 

69 

62 

50 

43 

34 

25 

19 

16 

13 

.1 

o. 

.15 

116 

101 

90 

80 

71 

59 

50 

40 

29 

23 

19 

16 

.15 

— +- 

.2 

126 

110 

97 

87 

78 

65 

54 

44 

32 

25 

21 

18 

.2 

V 

J-, V-. 

.3 

138 

120 

107 

96 

87 

73 

62 

50 

37 

30 

24 

21 

.3 

S.S 

.4 

148 

129 

115 

104 

94 

79 

68 

55 

42 

33 

27 

23 

.4 

•! 

.6 

157 

140 

126 

113 

103 

87 

75 

62 

47 

38 

31 

27 

.6 

©^ 

.8 

166 

148 

133 

121 

110 

93 

81 

67 

51 

42 

35 

30 

.8 

© .1 

1 

172 

154 

138 

125 

115 

98 

85 

70 

55 

45 

37 

32 

1 

w ^ 

• O 

1.5 

183 

164 

148 

135 

124 

106 

93 

78 

61 

50 

42 

37 

1.5 

II s 

2 

190 

170 

154 

141 

130 

112 

98 

83 

65 

54 

45 

40 

2 

Cl 

r# 

3 

199 

179 

162 

149 

138 

119 

105 

89 

71 

59 

51 

45 

3 


4 

204 

184 

168 

154 

142 

124 

110 

94 

76 

63 

55 

48 

4 

a’ -1 

6 

211 

191 

175 

161 

149 

130 

116 

99 

81 

69 

60 

53 

6 

©ii 

10 

219 

199 

183 

168 

157 

138 

123 

107 

88 

75 

66 

59 

10 


20 

227 

207 

190 

176 

164 

146 

131 

115 

96 

83 

73 

66 

20 

Wl 

50 

235 

215 

198 

184 

173 

154 

139 

123 

104 

91 

82 

75 

50 


L 100 

239 

219 

203 

189 

177 

158 

143 

127 

108 

96 

87 

80 

100 

o a ' 

.1 

110 

94 

83 

73 

65 

54 

45 

36 

27 

21 

17 

14 

-—* 

.1 

£ 

.2 

129 

113 

99 

89 

81 

66 

57 

45 

34 

27 

22 

18 


a & 

.3 

141 

124 

109 

98 

89 

74 

63 

51 

39 

30 

25 

21 

.3 


.4 

150 

131 

117 

105 

96 

80 

69 

56 

43 

34 

28 

24 

.4 

£=5 

.6 

161 

142 

127 

115 

104 

88 

76 

63 

48 

39 

32 

27 

.6 


.8 

169 

150 

134 

122 

111 

94 

82 

68 

52 

42 

35 

30 

.8 

© ii 

1 

175 

155 

139 

127 

116 

99 

86 

71 

56 

45 

38 

33 

1 

IH 

1.5 

184 

165 

149 

136 

124 

108 

93 

78 

62 

50 

43 

37 

1.5 

$o 

O o 

2 

191 

171 

155 

142 

130 

112 

98 

83 

66 

54 

46 

40 

2 

• 

3 

199 

179 

163 

149 

138 

119 

105 

89 

71 

59 

51 

45 

3 

II .5 

4 

204 

184 

168 

154 

142. 

124 

110 

93 

75 

63 

54 

48 

4 


6 

211 

190 

174 

160 

149 

130 

116 

99 

81 

68 

59 

52 

6 

ir II 

10 

218 

197 

181 

167 

155 

136 

122 

105 

87 

74 

65 

58 

10 

ft-s 

20 

225 

205 

188 

175 

163 

144 

129 

113 

94 

81 

72 

65 

20 

— M 

50 

232 

212 

196 

182 

170 

151 

137 

120 

101 

89 

79 

72 

50 

BB 

100 

236 

216 

200 

186. 

174 

155 

141 

124 

105 

94 

85 

77 

100 

.e 

.1 

110 

95 

83 

74 

66 

54 

46 

36 

27 

21 

17 

14 

.1 


.15 

122 

105 

93 

83 

75 

62 

52 

42 

31 

24 

20 

17 

.15 

C ’2 

.2 

130 

114 

100 

90 

81 

67 

57 

46 

34 

27 

22 

19 

.2 


.3 

143 

L5 

111 

100 

90 

76 

64 

52 

39 

31 

25 

22 

.3 

o « 

.4 

151 

133 

119 

107 

98 

82 

70 

57 

44 

35 

29 

24 

.4 

** ft 

.6 

162 

143 

129 

116 

106 

90 

77 

64 

49 

39 

33 

28 

.6 

G 

.8 

170 

151 

135 

123 

112 

95 

82 

68 

53 

43 

35 

31 

.8 


1 

175 

156 

141 

128 

117 

99 

87 

72 

56 

45 

38 

33 

1 

£<» 

1.0 

185 

165 

149 

136 

125 

107 

94 

79 

62 

51 

43 

37 

1.5 


2 

191 

171 

155 

142 

130 

112 

99 

83 

66 

55 

46 

40 

2 

© II 

3 

199 

179 

162 

149 

138 

119 

105 

89 

71 

59 

51 

45 

3 

a 

3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

"1 

4 

204 

184 

167 

154 

142 

123 

109 

93 

76 

63 

55 

48 

4 

« 3 

6 

210 

190 

173 

160 

148 

129 

115 

99 

81 

68 

59 

52 

6 


10 

217 

196 

180 

166 

154 

136 

121 

105 

86 

74 

65 

58 

10 


20 

225 

204 

187 

173 

161 

143 

128 

112 

93 

80 

71 

64 

20 

© II 

50 

231 

210 

194 

181 

168 

150 

135 

119 

100 

87 

78 

71 

50 


L ioo 

235 

214 

197 

184 

172 

153 

139 

122 

104 

91 

82 

75 

100 


For slopes steeper than .«! per unit of length, = 1 in 100 = 52.8 feet 
per mile, c remains practically the same as at that slope. But the velocity 

(being — c X T/mean radius X slope) of course continues to increase as the 
slope becomes steeper. 



















































































Slope = .0002 unit Slope = .0001 per unit of Slope = .00005 per unit of Slope = .000025 per unit of length 

of length, = 1 in 5000. length, = 1 in 10000. length, = 1 in 20000. — 1 -40000. 


HYDRAULICS 


277 


Table of coefficient c, for mean radii in metres. 


Mean 



Coefficients n of roughness. 




Mean 

rad K 













rad R 

metres 

.009 

.010 

.011 

.012 ’ 

.013 

.015 

.017 

.020 

.025 

.030 

.035 

.040 

metres 


c 

c 

c 

c 

c 

e 

c 

c 

c 

c 

c 

c 


.025 

34 

29 

25 

22 

20 

17 

14 

11 

9 

7 

6 

5 

.025 

.05 

44 

38 

33 

30 

27 

22 

19 

16 

12 

9 

8 

7 

.05 

.1 

58 

50 

44 

40 

36 

30 

26 

21 

16 

13 

11 

9 

.1 

.2 

72 

63 

56 

51 

46 

39 

34 

28 

21 

18 

15 

13 

.2 

.3 

82 

72 

64 

58 

53 

45 

39 

33 

25 

21 

17 

15 

.3 

.4 

89 

79 

71 

64 

59 

50 

44 

37 

29 

23 

20 

17 

.4 

.6 

99 

88 

80 

72 

67 

57 

50 

42 

33 

28 

23 

20 

.6 

1. 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1. 

1.50 

121 

109 

100 

92 

85 

74 

66 

57 

46 

38 

33 

29 

1.50 

2 

127 

115 

106 

98 

91 

80 

71 

61 

50 

42 

37 

32 

2 

3 

136 

124 

114 

106 

99 

87 

78 

68 

56 

48 

42 

37 

3 

4 

142 

130 

120 

111 

104 

93 

83 

73 

61 

52 

46 

41 

4 

6 

149 

137 

127 

119 

111 

100 

90 

80 

67 

58 

51 

46 

6 

10 

158 

145 

135 

127 

120 

108 

98 

88 

75 

66 

59 

53 

10 

15 

164 

151 

141 

133 

126 

114 

104 

94 

81 

72 

64 

59 

15 

20 

167 

155 

145 

137 

130 

118 

108 

98 

85 

75 

68 

62 

20 

30 

172 

160 

150 

142 

135 

123 

113 

103 

90 

81 

74 

68 

30 

.025 

40 

35 

30 

26 

24 

20 

17 

13 

10 

8 

7 

5 

.025 

.05 

52 

44 

39 

34 

31 

26 

22 

18 

13 

11 

9 

7 

.05 

.1 

65 

57 

50 

44 

40 

34 

29 

24 

18 

14 

12 

10 

.1 

j2 

79 

69 

62 

55 

51 

43 

37 

30 

23 

19 

16 

13 

.2 

.3 

87 

77 

69 

62 

57 

48 

42 

35 

27 

22 

18 

16 

.3 

.4 

93 

83 

74 

67 

62 

53 

46 

38 

30 

25 

21 

18 

.4 

.6 

102 

90 

82 

74 

69 

59 

52 

43 

34 

28 

24 

21 

.6 

1. 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1. 

1.5 

118 

107 

97 

90 

83 

73 

65 

55 

45 

38 

33 

28 

1.5 

2 

123 

111 

102 

94 

87 

77 

68 

59 

48 

41 

35 

31 

2 

3 

129 

117 

108 

100 

93 

83 

74 

64 

53 

45 

40 

35 

3 

4 

133 

121 

112 

104 

97 

86 

77 

68 

56 

49 

43 

38 

4 

6 

138 

126 

117 

109 

102 

91 

82 

72 

61 

53 

47 

42 

6 

10 

143 

131 

122 

114 

107 

96 

87 

78 

66 

58 

52 

47 

10 

15 

147 

; 135 

126 

118 

111 

100 

91 

82 

70 

62 

56 

51 

15 

20 

150 

137 

128 

120 

113 

103 

94 

84 

72 

64 

58 

53 

20 

30 

152 

| 140 

131 

123 

116 

105 

97 

87 

76 

68 

62 

57 

30 

.025 

47 

40 

35 

31 

28 

22 

19 

15 

11 

9 

7 

6 

.025 

.05 

59 

50 

44 

40 

35 

29 

25 

20 

15 

12 

10 

8 

.05 

.1 

72 

62 

55 

50 

45 

37 

32 

26 

19 

16 

13 

11 

.1 

.2 

84 

74 

66 

60 

54 

1 46 

39 

32 

25 

20 

17 

14 

.2 

.3 

91 

81 

73 

66 

60 

51 

44 

37 

.28 

23 

19 

17 

.3 

.4 

97 

86 

77 

70 

64 

55 

48 

40 

31 

25 

21 

18 

.4 

.6 

104 

92 

83 

76 

70 

60 

53 

45 

35 

29 

25 

21 

.6 

1. 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1. 

1.5 

117 

105 

96 

88 

82 

72 

64 

54 

44 

37 

32 

28 

1.5 

2 

120 

109 

100 

92 

85 

75 

67 

57 

47 

40 

34 

30 

2 

4 

128 

116 

107 

99 

92 

82 

73 

(54 

53 

46 

40 

36 

4 

6 

131 

119 

110 

102 

96 

85 

77 

67 

56 

49 

43 

39 

6 

10 

135 

123 

114 

106 

100 

89 

81 

71 

60 

53 

47 

43 

10 

15 

137 

126 

116 

109 

102 

92 

83 

74 

63 

00 

50 

46 

15 

30 

141 

129 

120 

112 

106 

95 

87 

78 

67 

59 

54 

50 

30 

noc: 


.025 

.050 

.1 

.2 

.4 

.6 


1 

2 

4 

10 

50 


52 

63 

75 

87 

99 

104 

111 

118 

124 

130 

135 


45 

55 

66 

77 

88 

93 

100 

107 

113 

119 

124 


40 

48 

59 

69 

80 

84 

90 

98 

104 

110 

114 


35 

43 

53 

62 

72 

77 

83 

90 

97 

102 

107 


39 

48 

57 

66 

71 

77 

84 

90 

96 

100 


32 

40 

48 

57 

61 

67 

74 

79 

85 

90 


27 

34 

41 

49 

53 

59 

65 

71 

77 

82 


21 

27 

34 

41 

45 

50 

56 

62 

67 

73 


16 

21 

26 

32 

36 

40 

46 

51 

57 

62 


12 

16 

21 

26 

29 

33 

39 

44 

50 

55 


10 

13 

17 

22 

25 

28 

34 

39 

45 

50 


8 

11 

15 

19 

22 

25 

30 

35 

40 

46 


.050 

.1 

.2 

.4 

.6 


1 

2 

4 

10 

30 


























































































278 


HYDRAULICS 


Table of coefficient c, for mean radii in metres.— Continued. 


pc 

■+-> 

mean 




Coefficients n of roughness. 



Mean 

fcfi 

c 

rad It 













rad It 

0) 

meters 

.009 

1.010 

.011 

.012 

.013 

.015 

.017 

.020 

1.025 

.030 

.035 

.040 

metres 

o 


c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 


*5 

.025 

55 

47 

41 

37 

33 

27 

22 

17 

13 

10 

8 

7 

.025 


.050 

66 

58 

51 

45 

40 

33 

28 

23 

17 

13 

11 

9 

.050 


.1 

78 

68 

61 

55 

50 

42 

35 

28 

21 

17 

14 

12 

.1 


.2 

90 

80 

70 

64 

59 

49 

42 

35 

27 

22 

18 

15 

.2 

*•5 

.3 

95 

85 

76 

70 

63 

54 

47 

39 

30 

24 

21 

17 

.3 

© rt 

.4 

99 

89 

80 

73 

67 

57 

50 

42 

32 

27 

22 

20 

.4 

C 11 

.6 

105 

94 

85 

78 

72 

62 

54 

45 

36 

30 

25 

22 

.6 

• 

1 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1 

II 

2 

117 

106 

97 

89 

83 

73 

65 

56 

45 

38 

34 

30 

2 


4 

123 

111 

102 

95 

88 

78 

70 

61 

50 

43 

38 

34 

4 

Si¬ 

ft 

6 

125 

114 

105 

97 

91 

81 

72 

63 

53 

46 

40 

36 

6 


10 

128 

117 

108 

100 

93 

83 

75 

66 

55 

48 

43 

39 

10 

to 

30 

132 

121 

112 

104 

98 

87 

79 

70 

60 

52 

48 

43 

30 

•pH 

.025 

57 

50 

43 

38 

34 

28 

23 

18 

13 

11 

9 

7 

.025 

5 © 

.050 

69 

59 

52 

47 

42 

34 

29 

23 

17 

13 

11 

9 

.050 

i-2 

.1 

80 

70 

63 

56 

50 

42 

36 

30 

22 

17 

14 

12 

.1 

CL ^ 

.2 

90 

80 

72 

65 

60 

50 

43 

35 

27 

22 

18 

16 

.2 

©*"" 

.3 

96 

86 

77 

70 

64 

54 

47 

39 

30 

25 

21 

18 

.3 

p* ^ 

.4 

100 

89 

81 

74 

67 

58 

50 

42 

33 

27 

23 

19 

.4 

C || ^ 

.6 

104 

94 

85 

78 

72 

62 

54 

46 

36 

30 

25 

22 

.6 

Zx: 

1 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1 

II M 

2 

116 

106 

97 

90 

83 

72 

64 

55 

45 

38 

33 

29 

2 

« g 

4 

121 

111 

102 

94 

87 

77 

69 

60 

50 

42 

37 

33 

4 


6 

124 

113 

104 

97 

90 

80 

71 

62 

52 

45 

40 

36 

6 

X o 

10 

127 

115 

106 

99 

92 

82 

73 

64 

54 

47 

42 

38 

10 

to 

30 

130 

119 

110 

102 

96 

86 

77 

68 

58 

51 

46 

42 

30 


.025 

59 

50 

44 

39 

35 

28 

24 

19 

14 

10 

9 

7 

.025 

G © 

.05 

69 

60 

53 

48 

43 

35 

29 

24 

18 

14 

11 

9 

.05 

s ° 

.1 

81 

71 

63 

57 

51 

43 

36 

30 

22 

18 

15 

12 

.1 

£ C 

.2 

91 

81 

72 

65 

60 

50 

44 

36 

27 

22 

18 

16 

_2 


.3 

97 

86 

77 

71 

65 

55 

48 

40 

31 

25 

21 

18 

.3 

sV 

.4 

101 

90 

81 

74 

68 

58 

50 

42 

33 

27 

23 

20 

.4 

O II 

• 

.6 

106 

95 

86 

78 

72 

62 

54 

46 

36 

30 

25 

22 

.6 

II £ 

1. 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1. 

o - 

1.5 

115 

104 

94 

87 

80 

70 

62 

53 

43 

36 

31 

27 

1.5 

Ski 

2 

117 

105 

96 

89 

83 

72 

64 

55 

45 

38 

33 

29 

2 

zZ 

4 

121 

110 

101 

93 

87 

76 

68 

59 

49 

42 

37 

33 

4 

to ° 

10 

126 

114 

105 

98 

91 

81 

73 

64 

53 

46 

41 

37 

10 


30 

129 

118 

108 

101 

95 

84 

77 

67 

57 

50 

45 

41 

30 


For slopes steeper than .01 per unit of length, = 1 in 100, the co¬ 
efficient c remains practically the same as at that slope. The velocity , however, 

being = cX l/mean radius X slope , continues to increase as the slope becomes 
steeper. 

To construct a diagram, fig 30 A, from which the values given 
by K.utier's formula may be taken by inspection. 

Draw xz hor, and say from 2 to 4 ft long; and oy vert at any point o within 
say the middle third of xz. On oy lay off, as shown on the left, the values of c 
for which the diagram will probably be used. If a scale of .05 inch or 002 
metre, per unit of c be used, and be made to include c = 250 for English meas¬ 
ure, or 150 for metric measure, oy will tie about 1 ft long. For the sakeof 
clearness we show only the larger divisions in this and in what follows. 

On oz lay off, as shown on its upper side, the square roots of all the values of 
the mean rad It for which the diagram is to be used. One inch per ft or .06 
metre per metre, of sq rt, is a convenient scale. Mark the dividing points with 
the respective values of the mean radii themselves. 

Having decided upon th e flattest slope to be embraced in the diagram, say 

0098 

W = 41,6 + SutMt slope per unit of length for English measure. 


or 




































































HYDRAULICS. 


279 



y — w 


1.811 


n 


for English measure ; or y — w 


n 


for metric measure. 


To each value of v — w, add w . thus obtaining values of y. We take .000025 
per unit of length as the flattest slope, ^aud .01, .02, .03 and .04 for n.f Hence 

(using English measure) w -= 41.6 + .^025 = 41,6 + 112 153 ' 6 ‘ 


1.811 1.811 1.S11 1 - 811 . _ 181.1.90.5,60.4, 45.3 respectively; 

y w .01 ’ .02 ’ .03 ’ .04 ’ __ 


# This is about as flat as is likely to occur i° practice, 
t In most cases many intermediate ones would be used. 


































279a 


HYDRAULICS. 


and y = 181.1 + 153.6, 90.5 + 153.6, 60.4 -f 153.6 and 45.3 -f 153.6; 

or 334.7,244.1,214.0 and 198.9 respectively. Layoff these values of yon oy in 
pencil, as at y, y', y", and y'", using the scale already laid off for c on oy. 

From each point, y, y' etc, draw a hor pencil line yt , y’t' etc, and mark on it, 
in pencil, the value of n used in determining its height oy etc. 

Next say x = w X greatest value of n. Make ox = x by the scale of sq rts of R 
on oz. In our case ox — 153.6 X -04 = 6.144 by the scale of sq rts of It, or = 6.144 2 
= 37.75 by the scale of R. 

Divide ox into as many equal spaces (4 in our case) as .01 is contained in 
greatest n. Mark the dividing points with the values of n, as in our Fig. 

From each dividing mark on ox erect a perpendicular, ( xt”' etc) in pencil, to 
cut that hor line (y"'V etc) which corresponds to the same value of n. The 
intersections are points in a hyperbola. Join them by straight lines t"’t”, t"t\ 
t't etc. 

From r in oz (corresponding to a mean rad of 3.28 ft, or 1 metre) draw radial 
lines ,rt,rt',rl" etc. Mark them “n = .01 ”, “ n = .02” etc, the same as their 
corresponding lines yl, y't' etc. 

For each slope (S) to be used in the diagram (except the flattest, for which 
this has already been done) say 


x', x" etc = 
x\ x" etc = 


/ .0028 \ 

\ 41 ' 6 + slope/ X 9 reatest 

( 23 + X orealesln. 


for English measure, 
for metric measure. 


Thus, our slopes are = .000025, .00005, .0001 and .01 per unit of length. Hence, 



41.6 + 


.0028 \ 
.00005 / 


X 


.04 = 3.904; 



41.6 + 


.0028 \ 

. 0001 / 


X .04 =— 2.784; 


— ^41.6 + X .04 = 1.675. 


Lay off each value of x', x" etc from oy on a separate hor pencil line o'x' etc, 
using the scale of sq rts of R as on oz. 

Mark each line o'x' etc in pencil with the slope used in fixing its length. 

Divide each dist o'x' etc into the same number of equal parts as ox. From 
the dividing points(whicl), like thoseof ox, represent the valuesof n) erect, perps 
to cut the radial lines rt'", rt" etc, each perp cutting that radial line which cor¬ 
responds to the value of n represented by the point at the foot of the perp. The 
intersections corresponding to each line o'x'etc form a hyperbolic curve. Mark 
each curve with the slope of its corresponding line, ox, o r x' etc. 

The drawing is now in the shape proposed by Mess Ganguillet and Kutter, and 
is ready for use in finding either c, n, R or S when the other three are given. 
Thus: 


1st. Having R, S and n, to find c. For example let R = 20 ft, S = .00005 
n = .03. From the intersection <1 of slope curve .00005 and radial line n = .03* 
draw*d-20 to the point # (20) in oz corresponding to the given R. At e where 
d- 20 cuts oy, is the reqd e, = 96 in this case. 

2d. Having R, S and c, to find n. For example let R = 20 ft, S = 00005 
c = 96. Through the points R = 20 in oz, and c = 96 in oy, draw * d-20 to cut. 
curve .00005. n (= .03) is found by means of the radial lilies nearest to the in¬ 
tersection, d. 


3d. Having S, v and c, to find It. For example let S = .00005, n — 03 
c = 96 Find curve .00005 and radial line n == .03. From their intersection d 
draw </-20 through the point eshowingc = 96. Its intersection with oz shows 
the reqd R, 20 in this case. 


* Instead of drawing these lines, we may use a fine black thread with a loop at one end. Drive a 
need e either into one of the points R or into one of the intersections, d etc. Slip the loop over the 
needle. The other end of the thread H held between the fingers, and the thread is made to cut the 
other points as reqd. The diagram should lie perfectly flat, and the string be drawn tight at each ob¬ 
servation, in order that friction between string and paper mav not prevent thestring from Terming a 
straight line. Or the free endof the string may rest on a pamphlet or other object about inch thick 
to keep the string clear of the diagram. Special care must then be taken to have the eye perp over 
the poiut observed. J * * 








HYDRAULICS. 


2796 


4ill. Having R, c and n, to find S. For example let R = 20 ft, c = 96, 
n = .03. Through R = 20 and c = 96 draw d-20. S (.00005) is found by means 
of the curves nearest to the point d, of intersection of d -20 with radial line 
n = .03. 

The following addition to Kutter’s diagram, proposed by Mr Rudolph Hering, 
Civil and Sanitary Engineer, Philadelphia,* enables us to read the veloc¬ 
ity from the.diagram. 


Fin d the sq rt of the recipr ocal of each slope to be embraced in the diagram 

-\j t-—tt— v-r-r . Lay off these sq rts on the right of oy, using 

\ slope per unit of length • ^ 

the scale of c already laid off on its left. In our fig we have so proportioned the 

c 1.5 

two scales that — .. = —j—. Mark the dividing points w T ith the slopes 

1 /recip of S 1 
per unit of length. 

On oz lay off the vels to be embraced in the diagram, using the scale of sq rts 

v^l £ 

of R already laid off on oz , and making —£ = — - 

1 /R i/recip of S 


1st. Having R, S andn; to find v. For example let R = 20 ft, S = .00005, 
n = .03. From R = 20 draw d-20 to the intersection d of curve .00005 with radial 
line n — .03. d -20 cuts oy at e, where c — 96. With a parallel ruler join R 
= 20 with S = .00005 on oy. Draw a parallel line through c — It cuts oz at 
■m, giving the reqd vel, 3.03 ft per sec. 

2d. Having R, S and v; to find n. For example let R = 20 ft, S = .00005, 
v = 3.03 ft per sec. With a parallel ruler join R = 20 and slope .00005 on oy. 
Draw a parallel line through v — 3.03. It cuts oy at. e, where c = 96. Through 
R = 20 and c = 96, draw <2-20 to cut curve .00005. The point d of intersection, 
being on radial line n = .03, shows .03 to be the proper value of n. 

Any line drawn to the curves from R = 3.28 ft or 1 metre, is one of the radial 
lines used in making the diagram. It therefore necessarily cuts all the slope 
{ curves at points showing the same value of n. 

3d. Having S, n and v\ to find R. For example, let S = .00005, n = .03, 
v = 3.03 ft per sec. Assume a value of R, say 10 ft. Find curve .00005 and radial 
line n = .03. Join their intersection d with R = 10 ft. The connecting line cuts 
oy a t c = 82. with a parallel ruler join c = 82 with v = 3.03. Draw a parallel 
line through slope = .00005 on oy. It cuts oz at. R = 27.3, showing tnat a new 
trial is necessary, and with an assumed R greater than 10 ft. 

If R thus found is the same as the assumed one, the latter is correct. If they 
are nearly equal, their mean may be taken. 

4th. Having R, n and v ; to find S. For example, let R = 20 ft, n = .03, 
v = 3.03 ft per sec. Assume a slope (sav .0001). Find its curve, and radial line 
n = .03. Join their intersection with R = 20, and note the value (89) of c where 
the connecting line cuts oy. With a parallel ruler join c = 89 with v = 3.03. 
Draw a parallel line through R = 20. It cuts oy at slope .000058, showing that 
a new trial is necessary, and with an assumed S flatter than .0001. If R is 3.28 
ft or 1 metre, the diagram gives the corrects at the first trial, no matter what 
S was assumed at starting. With any other R, if the diagram gives the same S 
as that assumed, the latter is correct. If the two differ but slightly, we may take 
their mean. 

----------- 


# Transactions of the American Society of Civil Engineers, January 1879. 


















279 c 


VELOCITIES IN SEWERS. 


Table of vels in Circular Brick Sewers when running full, by 

Kutter’s formula, p 272, but taking n at .015 instead of his .013, in consideration 
of the rougn character of sewer brickwork generally. 

When running: only half full the vel will be the same as when full •, 
but this is not the case at any other depth whether greater or less. At greater 
ones it increases until the depth equals very nearly .9 of the diam, when it it 
about 10 per cent greater than when either full or. half full. From depth of .9 off 
the djara the vel decreases whether the depth becomes greater or less. At depth 
of .25 diam the vel is about .78 of that when full; and then diminishes much 
more rapidly for less depths. All this applies also to pipes. 

The vel for any fall or diam intermediate of those in the table can be found by 
simple proportion. Original. 


Fall 

in ft 
per 
mile. 


.1 

.2 

.4 

.6 

.8 

1.0 

1.25 

1.50 

1.75 

2.0 

2.5 
3.0 

3.5 

4. 

5. 

6 . 

7. 

8 . 

9. 

10 . 

12 . 

15. 

18. 

21 . 

24. 

27. 

30. 

35. 

40. 

45. 

50. 

60. 

70. 

80. 

90. 

100 . 


2 

3 

4 

Diameters In fee 
6 | 8 

t. 

12 

16 

20 



Velocities In feet per second. 



.19 

.27 

.35 

.50 

.64 

.89 

1.10 

1 1.34 

.30 

.42 

.53 

.74 

.93 

1.26 

1.56 

1.84 

.46 

• .65 

.80 

1.08 

1.39 

1.81 

2.20 

2.60 

.59 

.81 

1.00 

1.35 

1.70 

2.22 

2.70 

3.18 

.69 

.95 

1.17 

1.57 

1.94 

2.56 

3.08 

3.60 

.79 

1.07 

1.32 

1.77 

2.16 

2.84 

3.43 

3.96 

.89 

1.21 

1.49 

1.98 

2.42 

3.17 

3.8 

4 5 

.98 

1.33 

1.64 

2.18 

2.64 

3.5 

4.2 

4.9 

1.06 

1.44 

1.78 

2.34 

2.85 

3.8 

4.5 

5.3 

1.15 

1.55 

1.91 

2.53 

3.1 

4.0 

4.8 

5.6 

1.32 

1.78 

2.18 

2.85 

3.5 

4.5 

5.4 

6.3 

1.44 

1.94 

2.38 

3.2 

3.8 

5.0 

6.0 

69 

1.58 

2.10 

2.58 

3.4 

4.1 

5.3 

6.5 

7.4 

1.68 

2.2 

2.7 

3.6 

4.4 

5.7 

6.9 

7.9 

1.90 

2.5 

3.1 

4.1 

4.9 

6.3 

7.6 

8.7 

2.06 

2.7 

3.3 

4.4 

5.4 

6.9 

8.3 

9.6 

2.2 

3.0 

3.6 

4.8 

5.8 

7.5 

9.0 

10.4 

2.4 

3.2 

3.8 

5.1 

6.2 

8.0 

9.7 

11.1 

2.5 

3.4 

4.1 

5.4 

6.6 

8.5 

10.3 

11.8 

2.7 

3.5 

4.3 

5.7 

6.9 

9.0 

10.8 

12.5 

2.9 

3.9 

4.8 

6.3 

7.6 

9.9 

11.9 

13.6 

3,3 

4.4 

5.4 

7.1 

8.5 

11.0 

13.3 

15.3 

3.6 

4.8 

5.9 

7.7 

9.3 

12.1 

14.5 

16.7 

3.9 

5.1 

6.3 

8.4 

10.0 

13.0 

15.7 

17.9 

4.2 

0.0 

6.8 

8.9 

10.8 

13.9 

16.8 

19.2 

4.5 

5.9 

7.2 

9.5 

11.4 

14.8 

17.9 

20.4 

4.7 

6.2 

7.5 

9.9 

12.0 

15.6 

18.8 

21.5 

5.0 

6.7 

8.2 

10.8 

13.0 

16.8 

20 4 

23.2 

5.4 

7.1 

8.7 

11.5 

13.9 

18.0 

21.7 

24.8 

5.6 

7.5 

9.2 

12.2 

14.8 

19.1 

23.0 

26.3 

5.9 

8.0 

9.7 

12.8 

15.5 

20.1 

24.2 

27.7 

6.5 

8.7 

10.7 

14.1 

17.0 

22.1 

26.5 

30.3 

7.0 

9.4 

11.5 

15.2 

18.4 

23.9 

28.5 

32.8 

7.4 

10.1 

12.3 

16.2 

19.7 

25.5 

31.0 

35.0 

7.9 

10.7 

13.1 

17.2 

20.9 

27.0 

32.3 

37.1 

8.4 

11.3 

13.8 

18.2 

22.0 

28.5 

34.1 

39.1 


Fall 

in ft 
per 
100 ft. 


.0019 

.0038 

.0076 

.0114 

.0151 

.0189 

.0237 

.0284 

.0331 

.0379 

.0473 

.0568 

.0662 

.0758 

.0947 

.1136 

.1325 

.1514 

.1763 

.1894 

.2273 

.2841 

.3409 

.3975 

.4546 

.5109 

.5682 

.6629 

.7576 

.8523 

.9470 

1.136 

1.326 

1.515 

1.705 

1.894 


A vel of 10 ft per sec = 600 ft per minute = 36000 ft. or 6.818 miles per 
hour. About 5 t per sec is as great as can be adopted in practice to prevent the 
lower parts of the sewers from wearing away too rapidly by the debris carried 
along by the water. u 


Art. 23. The rate at which rain water reaches a sewer or 

culvert, etc. may, according to the admirable “ Report on European Sewerage 
Systems by Mr. Rudolph Hering, Civ. and San. Eng. of Philada, he four?! 
Soc^E 1 ^'^' ^1881^ ,e l>y Mr Burkli-Ziegler. See Trans. Am 
















































HYDRAULICS. 


279 c? 


t second per A coef Av. citb. ft. of rainfall 4 |Av. slope of ground 

: acre reach- = ac . cor( iing X per second per acre, X \ / i n feet per 1000 ft 
ingsewer judgment daring heaviest fall. \ No. ofacresdrained 

, i 

1 His coefficient for paved streets is .75; for ordinary cases .625; and for 
! f suburbs with gardens, lawns, and macadamized streets .31. His average 
? heaviest fall is from ljj to 2J ins per hour. To this the writer will add 
that, each inch of rainfall per hour, corresponds closely enough to 1 cub ft 
“ per sec per acre: so that if we liberally allow for 3 or 4 , etc, ins per hour of 
p average heaviest rainfall, the third term of the above equation also becomes 
simply 3 or 4, etc. 

^ Example. If an area of 3100 acres (nearly 5 sq mites), with an average slope of 5 ft per 1000 ft, 
receives a rainfall averaging 3 ins per hour wheu heaviest, then, assuming a coefficient of .5, the rate 
at which the water would reach the mouth of a sewer at the lower end of the 3100 acres would be 


•5 X 3 X V 33^0 — *3 X 3 X *203 = .305 cub ft per sec per acre; 
or .305 X 3100 = 945.5 cub ft per sec, total. 

Now suppose the tall of the intended sewer to be say 4 ft per mile; and that for fear of the too rapid 
wearing away of its brickwork by debris swept along by the water, we limit its vel to 6.3 ft per sec, 
which may be permitted on occasions as rare as rains of 3 ins per hour, although for tolerublv constaut 
flow, where liable to debris, it should not exceed about 5 ft per sec. To Olid the diameter, look 
in the Table of Vels in Sewers, p 279c. for a diam corresponding as near us may be, to a vel of 6.3, 
and to a fall of 4 ft per mile. We find this diam to be 14 ft, the area of which is 154 sq ft. Hence, 
; 154 X 6 -3 — 370 cub ft per sec = capacity of sewer. This is a trifle more than our 945.5 cub ft per 

' sec of rainfall; nevertheless, to allow for deposits in the sewer, it would be advisable to increase the 
; diam say to 14.5 or 15 ft. 

Rem. Hr Wicksteed, an experienced English hydranli- 
fian, gives llie following; table of the least vels and grades 
i or falls, to be given to drain-pipes and sewers in cities, in order 

! that they may under ordinary circumstances keep themselves clean, or free from deposits. He re¬ 
commends that no drain pipe, even for a single common dwelling, shall be less than 6 ins diam. 


Warn, 
in Inches. 

Vet. in ft. 
per Min. 

Grade, 

1 in 

Grade. 
Feet per 
Mile. 

Diam. 
in Inches. 

Vel. in ft. 
per Min. 

Grade, 

1 in 

Grade. 

Feet per 

Mile. 

4 

240 

36 

146.7 

18 

180 

294 

18 0 

6 

220 

65 

81.2 

21 

180 

343 

15.4 

7 

220 

76 

6 ) 5 

24 

180 

392 

13.5 

8 

220 

87 

60.7 

30 

ISO 

490 

10 8 

9 

220 

98 

53.9 

36 

180 

588 

9.0 

10 

210 

11 !) 

41.1 

42 

180 

686 

7.7 

n 

200 

145 

36.7 

48 

180 

781 

6.8 

12 

190 

175 

30.2 

54 

180 

882 

6.0 

15 

180 

244 

21.6 

60 

180 

980 

5.4 


Weight per foot run of glazed terra cotta pipes for drains, etc.; made 
' jy Moorhead Clay Works, Spring Mill (office No. 11 South 7th S'.), Philadelphia. 
Net prices per foot run adopted by the United Sewer pipe Makers of the United 
States, Match 1887. Discounts, 18S8, front 40 to 50 per cent, on sizes smaller than 
15 inch, to 20 to 40 per cent, on 30 inch. For larger sizes, address the makers, 
as above. 


Drain pipe, with socket joint 


Bore 

Wt 

Price 

Bore 

Wt 

Price 

ins 

lbs 

$ 

ins 

lbs 

$ 

2 

4 

0.14 

6 

18 

0.30 

3 

7 

0.16 

8 

22 

0.45 

4 

10 

0.20 

10 

30 

0.65 

5 

12 

0.25 

12 

33 

0.85 


Sewer pipe, with sleeve joint 


Bore 

Wt 

Price 

Bore 

Wt 

Price 

ins 

lbs 

$ 

ins 

lbs 

$ 

15 

45 

1.25 

30 

150 

5.50 

18 

65 

1.70 

36 

195 

7.00 

21 

89 

2.50J 

42 

203 

8.50 

24 

100 

3.25 

48 

230 

10.50 


The joints are filled with cement mortar; or, when used for drainage only, 
with clay. Drain pipes (3 to 12 ins bore) are about | inch thick. A bend or 
branch costs about as much as from 3 to 5 feet of pipe. The 48-inch pipes are 
about 2 ins thick. 

Art. 24. When the area of cross section of channel is re¬ 
duced at any point, as by a dam (Fig 33, p 279 e), or by narrowing it, either 
at its sides (Fig 32) or by placing in it a pier etc, Fig 34; a portion at least of 
the force of grav (which would otherwise he giving vel to the water up-stream 

( from the point where the obstruction takes place), causes pressure against the 
dam etc. This pres maintains the up-stream water at a higher level than it 

19 


















































279 e 


HYDRAULICS. 


would otherwise have. Said water is then practically in a reservoir ', i e, it has 
less vel and greater pres than before. If the reservoir has no oullet, there is no 
vel; and all of the head, or force of grav, acting on the water is expended in pres. 

But if there is an outlet, as over the dam, or between the piers etc, a portion 
co, Figs .31, 33, 34, of this pres or head, is expended in giving vel (or an accelera¬ 
tion of vel) to the water escaping by that outlet; alter which only so much t 
head tin the shape of surface slope) is needed as will overcome the resistances 
of the channel down-stream from the obstruction, and so maintain uniform the j 
vel given to the water by the head co. 

Where a large canal, such as those intended for navigation, is fed from a reser¬ 
voir, the fall co in feet is approximately 

= mean velocity 2 in canal, in feet per second, X .017; 

and in smaller canals, such as mill courses, 

= mean velocity 2 in canal, in feet per second, X -02. 

The abruptness of the fall may be diminished by rounding off or sloping the * 
edges of the piers, or the corners at the sides of the channel (Fig 32) or the 
approach to the dam (Figs 1 to 4, pp 283, 284). 

Fig 33 is a cross section of CTejjrg's (lam, across Cape Fear River, N. C. It 
is from measurements made by Elhvood Morris, C E; by whom they were com¬ 
municated to the writer. The dam is of wooden cribwork ; and its level crest, 

8 ft 5 ins wide, is covered with plank ; along which the water glides in a smooth 
sheet, 6 ins deep, (at the time of measurement). At the upper end of this 
sheet, and in a dist of about 2 ft, a head co of 9 ins forms itself, as iu the fig. 



For Construction of Dams, see p 282, etc. 



! 





) 


' 




















































HYDRAULICS. 


279/ 


Scour. In a channel of uniform and constant slope and cross 
section, the yel ot the particles of water immediately adjoining the bottom and 
i 1 sides is very slight.; and hut little scouring takes place. But when irregular- 
. ities in the slope or cross section occur, as in the last article, the scour is greatly 
i increased in their immediate neighborhood. 

j 1 he erection ot one or two piers in a quite large stream, will frequently pro- 
i I duce an almost incredible amount of scour, if the bottom is at all of a yielding 
IB nature. The greatest scour of course takes place during freshets: and near the 
. ! obstruction. 


Scouring* action is supposed! to H»e as square of vel. 

to Snieaton, a vel of 8 miles an hour will not derange quarry rubble stones, not exceed" 
g alt a cub ft, deposited around piers, <fcc; except by washing the soil from under them. 

1 inch per sec, — 5 ft per min, ir .056818 of a mile, or 800 ft per hour. 

1 foot, per sec, — 60 ft per min, =: .681816 of a mile, or 3600 ft per hour. 

To reduce inches per sec, to feet per minute, multiply by 5. 

„ ;; “ .hour. “ “ 300. 

r- . ‘ “ to miles per hour, divide by 17.6. 

One mile per hour =r 88 ft per min = 1.4667 ft, or 17.6 ins per sec. 


The two following tables are (with many corrections) from Nicholson’s 
Architecture; and must be looked upon merely as probable approximations. They suppose the piers, 
Ac, to be properly rouuded or pointed at their upstream ends, so as to give as free a passage as pos- 
1 sible to the water. He says that if they are square-ended, the head vyill be increased about 50 per ct. 

J The subject is an extremely intricate one, and admits of no precise solutfbn. If the increased vel 
scours away the bottom until the area of water-way becomes as great as it originally was, the head 
i i disappears; and the vel also becomes reduced to its original rate. This is common in soft bottoms. 


TAULE Of heads produced by obstructions to streams. 


Kind of Bottom 
which begins to ' 
Original Vel. wear away under 

of Stream.* Bottom Vel. equal 

to those in the 
first three cols. 


Proportion of Area of original Water-way, 
occupied by the Obstructions. 


1 

1 

1 

1 1 1 

1 1 1 

5 

3 

1 1 

TO 

8 

6 | 4 

H | 1. 

■8 

4' 


Head of Water produced at the Obstructions 5 in 



Ins. 

3 

Ft. 

y* 

Miles. 

.170 

Ooze, and Mud... 

.0003 

.0004 

.0004 

.0006 

Fet 

.001 

3t. 

.0014 

.0033 

.0067 

.0162 


6 

y 

.341 

Clay. 

.0011 

.0014 

.0017 

.0023 

.001 

.0058 

.0133 

.0267 

.0616 


12 

i 

.681 

Sand. 

.0045 

.0056 

.0069 

.0091 

.015 

.0231 

.0532 

.1069 

.2581 


24 

2 

1.36 

Gravel. 

.0182 

.0225 

.0276 

.0361 

.060 

.0924 

.2128 

.4276 

1.036 


36 

3 

2.04 

Small Shingle.... 

.0409 

.0507 

.0621 

.0819 

.135 

.2079 

.4788 

.9621 

2.326 


48 

4 

2.72 

Large “ .... 

.0728 

.0902 

.1104 

.1456 

.240 

.3696 

.8412 

1.710 

4.144 


60 

0 

3.41 

Soft Shistus. 

.1137 

.1410 

.1725 

.2275 

.375 

.5775 

1.320 

2.672 

6 475 

> 

72 

6 

4.09 

Stratified Rocks.. 

.1638 

.2030 

.2484 

.3276 

.540 

.8316 

1.915 

3.848 

9.304 


T20 

10 1 

6.81 

Hard Rocks. 

.4550 

.5640 

.6901 

.9100 

1.50 

2.310 

5.280 

10.69 

25.9 


TABLE Increased velocities produced at. and by 

rounded, or pointed obstructions. If square, these vels must, accord¬ 
ing to Nicholson, be increased % part. 


Original Vel. 
of Stream.* 

Proportion of Area of Water-way, occupied by the Obstructions. 

11111111111111 ft 7 3 


Per 

TS 

TO 

8 1 

¥ ! 

¥ 1 

3 1 

2 1 

8" i 

¥ 

Ins. 

Ft. 

Hour. 

Miles. 

Velocity produced at the Obstruction in Feet per Second. 

3 

y< 

.170 

.28 

.29 

.30 

.32 

.35 

.394 

.52 

.7 

1.05 

6 

hi 

.341 

.56 

.58 

.60 

.64 

.70 

.788 

1.05 

1.4 

2.1 

12 

i 

.681 

1.13 

1.16 

1.20 

1.26 

1.40 

1.58 

2.1 

2.8 

4.2 

24 

2 

1.36 

2.27 

2.33 

2.40 

2.52 

2.80 

3.16 

4.2 

5.6 

8.4 

36 

3 

2.04 

3.39 

3.48 

3.60 

3.78 

4.20 

4.74 

6.3 

8.4 

12.6 

48 

4 

2.72 

4.54 

4.66 

4.80 

5.04 

5.60 

6.32 

8.4 

11.2 

16.8 

60 

5 

3.41 

5.60 

5.80 

6.00 

6 40 

7.00 

7.88 

10.5 

14.0 

21.0 

72 

6 

4.09 

6.78 

6.96 

7.20 

7.56 

8.40 

9.48 

12.6 

16.8 

25.2 

120 

10 

6.81 

11.3 

11.6 

12.0 

12.6 

14.0 

15.8 

21.0 

28.0 

42.0 


* A very vague expression. Does it refer to the greatest surface vel at mid-channel; or to the mean 
vel of the entire cross-section 7 


























































280 


HYDRAULICS, 


Art. 26. The resistance of water against a flat surface mov¬ 
ing through it at right angles, is nearly as the squares of the vel; and, 
accordiug to Hutton, its amount in lbs per sq ft approx = Square of vel in ft per 
sec. Or like the pres of a running' stream against a perp fixed flat 
surface, it is—wt of a col of water whose base = pressed surf, and whose ht = head due to the vel as 
per table p 258. 

The resist of a sphere is to that of its great circle about as 1 to 2.9. 

When the moviDg surf, Instead of being at right angles to the direction in which it moves, forms , 
another angle with it. the resistance becomes less in about the following proportions. Therefore, 
when the surf is inclined, first calculate the resistance as if at right angles ; and then mult by the I 
following decimals opposite the angle of inclination : 


90°. ...1.00 

60°.88 

40°.58 

20°.16 

80.98 

55.83 

35.46 

15.10 

70.95 

50.76 

30.34 

10.06 

65 .... .92 

'«-■ ——------- 

45.68 

25.24 

5.02 


The scour, or abrading power of moving water is considered to 

be as the square of its vel. 


Art. 27. To calculate the horse-power of falling w r ater, on 

the ordinary assumption that a horse-power is equal to 83000 lbs lifted 1 foot vert per min. That of 
average horses is really but about % as much, or 22000 fibs, 1 foot high per min. Mult together the 
number of cub ft of water which fall per min ; the vert height or head in feet, through which it falls; | 
and the number 62.3, (the wt of a cub ft of water in lbs ;) and div the prod by 33000. Or, by formula, 

cub ft ^ vert — lbs 
The number of __ per min A height in ft * 62.3 
horsepowers -:===- 


Ex. Over a fall 16 ft in vert height, 800 cub ft of water are dischd per min. 
powers does the fall afford ? 


Here, 


cub ft ft Tbs 
800 X 16 X 62.3 


33000 


797440 

33000’ 


24.17 h-pow. 


How many horse- 

I 

I 


Water-wheels do not realize all the power inherent in the water, as found by out 
rule. Thus, undershots realize but from % to %; breast-wheels, %; overshots, from % to % ; tur¬ 
bines, % to .85 of it ; according to the skill of design, and the perfection of workmanship. Kven when 
the wheel revolves in a close-fitting casing, or breast, elbow buckets give considerablv more power 
than plain radial or ceuter-buckets. Of the power actually received by a wheel, part is expended in 
friction, &c; while the remainder does the useful or paying net work of raising water, grinding 
grain, sawing, &c. 


Observations by Cenl Haupt, in 1866, gave the following results for a 
small hydraulic ram. Head of water to ram = 8.812 It; diam of drive-pipe = 
ins; length 15 ft. Diam of delivery-pipe - % inch; length 200 ft. Vert height to which the 
water was raised by the delivery-pipe, 63.4 feet. Strokes of ram per min, 170. Quantity of water 
which worked the ram — 768 cub ins, =3.31 galls, = 27.73 fibs per min. Quantity raised 63.4 ft high 
. Ibs water ft ft-lbs 

per min, — 48 cub ins, — 1.736 Ibs. Hence the power expended per min, wa» 27.73 X 8.812 = 244.35. 

lbs water ft ft-Tba 


And the useful effect, was 1.736 X 63.4 — 110.06. Hence the ratio which the useful effect bears to the 

... 110.06 /i i 

power in this instance, is ~ , or .45. The actual power of the ram is, however, greater than this, 


inasmuch as it has to overcome the friction of the water along the delivery-pipe.* 

To dud the horse-power of a running stream. Water-wheels 

with simple float-boards,t instead of buckets, are sometimes driven by the mere force of theordinarv I 
natural current of a stream, without any appreciable fall like that in the foregoiug case. In such 
cases, we must substitute the virtual or theoretic head ; which is that which would impart to it the 
same vel which it actually has. This virtual head may be taken at once from Table p 258. Thus, a 
stream lias a vel of 2.386 miles per hour; or 210 ft per min ; or 'i% ft per sec ; and in the column of 
heads in Table 10, opposite to 3.5 vel per sec, we find the reqd head .190 of a ft. Having thus found 
the heau, we must now find the quantity of water which passes any given area of the stream in a 
min. Thus, suppose that the immersed part of a float when vert is 5 ft long, and 1 ft wide or deep; 
then the area of this part which receives the rorce of the current, is 5 X 1 = 5 square feet. Hence. 

art 'a vol 


7 * S< 1 ^ X 210 — 1050 cub ft per min. Having now the cub ft per min, and the vert height or head, 
the number of horse-powers of the stream of the given area, is found by the foregoing rule, or formula. 


*4 conimittee of the Franklin Institute, in 1850. gave .71 as the 

coefficient for a ram at the Girard College, iu which the diam of drive-pipe was 2% ins; its length 
160 ft; fall, 14 ft. Delivery-pipe, 1 inch diam; 2260 ft long: vert rise, or height to which the water 
w^as raised. 93 ft. No details of the experiment are given. Some large rams in France give a useful 
effect of from .6 to ,6o of the whole power expended. It is an excellent machine for many purposes- 
and is^sometimes used for filling railway tanks at water stations. * ’ 

f Such wheels, for floating mills, in Furope, rarely exceed 15 ft 
diam. Whatever the diam, they may have about 18 to 20 floats. The floats are from 8 to 16 ft long; | 
and about i to ^ as deep as the diam of the wheel. They should not dip their entire depth into 
the water, but nearly so. They should not be in the same straight line with the radii; but should 
incline from them 30° up stream, to produce their full effect. All these remarks apply to w heels 
moving freely in a wide or indefinite channel; as in the case of a floating mill, built on a scow and 
anchored out in a stream ; but not to wheels for which the water is dammed up, and acts with a prnc- 
ticaUall. No great exactness is to be expected in rules on this subject. The best vel for the wheel 





















HYDRAULICS, 


281 


Thus, 


cub ft per min v vert ht in ft _ lbs 
of _ 105 0 x .190 X 62.3 

' Po “'' 33000 


12429 

— 33000 = - 37T °f a Pow - 


But in practice the wheel* actually realize but about 4 of this power of the stream, when working 

chaunTbuUirUe wid^.h 111 It" ‘he water flows with the same vel through a narrow artificial 

4 -Tis b the WheeU 1 beretore, the actual power of our wheel will be but 377 X 

.4 - .la08 , or about i of a horse power; or 33000 X .1508 = 4976 ft. lb. per min. Making a rough 

allowance for the friction of the machine at its journals, &c, we should have say about 4400 ft-Ibs of 
oer^n ° W ThL ] S A* he wheel W0U * d actually raise about 440 tt»s 10 ft high ; or 44 fts 100 ft high, &c 
wh ! he ve of th '? stream must not be measured at the surface ; but at about U of the depth to 

w hich the floats are to dtp, or be immersed. This, however, is ehieflV necessary in shallow streams 
in which the depth of the float bears a considerable ratio to that of the water. ’ 

This power of a, running 1 stream, (for any given area of 
transverse section,) increases as the cubes of the vels: for as 

we have seen, the power in ft-lbs per miu is found by mult together the weight of water w hich passes 
through the section in a min, and the virtual head in ft; and since this wXht inwelses as the vel 
and this head as the square of the vel, the prod of the two (or the • ower) mus be as the cube or the 
7 • Therefore, if the vel in the foregoing case had been 10.5 ft per sw. or 3 times I 5 ft the power 
•1 the wheel would have been 27 times as great, br .1508 X 27 = To7 horse-powe™ ’ P 










282 


DAMS. 


DAMS. 

4 


We can devote but little space to this subject, in addition to what is said on 
earthen dams for reservoirs, p 287 ; and on stone ones, p 229, Ac. Those we shall 
now describe will.also answer for such reservoirs, when the perishable nature of 
timber is not an objection. 

Primary requisites, in the erection of dams, are, a foundation suffi¬ 
ciently tirni to prevent them from settling, and thus leaking; the prevention 
of leaks through their backs, or under their bases; and the prevention of wear 
of the bottom of the stream in front of the dam, by the action of the falling 
water. For the first purpose, hard level rock bottom is of course the best; and 
should be chosen, if possible. In that case, thick planks, tt, Fig 6, (single or 
double, as the case may be,) closely jointed, and reaching from the crest, e, to 
the back lower edge w, (where they should be scribed down to the rock ;) with a 
good backing, b, of gravel, will suffice to prevent leaks. Gravel, or rather verv 
gravelly soil, is far better than eartli for this purpose; for if the water should 
chance to form a void in it, the gravel falls and stops it. To prevent this back¬ 
ing from being disturbed near the crest of the dam, by floating bodies swept 
along by freshets, a rough pavement of stones, about 15 to 18 inches deep, as 
shown in Fig 7, should be added for a width of about 10 to 20 feet: or until its 
top becomes 3 to 5 feet below the crest c of the dam, according to circumstances. 


c 



In Fig 1, (a dam on the Schuylkill navigation,) the upper timbers, e, are all 
close jointed, and laid touching, so as not to require planking in addition 
But if the bottom of the stream is gravel or earth, there must in addition to* 
these be used two thicknesses of sheet piles, p, Fig 2, &c, close driven, breaking 
joint, to a depth of several feet, to prevent leaking through the soil beneath 
the base of the dam. Frequently but one thickness is used. If the bottom is 
soft or open for a depth of only a few feet, it is at times better to remove :t and 
base the dam on the firmer stratum below ; still, however, using the sheet piles 
Old decayed timber and other rubbish should be removed from the base In 
very bad soils of greater depth, it may be necessary to support the dam entirely 
upon a platform resting on bearing piles. Here'great precautions are neces¬ 
sary against leaks; but the case occurs so rarely, that we shall not stop to con¬ 
sider it. ^ 

As to the wearing away of the bottom of the stream by the water falling over 
the front o i the dam, precautions should be used in all cases except that of very 



hard rock, or of medium rock protected by a considerable depth of water. The 
dam, Fig 1, was built upon a tolerably firm micaceous gneiss in nearly vertical 
st rai a, covered by!about2 feet of water in ordinary stages. In 39 years the rock was 
















DAMS, 


283 


worn away id front of the dam, as shown in the fig, to the average depth of 3 feet; or 'very nearly 1 
inch per year. The depth of water on the crest c, was usually from 6 to lb ins; rarely 5 or 6 ft dur¬ 
ing freshets; and but a few times duriug the whole period, 8 or 9 ft. 

At Jones's dam, on Cape Fear River; height of dam, 16 ft; front vert; 

fall, usually 10 ft, into 0 ft depth of water; the soft shale rock, iu vert strata, was, in the course of 
a few years, worn away 10 ft; and the dam was undermined to such an extent as to fall into the cavity. 

In another case, dam 36 ft high ; front vert; the water falling upon nearly vert strata of hard shale 
rock, usually covered by hut about 2 ft of water : iu about 20 years wore it to an irregular depth of 
from 10 to 20 ft; and extending from the very face of the dam’, to 70 or SO ft in front of it. 

In Fig 2, upon a stream subject to very violent freshets, the gravel was washed awav for a consid¬ 
erable width and depth beyond the apron, as at A. To prevent a repetition, the cavity was tilled 

A deposit of blocks of loose stone, of even 
a ton weight or more, will not serve as a pro¬ 
tection in front of a dam exposed to high 
freshets; but will soon be swept away. A 
common precaution against this wear, in low 
dams, is an apron, a a. Fig 2; or dd, Fig 3; 
of either rough round tree trunks, or of hewn 
timber, laid close together; extending under 
the entire base of tbe dam, and from 15 to 30 
ft in front of its face. These are sometimes 
bolted to pieces, s s, Fig 2 ; or y y, Fig 3; laid 
under them across the stream. Iu Fig 3, 
with very soft bottom, these pieces yy are 
supposed to be bolted to short piles It, driven 
for that purpose. 

At times a distinct wide low timber crib, filled with stone, and covered on top with stout plank, 
has been placed in front of the dam, to receive the fall of the water; and is effective iu protecting 
the bottom. Also, iu some cases, a dam of less height, and of cheap character, has beeu built at a 
short distance down stream from the main one, in order to secure at all times a deep pool in front 
of the latter for breaking the force. 





Another precaution is to 
substitute a sloping front like 
el, Fig 4, or such as Figs 1 
and 2 would form if reversed, 
for the nearly vert one of the 
other figs; thus ta someextent 
reducing the foree of the wa¬ 
ter. _ This, however, is but a 
partial remedy, especially 
for soft bottoms in shallow 
water; for the sliding sheet 
still descends with great foree. 
The best form of dam, per¬ 
haps, in such eases, is that 
shown in Fig 5, in which the 
front consists of a series of steps of 
about 1 vert, to 3 or 4 hor. These ef¬ 
fectually break the force of the water; 
and, with the addition of an apron oo, 
secure a satisfactory result. It is ob¬ 
jected against this form, as also 
against Figs 4 and 6, that their fronts 
are liable to he torn by descending 
trees, ice, and other bodies swept 
along during freshets; but experience shows that this objection has but little weight; for when such 
bodies pass, the sheet of water is thicker than usual; and protects the front timbers. On tbe Sch 
Nav, the timbers cl, Fig 6, scarcely wear thin at the rate of an inch in 10 to 15 years. 



The forms of wooden clams are many; (see the figs, which show 

those most used :) varying with the circumstances of the ease, and with the fancy of the designer. 
In the United States they are usually of cribwork. of either rough round logs with the bark on, or of 
hewn timber; in either case about a foot through. These timbers are merely Ja,id on top of each 
























































































DAMS. 


284 


? th ei\ forming in plan a series of rectangles with sides of about 7 to 12 ft. They are not rotched 
together but snnply bolted by 1 inch square bolts (often ragged or jagged) about 2 to 2% feet long, 
through two timbers at every intersection. These are not found to rust or wear seriously, even when 
exposed to a current Square bolts hold best. Round logs are flattened where they lie upou each 
> er. experience shows that firmer but more expensive connections are entirely unnecessary. The 

cnhs are usually, but not always, tilled with 3 3 

IVtlll/h utAllo In /l ,. ... _-__ J 

C 




Fig. 8. 


rough stone. In triangular dams, disposed 
as in Figs 1, 2, and 7, this stone tilling is 
not so essential as in other forms ; because 
the weight of the water, and of the gravel 
backing, tends to bold the dam down on its 
base. Still, even in these, when the lower 
timbers are not bolted to a rock bottom, or 
otherwise secured in place, some stone may 
be ueeessary to prevent the timbers from 
floating away while the work is unfinished, 
and the gravel not yet deposited behind it! 

Ou rock, the lowest timbers are often bolted 
to it, to prevent them from floating away 
during construction ; and when the water 

platform or flooring wil be rJn *> 6 tbem w,th s,one ; for the reception of which a rough 
^k may theX di.wnsed^Uh tL I ?’ " Utt,e « Bbo ? their lowest The bolting to the 

highernei^ ■ rttoESTwai «water may flow through the open cribwork as the building 

should happen.’ Or, cribs TolZln pUn ^VcX,TuS*?™* “ ,f “ ,reShet 

stone, may be sunk, leaving oue or more intervals, like that at oooo 
between them, for the free escape of the water. These openings to 

^The^-orkn a** 1 n- int .° them closing-cribs shaped like g »». 

The workmanship of a dam in deep water can of course be much 
better executed in coffer-dams, than by merely sinking cribs. The 

K more'closefy 8t ° Ue ^ PaCked ; lhe ^ 

W hen a very uneven rock bottom in deep water, or the introduc- 

ncdi °nr K , U1 . CeS i 1 ] n , t ‘‘ e dani - or a "- v other considerations, make it ex- 
pedient to build dams within coffer-dams, both should be carried on 
m sections; so as to leave part of the channel-way open for the es- 

iisSIPPPisgi 

backed with earth or gravel aBswer ,or coffer-dams, or rough stone mouuds, 

quarry stones with which the cribs are tilled either nartJv^rl^!* an ! OBg , tbe ® nb timbers, and rough 

a Asu°Lt°r-, *■v hed ™-*£'■™ 

cement, ^me’he".^!^?.*’E3dE fl^.y'bum hoHz'on^f f ° f * B 

back at a few feet apart with theTr Vops level witS he su^o^^ T ° f v. the slo P iBg 

rz c X g ,h r tt n rF “T r7m i# 

apro„ s in front of the 

on the 8ch Nav, was built in 1819, and served f r,8Cn 40 ft above the crests. Fig 1, 

of its timber, especially of the close-laid top ones c. rJndhTre^ T ears ’ UBtl1 in 185 . 8 the decay of much 
frontofit. It was of extremely simple construct iVn ’ e ? dered 11 necessary to build a new one just in 
hers, o o, 10 ft apart. *"*1 w ‘ lh atone - The bottom tim- 

inclined timbers as is shown in the fig The ton onl" , hn T ° Ver e " C , h ° f tbeiB ’ was Buch a seric8 of 
so as to form the top sheeting, Instead of thinner nbinhi STf’ C ° 8e J oin,t ' d ' and la > d touching, 
way. No coffer-dam was used ; but th etouZXZLJn* ?? ,eCes Ht * were ,ai '' in tbe 8nB >« 
the stringers and the sloping pie^s were added flm bo, ! ed to the rork : 10 ft a P a ~‘ 1 then 

each end of tbe dam, until at last a snaee of „,,| d ' J* B coverin * («) was carried forward from 

J-aas. Tbe close ooVering for this 8p P a « teing Then°°Jl'“ T V? Ieft in tbe ceBtcr ' for «•* "ft*? S 
work, and the space was covered go rauidlv ib.t tn. Li* 1 T ead?1 a 1 *rong force of men waJset to 
impede the operation. ^ ^ bal tbe r * ver bad “et time to rise sufficiently high to 


dropped into pl»M means iffg^ves oTgChies'of eome'ki ' Wate , r '* and^ddenl^ 

eral such timbers mav at times he flrmlv framed taLth? * ", d .l° r r « tainin 8 them in position. Sev- 
the opening or sluice at one " ‘ dr °?» ,ed at once; closing 

■ri^urs 7heXz;. t k r :rv 8ld i e ° f - & MnW 8iie - ,n soiue caBeB - a 

engineer and snperimendent or't&'ZZr™ofiHr*rUZJuflS* 1 F ~ 8m ‘th. Esq, chief 

be found in different parts of this volume. ‘ ' valuable ,nformat ion from the same source will 























DAMS, 


285 


F'g 2 is a canal feeder dam on the Juniata. Here s a are timbers stretching clear across the stream, 
(about .100 ft,) and sustaining the apron a a, of stout hewn timbers laid touching. This dam was filled 
with stone, for the retention of which the front sheeting planks were added. 

_ Fig 6 is ou the Sell Nav ; was built iu 1865. It is a form much approved of on that work, for such 
situations; namely, firm rock foundation, with a considerable depth of water in front. The highest 
♦ dam (32 ft) on the Sell Nav, is very similar to it; built in 1851. All the dams ou this work are of 
'/ hewn timber, chiefly white and yellow pine. The water occasionally runs from 8 to 12 feet deep over 
their crests; and theu overflows and surrouuds many of the abuts. The vertical back allows the 
overflowing water to leak down among all the lower timbers of the dam, and thus tend to their 
preservation. 

Fig 4 shows the dams on the Monongahela slackwater navigation ; W. Milnor Roberts, eng. They 
are of round logs, with the bark ou; flattened at crossings. The longest oues in the fig are 10 feet 
apart along the length of the dam. Experience shows that such dams possess all the strength neces¬ 
sary for violent streams. On rock, the lowest timbers are bolted to it. 

Fig 7 has been successfully used to heights of 40 ft.* 

Fig 3 is intended merely as a hint for a very low dam on Yielding bottom. Its main supports are 
piles ft, from 4 to 8 ft apart, according to the height of the'dam; and other circumstances ; and tt 
are short piles for sustaining the apron dd. It may be extended to greater heights by adding braces 
in front; which may be covered by stout planks, to form an incliued slide for the overtoiling water. 
^ Many effective arrangements of piles, and sloping timbers for dams on soft ground, will suggest them¬ 
selves to the engineer. Thus, at intervalsof several feet, rows of 3 or more piles may be driven trans¬ 
versely of the dam; the top of the outer pile of each row being left at the intended height of the crest 
while those behind are successively driven lower and lower; so that when all are afterward con¬ 
nected by transverse and lougitudiual timbers, and covered by stout planking, and gravel, they will 
form a dam somewhat of the triangular form of Fig 7. It would be well to drive the piles with an 
inclination of their tops up stream. 

There is much scope for ingenuity both in designing, and in constructing dams under various cir¬ 
cumstances ; and in turning the course of the water from one channel to another, by means of ditches, 
pipes, or troughs, Ac., at diff heights; aided at times by low temporary dams or mouuds of earth ; or 
of sheet piles, Ac; or by coffer-dams ; so as to keep it away from the part being built. Each locality 
will have its peculiar features; and the engineer must depend on his judgment to make the most of 
them. 

Abutments of (lams as a general rule should not contract the natural 

- width of the stream ; or, if they must do so, as little as possible; for contractions increase the height, 
and violence of the overflowing water in time of freshets; during which a great length or overfall is 
especially desirable. They should be very firmly connected with the ends of the dams; and should, 
if the section or the valley admits of it, be so high, and carri d so far inland, that the high water 
of freshets will not sweep either over them, or around their extremities; and thus endanger under¬ 
mining, and destruction. In wide, flat valleys they cannot be so extended without too much ex¬ 
pense; aud the only alternative is to founl them so deeply and securely as to withstand such 
action ; making their height such that they will, at least, be overflowed but seldom. Their ends 
adjaceut to the dam, should be rounded off, so as to facilitate the flow of the water over the crest. 

They are best built of large stone in cement; for although sufficient strength may be secured by 
timber, that material decays rapidly in such exposures. If of earth only, they are very apt to be 
carried away if a freshet should overtop them. 

Sluices should be placed in every important dam. in order that 

all the water may be drawn off, if necessary, for the purpose of repairs; or of removing mud deposits; 
or finding lost articles of importance, Ac. They may be merely strong boxings, with floor, sides, and 
top of squared timbers; and passing through the breadth of the dam, just above the bottom. To pre¬ 
vent trees, Ac, from entering and sticking fast in them, some kind of strong screen is expedient. In 
common cases a sluice should not exceed about 3J^ ft by 5 ft in cross-section ; otherwise it becomes 
■ r- ird to work. Two or more such openings may be used when much water is to be voided. They 
should be near the abutments. The gates or valves for opening and shutting them, should be at the 
up-stream end; for if at the lower one, accumulations of mud, Ac, will fill the sluices, and prevent 
them from working. They are usually of timber; and slide vertically in rebates; being raised and 
lowered by rack and pinion; but in very important dams they may be of cast iron. Two sets of sluices 
are desirable; that one may be always ready for use if the other is stopped for repairs. 

The part of the apron in front of the sluice should be particularly firm, so as not to be deranged by 
i the water rushing out under a high head. 

Warns are sometime*, but rarely, built in the form of an 

arch ; convex up stream. This form is strong; and when the shores are of rock 

- may be expedient to use it; but if the banks are soft, they will be exposed to wear by the curreni 
thrown against them at the abuts of the arch. 

At time* dam* are built obliquely aero** the stream, with 

the object of increasing the length, and consequently reducing the depth of water over the crest in 
times of freshets. The argument, however, appears to the writer to be of but little weight, inasmuch 
as the reduction of depth would extend but a trifling distance up stream from the dam; and would 
therefore scarcely have an appreciable effect in diminishing the injury to the overflowed district above. 

| Moreover, the increased expense is probably always more than oommensurate with any ad van tag* 
gained. 

Home dam* are subject to “tremblings,” which have not been 

satisfactorily accounted for. They exhibit themselves chiefly a9 undulations of the air, produoed by 
the falling water; and which occasionally cause a rattling of windows within a distance of % a mile 
or more. We have known this to be stopped unintentionally in one case, by building a well-oovered 


* Cost of crib dam*. With common labor at $1.50 per day ; $20 per 1000 ft 

board measure, delivered ; stone for filling. $1 per cub yard : gravel 50 cts per cub yd; iron for bolts, 
Ac, 4 cts per lb; such dams in shallow water usually cost complete, from 9 to 12 cts per cub (t; or 
$2.43 to $3.24 per cub yd of crib. 











286 


DAMS 


wide crib apron, a few feet high, against the front of the dam, for preventing the abrasion of the bot¬ 
tom. Iu other cases a series of oblique timbers placed against the front of the dam, and part way up 
it, at a slope of about ] % to 1, and covered with plank, has been perfectly effective in stopping it. 

The proper t ime for building- dams is of course at the longest period 
of low stage of water. f 

To ascertain in advance approximately, the height to 
which the water will rise above the crest of a dam: or rather. 

a little back from it; the crest being above tbe level of the original water. This will vary with the shapf 
of the crest, as may be seen by refeience to Fig 26J4, page 268, which, however, is a very peculiat 
case. Still, until we have more experiments, appreciable deviations from tbe results of such rules 
must be expected in practice. Square the disch of the stream in cub ft per sec. Call this square, s. 
Square the length of the overfall in ft. Mult this square by 7. Call the prod p. Divide s by p. Take 
the cube r't of the quot.* This cube rt will be the reqd approximate height of rise in ft. 

When, in times of freshets, the water rises above the crest to a height equal to that of the dam 
itself, there is no perceptible fall at the dam; and boats may pass in safety over the crest. 

For measuring the disch over dams, see pp 264, 267. See also Art 1 of Hydrostatics. In 

shape of a formula, the foregoing rule will be, 

Rise ~ cu i e .( 0 f /discharge in cub ft per sec 2 \ 
in ft cu \7 (length of overfall in ft 2 ). / 

When tlie (Inin is originally a 
submerged, or drowned one. as D, 

Fig 9 ; the following is a rough approximation, probably 
somewhat in excess; og being the natural level of the 
water previous to building the dam; and oc the natural 
depth in ft, of the water above the intended crest. Then 
the required depth ac. of the up-stream water, above the 
crest when built, will be, approximately, 

a c — o c-\- cube root of/ dis charge 2\ 

\ 7 X length 2/ 



Having a c, deduct o c ; and the rem will be a o, or the required rise produced by the dam.* 

The rnles given for the varying rise of surface for eonsid 
erable distances lip stream front dams ; as well as for some allied sub 

jects in hydraulics; are extremely complicated : and require much greater knowledge of mathematic 
than is usually found among civil engineers ; and so far as regards their applicaliou to theactualiue ! 
of common occurrence, the}' are probably no less useless than complicated. i 


i 

1 


Table of thickness of white pine plank required not to bent 
more than part of its clear horizontal stretch, untie t 
diiferent heads of w ater. (Original.) a 


Stretch 


Ilea 

ds in feet. 


iu Ft. 

40 

30 

20 

10 

5 



Thickness iu Inches. 


3 

3% 

3 

2% 

2% 

W, 

4 

i 

4 

3% 

2k 

2% 

6 

6 

5 % 

4% 

8 % 

8 

9 

8 

7 

3% 

4% 

10 

11 % ■ 

10 


7 

5% 

12 

13 H 

12% 

log 

8 K 


15 


15 

13 

17% 

10 % 

8% 

20 

2 M 

20 

14 

11 


* These two rules are from Kankiue, who says they apply to " crests either flati 
But that, iu itself, is very vague. 


it 


/slightly rouudec d; 


ti 
































WATER SUPPLY. 


287 


b 


WATER SUPPLY. 

p r day, to each inhabitant, is usually considered a lair ample allowance Manv 

“iTh' 1 ; * while Y«k“Kl" a „ n the” a „7 e 

pllona would probably sufiico'for any ’*3*“ 

[St 2,5 * “» - fe^rStriSs C i a p S „Se 

r *£^ r City us ? sI ? OH,<1 ««t *>e drawn from the very bot- 
S reservoir, because it will then be apt to carry alongthe sedi- 

1 lent, which not only injures the water, but creates deposits within the pipes- 
thus obstructing the flow. In fixing upoii the necessary capachy of a resenmir’ 

fii d?awin^off k .miif 1 °. consid 7:' lion ; inasmuch as all the Sate/below the level 
S,n.I a o i g n’ be re S a , rded as lost. When circumstances justify the ex- 
pen^e, it is well to curve up the reservoir end of the service main so as to nro- 
vide it with valves at different heights; for drawing off only the purest stratum 

Sallr hL b<? Ih th f reservoir - W» th this view, the valve-tower (page 289) gen- 
fu; a y haS such , va j ves communicating with the water in the reservoir; and by 
this means only the purest is admitted into the tower: and from it into the 
city pipes. This refinement, however, is rarely practicable. Such valves must 
of course beworked by watchmen. For rainfall, see p 220. 

Art. 1. Reservoirs. In important reservoirs of earth, for storing water 
to moderate depths for cities, experience appears not to sanction dimensions 
bolder than 10 feet thick at top ; inner slope 2 to 1; outer slope \y o to 1 * A toil 

norfLf “ t ? M 20 fe ? f ’ inside sl °^ s of 3 !« are ad P opt7d in some im- 

portant cases, with outer slopes of 2 to 1. Both slopes, however, are at times 

> elnw^h i The . lev . el water surface should be kept at least .3 or 4 feet 

below the top of the embankment; or more, if liable to waves. In a large 
reservoir, a quite moderate breeze will raise waves that will run 3 feet (measured 
vertically) up the inner slope. A low wall, or close fence, w, Fig. 37, is some¬ 
times used as a defence against them. The top and the outer slopes should be 
protected at least by sod or by grass. To assist in keeping the top dry, it 
should be either a little rounding, or else sloped toward the outside.f The soft 
soil and vegetable matter should be carefully removed from under the entire 
base o. the embankments; which should be carried down to soil itself imper¬ 
vious to water, in order that leakage may not take place under them. To aid in 
this, a double row of sheet piles, or a sunk wall of cement masonry, carried to 
a suitable depth below the bottom, may be placed along the inner toe in bad 
cases. If there are springs beneath the base, they must either be stopped or • 
led away by pipes. Ihe embankment should be carried up in layers slight!v 
hollowing toward the center, and not exceeding a foot in thickness; and ail 
stones, stumps, and other foreign material, such as clean gravel, sand and de¬ 
composed mica schists, Ac, that may produce leakage, carefully exclude! These 
layers should be well consolidated by the carts; and the easier the slopes are 
the more effectively can this be done. The layers, however, should not be dis— 
tinct, and separated by actual plane surfaces; but each succeeding one should 
be well incorporated with the one below. This has sometimes blen done by 
driving a drove of oxen, or even sheep, repeatedly over each laver; in addition 
to the carting. Rollers are not to be recommended, as they fend to produce 
seams between the layers. This might possibly be obviated by projections on 
the circumference of the roller. J 

Gravelly earth is an excellent material, perhaps the best. The choicest 
material should be placed in the slope next to the water; and should be de¬ 
posited and compacted with special care in that portion, so as to prevent the 
water from leaking into the main body of the dam, and thus weakening it It 
is not amiss to introduce a bench, b , Fig 37, in the outer slope, to dFminish 
danger from rainwash by breaking the rapidity of its descent. 

If the bottom of the reservoir itself is on a leakv soil, or on fissured rock 
through the seams of which water may escape, it must be carefully covered 
with from 1% to 3 feet of good puddle; which, in turn, should be protected from 
abrasion and disturbance, by a layer of gravel; or of concrete, either paved or 
not, according to circumstances. 


* The writer suggests that a top width equal to 2 feet + twice the square root 
of the height in feet, will be safe for any height whatever of reservoir properly 
constructed in other respects. ^ 1 J 

fSome engineers slope the top toward the inside. 












288 


RESERVOIRS. 




Reservoirs constructed with the foregoing dimensions, and with care, may 
remain safe lor an indefinite period; but where serious damage would result 
from failure, the following additional precautions should be taken. 
The inner slopes should be carefully faced up to the very top, with at least a 
close dry rubble-stone pitching, not less than 15 to 18 inches thick ; as a protec¬ 
tion against wash, and against muskrats. These animals, we believe, always i 
commence to burrow under water. If the slopes are much steeper than 2 to 1, I 
this dry pitching will be apt to be overthrown by the sliding down of the soft-''; 
ened earth behind it, if the water in the reservoir should for anv cause be 
drawn down rather suddenly. It will be much more effective, but of course 


.. -y..v,. . it, wm uc niucii more eiieeuve, out oi course 

more costly, if laid in hydraulic cement; and still more so if la d upon a laver 
a few inches thick of cement-and-yxavd concrete; especially if this last'be 


1 vuuivui anu-,lrt> r 1 tUUUClC j C3]JcUdllj II llllb JtlSl UG 

underlaid by a layer about 1V£ to 3 feet thick of ^ood puddle, spread over the 
face of the slope ; the great object being to protect the inner slope from actual 
contact with the water. n--* _ . *. 


.1 ~ ~ o vv ' i'* wvvv imu/i oiv|/v ii vui actual 

—iTVl —V . can he effectually accomplished, slopes as steep 

as 1 ^ 2 _to 1 w ill be perfectly secure; for the danger does not arise from any want 
of weight of the earth for resisting overthrow. Special care should be 
bestowed upon the inner toe of the slope, to prevent water from 
finding its wav beneath it, and softening the earth so as to undermine the stone 
pitching. Is ear the top. reference should be bad to danger of derangement by 
ice, trost, rain, and waves. Flat inner slopes tend not onlv to prevent the dis¬ 
placement of the pitching; but increase the stability of the embankment by 
causing the pressure of the water (which is always at right angles to the slope) 
to become more nearly vertical; and thus to hold the embankment more firmly 
to its base than if there w*ere no water behind it. Sometimes the toes of both 
the inner and outer slopes abut against low r< taining-walls in cement. This 
gives a neat-finish, and tends to preservation from injury. 

Many engineers, in order to prevent leaking, either through or beneath the 
embankment, construct a puddle-wall, p, Fig. 37, of well-rammed imper¬ 
vious soil, (gravelly clay is the 
best,) reaching from the top 
to several feet below the base. 
This wall should not be less 
than 6 or 8 feet thick on top, 



for a deep reservoir; and 


should increase downward by 
offsets (and not by slopes or 
batters) at the rate of about 
1 in total thickne-s, to 3 or 4 


in depth Other engineers object, to these puddle-wabsTand conten^thaUeTk- 
age should be prevented by making both the inner slopes, and the bottom of the 
reservoir water-tight, by means of puddle, concrete, and stone facing in cement 

is fflf'a if the embankmenl ls we " “Swructed, it 

i he other 93 high, 25 on top, inner slope 3.5 to 1, outer 3 to 1. In each the Dud 
dJe-wall is carried 47 feet deeper than the base. No stone facing. P 

It is difficult to prevent water under lii«l» pressure from 
fiudins; its way through considerable distances alon~ seams 

nil! ere *i earth e S in c '* n . tact w jth smooth rock, wood, or metal; as for* in stance 
a ong the surfaces of u-on pipes laid under reservoir embankments • or alom? 
the tie-rods sometimes used through the puddle of coffer-dams • mH iho aomf 

SSSSisJsliilpS 

tides in muddy water; depending on the depth of the reservoir par ‘ 

t0 thG b ? ttom of the reservoir should be^rervided °" e 0r 

H , n Reservoirs. The reservoirs of the New River Water To Inn 

don, England, were unc.eaned for 100 years, during wh ch mud 8 f^t deep was 


« 


i: 

e 

It 


t 


h 













RESERVOIRS. 


289 


fmin S m e f d rn?n W, an “ uaI . 1 3 r * At Philadelphia it is about .25 inch per 

«.“" m fr0I , u the Schuylkill, and 1 inch from the Delaware River. At St Louis 

reservoirs 4 f £ 6et £ Gr ye fj Ve S etation is a P fc to take place in shadow 

reservoirs and near the edges of deep ones, especially in veiy warm weather- 

\ a "d the plants, on decaying, injure the water. e> 

ins 1 nJlSo r w 0 ThJ , w thr0 * ,,?h *«arsh lands is sometimes unfit for drink- 
« 1 h I°» r in . stailce - 111 Solue sections of the Concord River, Massa- 

} Bosto^ L* Kf by the enuT l ent hydraulic engineer, Loammi Baldwin, of 

Boston, to be absolutely poisonous from this cause. 

i ie construction of a large deep reservoir is not only a verv costly but a 
Sn^fbuTlmi u , ndertakln S- With every watchfulness and cafe, it. hf almost 

P m , 2 ° P reven t leaking; although this may not manifest itself 

^r months, or even years Should a break occur, especially near a city, it 
ld fi p ™ bab ^ly be attended by great loss of life and property. If the water 
dnd8 2 ts wa y ! n a s t tr f a,1 b either across the unpaved top, or through the 
certain 16 em ^ ank,ueD ^» ^he rapid destruction of the whole becomes almost 

la ‘ . 1 Storing Reservoirs. The entire annual yield of a stream 
be n a UCh . - m ° re 1 ian s , ufficient f °r supplying a certain population with 
water, and yet in its natural condition the stream may not be available for this 
purpose, because it becomes nearly dry in summer, when water is most needed: 
white, at other seasons, the rains and melted snows produce floods which supply 
vastly more than is required; and which must be allowed to run to wastef A 
scoring reservoir is intended to collect and store up this excess of water, so that 
it may be drawn off as required during the droughts of summer, and thus 
equalize the supply throughout the entire year. This, when the locality per¬ 
mits, is effected by building a dam across the stream, to form one side of the 
reservoir; while the hill-slopes of the valley of the stream form the other sides. 
I he stream itself flows into this reservoir at its up-stream end. When the 
stream is liable to become nearly dry during long summer droughts experience 
shows that the capacity of the reservoir should be equal to from 4 to 6 
months supply, according to circumstances. During the construction of the 
dam, a free channel must be provided, to pass the stream without allowing it 
to do injury to the work. If the dam were built precisely like Fig 37, entirely 
of earth, it would plainly be liable to destruction by being washed away in case 
the reservoir should become so full that the water would begin to flow over its 
top. To provide against, this we may, by means of masonrv, or of cribs filled 
with broken stone, or otherwise, construct either the whole, or part of the dam, 
to serve as an overfall, or a waste-weir. Or a side channel (an open cut, 
pipes, or a culvert, &c) may be provided at one or both ends of the dam, and in 
the natural soil, at such a level as to carry away the surplus flood water before 
it can rise high enough to overtop the earthen dam. Besides these, and the 
pipes for carrying the water to the town, there should be an outlet, with a valve 
or gate, at the level of the bottom of the reservoir; in order that, if necessary 
for repairs, or for cleaning by scouring, all the water may be drawn oft'. The 
entrances to the city pipes should be protected by gratinsrs, to exclude fish Ac 
To facilitate repairs or renewals of all valves, «fcc, which 
are under water, the reservoir ends of the pipes or culverts to which they 
are attached, may be surrounded by a water-tight box or chamber, which will 
usually be left open to the reservoir ; but may be closed when repairs are re¬ 
quired. Access may then be had to them by entering at the outer end, after 
the water has flowed away from inside. In case the outlet is through a long 
line of pipes which cannot thus be entered, a special entry for this purpose may 
be cast in the pipe itself, near the outer toe of the embankment.; to be kept 
closed except in case of repairs. Sometimes a better, but more expensive means 
of access to such valves, is secured by enclosing them in a valve-tower of 
masonry. This is a hollow vertical water-tight chamber, like a well; but near 
the toe of the inner slope; having its foundation at the bottom of the reservoir; 
whence the tower rises through the water to above its surface. This chamber 
is provided with valves or gates usually left open to the reservoir; but which 
may be closed when repairs are needed ; and the water in the tower allowed to 
escape from it through the open valves of the outlets. This done, workmen can 
descend through the tower by ladders from the aperture at its top. 

At times the outlets for the discharge of surplus flood water are, like those for 
scouring, placed at, or just above, the level of the bottom of the reservoir. In 
order that these may work in case of a sudden flood at night, &c, they must be 
furnished with self-acting valves, which will open of their own accord when the 
flood is about to rise too high. This may be effected by attaching them to floats, 
the rising of which, w-hen the water is high, will pull them open. All such out¬ 
lets should be large enough to let men enter them for repairs. They should by 






290 


WATER-PIPES. 


no means be laid through the artificial earthen body of the dam itself, without 
being supported upon masonry reaching down to a firm natural foundation ; 
otherwise they are very apt to be broken by tlie subsidence of the embankment. 

It is usually safer to carry them through the firm natural soil near one end of 
the dam. i^heir valves, if only single, should be at their inner or reservoir end, 
so as to leave the outlets themselves usually empty, for inspection ; but it is 
better to have two valves, so that one may be used when the other needs repair ; 
and in this case one may be placed at each end. Reservoirs which are supplied 
bv pumps, need no precautions against overflow; because the pumping is 
stopped when they are filled to the proper height. Large storing reservoirs 
necessarily submerge more or less land, which has therefore to be purchased. 
Bv intercepting the descending water, they frequently prevent spring floods 
from injuring low lands farther down stream. If there are mills down stream 
from the reservoir, they would evidently be deprived of water for driving them, 
unless a portion of that stored in the reservoir be devoted to that purpose. 
Water thus applied to compensate for the loss of the natural stream, is called 
compensation water; and the reservoir, a compensating one. 

Art. lb. Distributing: reservoirs. Frequently a valley fit for a storing 

reservoir can be found only at a long dist (sometimes many miles) from tlie town; and it then be- 
comes expedient to construct also an additional one of smaller size than the storing one, near the 
town ; and at as great an elevation above it as circumstances will permit; but lower than the storing 
one. This is called, by way of distinction, a distributing reservoir, because from it the water, after 
having flowed into it from the storing reservoir, through the long supply pipe w liich connects them, is 
distributed in various directions through the town, by means of the street mains, or pipes. This 
small reservoir should hold a supply sufficient at least for a few days; a. few weeks would be better; 
and the end of the supply pipe which terminates in it, should be provided with a valve for shutting 
off the supply from the storing reservoir. These precautions permit repairs to be made along the line 
of supply pipe without depriving the town of water in the mean time. With a view to such repairs; as 
well as to scouring out sediment from the supply pipe, this last should be provided with OH t lot 
valves at various low points along the entire interval between the two reservoirs ; especially at 
those at which the valves may disch into natural watercourses. On opening these valves, the out- j 
rush of the water carries away sediment; and leaves the pipe empty for inspection. 

In fixing: upon tlie <liams of pipes for supplying cities, it is necessary 
to bear in mind, that by far the greater portion of the 24 hours' yield is actually drawn from them 
during only 8 to 12 hours of daylight; and therefore the capacity of the pipes must be sufficient to 
furnish the dailv supply in much less than 24 hours. Again, during the hot summer months, much 
more water is used than during the winter ones; and this consideration necessitates a still larger diam. 

Art. 2. Systems of street pipes for supplying: cities. The 
writer knows of no practical rules for proportioning the diams for such systems. The various com¬ 
plications involved, render a purely scientific investigation of little or no service. With much hesi¬ 
tation, he veutures the following purely empirical rules or his own ; based on such limited observa¬ 
tions as have casually fallen under his notice. ..... r 

Rule 1. When, at no point in a system of city pipes, is the head, or vert dist below the. surface 
of the reservoir, compared with the hor dist from the reservoir, less than at the rate, oj bO ft per mile, 
then the population in the last column of the following Table A, may be abundantly supplied, for all 
city purposes, by either one pipe of the inner diam or bore in the 1st col; or by 2, 'i, <6c, pipes of the 
diams in the other cols. These diams are given to the nearest safe % inch. The supply is assumed 
to be about 60 gallons per day to each Inhabitant. 


TABLE A. (Original.) 


1 

2 

NUMBER 

3 | 4 

DF PIPE 
6 

S. 

8 

12 

24 


Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 


Ius. 

Ins. 

Ius. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 


6 

4% 

3% 

3% 

3 

2% 

2% 

1% 

1647 

8 

6% 

5% 

4% 

4 

3% 

3% 

2% 

3465 

10 

7% 

6% 

5% 

5 

4% 

3% 

3 

5908 

12 

9% 

7% 

7 

5% 

5% 

4% 

3% 

9324 

14 

10% 

9% 

8% 

7 

6% 

5% 

4% 

13706 

16 

12% 

10% 

9% 

7% 

6% 

6 

4% 

19141 

18 

13% 

11% 

10% 

8% 

7% 

6% 

5% 

25677 

20 

15% 

13 

11% 

9% 

8% 

7% 

5% 

33426 

22 

16% 

14% 

12% 

10% 

9% 

8% 

6% 

42433 

24 

18% 

15^ 

13% 

11% 

10% 

9 

6% 

52671 

26 

19% 

16% 

15 

12% 

11% 

9% 

7% 

64447 

28 

21% 

18% 

16% . 

13% 

12% 

10% 

8 

77565 

30 

22% 

19% 

17% 

14% 

13% 

11% 

8% 

91580 

32 

24% 

20% 

18% 

15% 

14 

11% 

9 

108160 

34 

25% 

22 

19% 

16% 

15 

12% 

9% 

125840 

36 

27% 

23% 

20% 

17% 

15% 

13% 

10% 

144480 

40 

30% 

25% 

23% 

19% 

17% 

15 

1 l ^ 

188320 

44 

33% 

28% 

25% 

21% 

19% 

16% 

12% 

239600 

48 

36% 

31 

27% 

23% 

21 % 

18 

13% 

297600 

54 

41 

34% 

31% 

26% 

23% 

20% 

15% 

391200 

60 

45% 

38% 

34% 

29% 

26% 

22% 

17 % 

511200 

66 

50% 

42% 

38% 

32% 

29% 

24% 

18% 

650400 

72 

54% 

46% 

41% 

35% 

31% 

26% 

20% 

800000 

80 

60% 

1 51% 

46% 

1 39% 

35% 

29% 

22% 

1064000 










































WATER-PIPES. 


291 


It is well to allow in addition from % inch to 1 inch, or more, (depending on 

Tim waSrXW tu e * C * h d l ,ameter 5 for ^posits and concretions? 

I lie water, after reaching the city through one or more large main pipes from 

the reservoir, must be distributed through the streets by means Sf smalle 
-ijnains branching from the larger ones. The diameters o/these smaller ones 
a so may be found by Table A. Thus, if a street with its alleys X , 

about. 6000 persons, (the rate of head being, as before, not less than 50 feet to a 
mile at any point of the system,) then wl’see by the tableThat a 10-inch p?pe 

diameten er ' 1 W ° l ‘ ld ^ W6 l ° lay D ° City S ' reet pipes of less than 6 in « hes 
Mains which cross each other should he connected at snmo 
® ,eir intersections, to allow the water a more free circulation through¬ 
out the entire system ; so that if the supply at any point is temporarily cut off 
from one direction by closing the valves ior repairs, or is diminished by exces¬ 
sive demand it may be maintained by the flow from other directions 

nnt holesomf a<1 e,#ds wben posslb,e ’ as tl *e water in them becomes foul and 

2 ‘ , ] f. Uh S( ] me diameters , different rates of head will supply the proper - 

m C0 ! M7 ?f n 3 °f Table B - 0r > 10 fi nd lhe diameters which at different 
\ales of head will supply the same populations given in the last column of Table A 
nultiply the diameter given in Table A, by the corresponding number in col¬ 
umn 4 of Table B; or (approximately) do as directed in column 5. 


TABLE B. (Original.) 



Col. 1. 

Col. 2. 

Col. 3. 

Col. 4. 

Col. 5. 

Rate of Head, 
in Feet per Mile. 

Rate of Head, 
compared with 
that in Table A. 

Proportionate 

Populations. 

Proportionate 
Diam. to supply 
the Populations 
in Table A. 

Remarks. 

5 

.1 

.32 

1.58 


10 

.2 

.45 

1.37 


121* 

.25 

.50 

1.32 

Add one-third. 

15 

.3 

.55 

1.27 

Add full one-fourth. 

20 

.4 

.64 

1.20 

Add one-fifth. 

25 

.5 

.71 

1.14 

Add one-seventh. 

30 

.6 

.78 

1.11 

Add one ninth. 

35 

.7 

.84 

1.07 

Add one fourteenth. 

37^5 

.75 

.87 

1.06 

Add one-sixteenth. 

40 

.8 

.90 

1.05 

Add one-twentieth. 

45 

.9 

.95 

1.02 

Add one-fiftieth. 

50 

1.0 

1.00 

1.00 


75 

1.5 

1.23 

.92 

Deduct one-thirteenth. 

100 

2.0 

1.41 

.88 

Deduct one eighth. 

125 

2.5 

1.59 

.83 

Deduct full one-sixth. 

150 

3.0 

1.73 

.80 

Deduct one-fifth. 

200 

4.0 

2.00 

.76 

Deduct nearly one-fourth. 

250 

5.0 

2 25 

.73 

Deduct nearly two-sevenths 

300 

6 0 

2 46 

.69 

Deduct three-tenths. 

400 

8.0 

2.83 

.66 

Deduct full one-third. 

500 

10.0 

3.18 

.63 



Example. By Table A we see that with the rate of head of 50 feet per 
mile, a 30-inch pipe will supply a population of 91580; but with three times that 
ate of head, or 150 feet per mile, we see by column 3, Table B, that the same 
pipe will supply 1.73 times as many persons, or 91580X 1.73= 158433 persons. 
But if, at this greater rate of head, we stiff wish to supply only 91580 persons, 
t hen we find in column 4, Table B, that we may diminish the diameter of the pipe 
trom 30, down to 30 X .80 = 24 inches; or, by column 5, we have 30 — 6 = 24 
inches. 

Again, after the water has reached the city by the 30-inch pipe of Table A, 
if we wish to distribute it through the city by say eight branches or smaller 
mains, we see by column 6,’Table A, that each of them must have at least 13% 
inches diameter, p’rom these eight, other smaller ones may branch off into the 
cross streets, alleys, Ac; and in estimating the supply required for any partic¬ 
ular street main, w r e must evidently add what is required also for such cross 
streets, Ac, Ac, as are to be fed from said main. 

If certain limited parts of a city pipe system have considerably less rates of 
head than most of the remainder, it may become expedient to supply the former 
by a special separate main of larger diameter; which may start either directly 

























292 


WATER-PIPES. 


1 


from the reservoir; or as a branch from the grand leading main which feeds the 
lower parts, according to circumstances. 

It must be remembered, that although by increasing the diameters, an abun- ; 
dant supply may be obtained under a small rate of bead, as well as under a great, i 
one, yet the water will not rise to as great a height in the service pipes for sup-^ 
plying the different stories of dwellings. &c. Even with the diameters in Table ' 
A, the water, under ordinary use, will not rise in these pipes to the full height 
of the surface of the reservoir; and if an unusual drawing-off is going on at 
the same time at many parts of the system, as in case of an extensive fire, or 
frequently during the hot summer months, it may not rise to even one-half of 


that height. 

Art. 3. Tlie following' has been found very effective for 
preventing concretions in w r ater pipes. Formerly in Boston, cast 
iron city pipes, 4 inches diameter, became closed up in 7 years; and those o 
larger diameter became seriously reduced in the same time. But later, during 
8 years, in which this varnish was used, no concretions formed. 


Coal-pitch varnish to be applied to pipes and castings, 

made for the Water Department of Philadelphia, under 

the following conditions: 

First. Every pipe must be thoroughly dressed and made clean, free from the 
earth or sand which clings to the iron in the moulds: hard brushes to be used 
in finishing the process to remove the loose dust. 

Second. Every pipe must be entirely free from rust when the varnish is ap¬ 
plied. If the pipe cannot, be dipped immediately after being cleansed, the sur¬ 
face must be oiled with linseed oil to preserve it'uutil it is ready to be dipped: 
no pipe to be dipped after rust has set in. 

Third. The coal-tar pitch is made from coal tar, distilled until the naphtha 
is entirely removed, and the material deodorized. It should be distilled until it 
has about the consistency of wax. The mixture of five or six per cent of linseed 
oil is recommended. Pitch which becomes hard and brittle when cold, will not 
answer for this use. 

Fourth. Pitch of the proper quality having been obtained, it must be care¬ 
fully heated in a suitable vessel to a temperature of 300 degrees Fahrenheit., and 
must be maintained at not less than this temperature during the time of dip¬ 
ping. The material will thicken and deteriorate after a number of pipes have 
been dipped; fresh pitch must therefore be frequently added; and occasionally 
the vessel must be entirely emptied of its old contents, and refilled with fresh 
pitch ; the refuse will be hard and brittle like common pitch. 

Fifth. Every pipe must attain a temperature of 300 degrees Fahrenheit, before 
it is removed from the vessel of hot pitch. It may then be slowly removed and 
laid upon skids to drip. 

All pipes of 20 inches diameter and upward, will require to remain at least 
thirty minutes in the hot fluid, to attain this temperature; probably more is» 
cold weather. 

Sixth. The application must be made to the satisfaction of the Chief Engineer i 
of the Water Department: and the material be subject at all times to his ex¬ 
amination, inspection, and rejection. 

Seventh. Payment for coating the pipes will only be made on such pipes as 
are sound and sufficient according to the specifications, and are acceptable inde¬ 
pendent of the coating. 

Eighth. No pipe to be dipped until the authorized inspector has examined it 
as to cleaniug and rust; and subjected it thoroughly to the hammer proof. Ie ! 
may then be dipped, after which, it will be passed to the hydraulic press to meet 1 
the required water proof. ; < 

Ninth. The proper coating will be tough and tenacious when cold on the 
pipes, and not brittle or with any tendency to scale off. When the coating of \ 
any pipe has not been properly applied, and does not give satisfaction, whether 
from defect in material, tools, or manipulations, it shall not be paid for; if it 1 
scales off or shows a tendency that, way, the pipe shall be cleansed inside before 
it can be rccoated or be receivable as an ordinary pipe. 




_I 



WATER-PIPES. 


293 


Art. 4. The pipes are laid to conform to the vert undulations of the street sur¬ 
faces. The tops of the pipes are laid not less than 3% feet below the surface of the 
street; but in 3-inch pipes the water has at times been frozen at that depth. 

In Philada, in 1885, there are about 784 miles of street 
pipes; or about 1 mile to every 1100 inhabitants. The population is about 
860,000; residing in about 150,000 dwellings. 

No galvanic action has been observed where lead pipes or brass unite with 
cast-iron ones. No pipe less than 6 inches diam should be laid in cities; and 
even they only for lengths of a few hundred feet. Their insufficiency is chiefly felt in 
case of fire. 8 ins would be a better minimum. No more leakage occurs in winter 
than in summer; except from the bursting of private service-pipes by freezing. 

To compact the earth thoroughly against the pipes excludes air, and greatly im¬ 
pedes rust. Pipes may be corroded by the leakage of gas through the body as well as 
through the joints of adjacent gas-pipes. 

For thickness of metal pipes to resist safely the pressure of various 
heads, see p 233 of Hydrostatics. 


WEIGHT OF CAST-IRON WATER-PIPES 

As used in Phila, and tested by hydraulic press bofore delivery to an internal 
pres of 300 lbs per sq inch. This table includes spigots, and faucets or bells. The 
pipes are required to be made of remelted strong tough gray pig iron, such as may 
be readily drilled and chipped; and all of more than 3 ins diam to be cast vertically, 
with the bell end down. Deviations of 5 per cent above or below the theoreti¬ 
cal weights, are allowed for irregularities in casting, which it seems impossible to 
avoid. • 

The pipes are in lengths from 3 to 3% ins longer than 12 ft; so that when laid they 
measure 12 ft from the mouth,/, Fig 38, of one bell to that of the next. 


Diam. 

Thick¬ 

ness. 

Wt per 
length. 

Diam. 

Thick¬ 

ness. 

Wt per 
length. 

Diam. 

Thick¬ 

ness. 

Wt per 
length. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

3 

5 

153 

16 

% 

1322 

36 

u 

4334 

4 

211 

20 

it 

1654 

36 


4862 

6 

7 

385 

20 


1798 

36 


5366 

8 

»< 

460 

30 

IS 

3313 

48 


7282 i 

10 


667 

30 

.9 

3610 

48 

m 

8667 

12 


899 

30 

1 

3964 

48 

9378 


Price, Phila, 1888, about $30 to $36 per ton of 2240 pounds, depending on size 
and quantity ordered. Elbows, Connections, Ac, about 75 to 100 per cent. more. In 
ordering anything by the ton, be careful to specify the number of lbs. (2240). 
This prevents mimnderstandings. 

The following sizes of lap-wel«le<l wronght-iron water-pipe are 

made by the National Tube Works Co, McKeesport, Pa, and fitted with their 
44 Converse lock-joint.” One end of each leugth of pipe has the lock-joint 
permanently attached (leaded) to it at the works before shipping. The “weights 
per foot” include these joints. The weight of “lead per joint” given is that re¬ 
quired to be poured iu laying the pipe, or that for one side only of the joint. 


Outer diam, ins. 2 3 

Weight per ft, lbs. 1.86 3.48 

Ecad per joint, lbs. % 1^ 

Average car load t 

Number of lengths. 800 380 

“• “ feet.. 11500 5600 


4 

5 

6 

8 

10 

12 

16 

5.26 

7.33 

8.76 

13.20 

17.08 

25.12 

47.70 

2/4 

3% 

3& 

5% 

6 

8-% 

16 

275 

145 

126 

128 

80 

56 

40 

4500 

2600 

2000 

2000 

1200 

800 

630 


The pipes are tested for a bursting pressure of 500 lbs per square inch, or higher 
if desired. They are furnished either coated with asphaltum, or “kala- 
meined;” or, if desired, first kalameined and then coated with asphaltum. 
Kalameining consists in “incorporating upon and into the body of the iron a non- 
corrosive metal alloy, largely composed of tin.” The surface thus formed is not 
cracked by blows, or by bending the pipe, either hot or cold. 

20 































294 


WATER-PIPES. 


The joint, oroonphng, is of cast-iron, and has interna? recesses which receive and 
hold Inga on the outside of each length of pipe, near each of its ends. The joint is 
then poured with lead m the usual way (see next page), either with clay collars, or 
.£T ing clH “P fu r n ! shed tl)e Co. This clamp resembles the 
, J .1 * V ***?*. '* 9 Ac ’ exce P t tljat 14 is in two rigid semi-circular pieces, connected 
toge her by a hinge-joint, and furnished with handles like those of a lemon-squeezer 
and has a hole in one side for pouring. The coupling forms a flush inner surface 
villi the pipe at the joint, thus avoiding much of the resistance of cast-iron pines 

Where rt may b . e news8ar y to make frequent changes the coup- 
Jings are made in two pieces, which are bolted together by flanges. P 

nf ii T? USri, . t " a ^ 0n ’ for , > > . i,,es * has the P r, ' at advantages over cast-iron 

di r-fhim ^,W g toT 88 m nd P l Ia,i ' ity - The hghtness of wrong!,t-iron pipes ren- 
I r them easier to handle, and cheaper per foot notwithstanding that their cost per 

or f om ,l;h P h r C n nt S ™'T\ T,, » y are not liable b reaka|e in transposition 
Tl.Iv ?har ? Kh hivndlll, g’ and they may he bent through angles up to about 25° 
m th . p re re< l ul *. e no special bend castings for such angles. The National Co 
1 1!!. , ,liL be ,!l d 1 I .o g i mac . h 1 n a S ’ 10 bewo,ked b y two n,eu - One machine can, by changing 
18 Winst^If of^ffp d t" g *• m 208 of p,pe ’ TI,e p ‘J ,es are lengths of from 15 to 
required pel- mile. 1U ° ^ ca8t * iron ’ 80 that ftwer are 

Hie ( ° furnish special “service clamps” and tapping machines for nttacliinz? 
service pipes to mains. This may he done (as in the case of the Payne 

?ron h saddlT M , h r e W V e . Inain is ,lnder pressure. The service clamp is a cast- 

iion saddle which, before the main is tapped, is attached to it by means of a U 

holt and which remains permanently so attached after the tapping. A sheet-lead 
placed between clamp and main. The clamp has a tapped cylindrical 
opening through it, into which the corporation stop (see page 299) is screwed before 
the pipe is tapped The drill of the tapping machine parses through the stop ami 

asr&sisssss K n tl,e c,awp ’ aiid ddi,s u - ugh ^‘iiaM 

c^ S X!l^U^ and Special (reducers, 


**?.*.?#*« ^ ro "S ht : iron Pipes corrode much more rapidly than cast 

an imcruaipf^I f'nll'ttf 'K■ ^r's^iu ’I”' 1 % ,ncl ' lK)re » has sustained safely 

siightly at 837 lbs. iT^l a mlteria bead , . merely swelled 

was sunk in the East River. New York to trim »)'«» r*? 8 >ore ; al)0ut n ^ thick, and 1350 ft long, 
down by weights. It proved unsatisfootorv nu i. r \ (ro ' ou . Wi4 ter to Blackwell s Island. It was held 
from the anchors of dragging vessels wranninirorSanv °° ca, « ed . b y lidal currents, and injury 

pr, ^nV 

and (las Pipe^o. of Jersey City, N*j*'** tTaOornmd^ rVvMef’h ** “* a * ie The Patent w »ter 
into and coated with, a hot mi.xlureof coal tar and asphaU^ThHinme or htaM 01 !- le ' lpth is . di,,,>ed 
applied. This ranges, in thickness, from ^ inch for i\» i„.h .; ? dra ulic cement is then 

pipe is made up to diams of 36 ins it is laid in.iia , ,uch PM** to 1 inch for 20-inch pipe. This 
the same. Suitable means are provided for makim/nlfThe'ht mortar, and completely covered with 
for water and gas. More than 1300 miles or it aie in i.sem f tachmcnts - *c, required in city pipes 
ami it appears to give general satilS^ «* for Solars ; 

do in cast-iron ones. There is every reason to , ln * in tliese pipes, as they are apt to 

dug the Jersey City Co furnish pipes and lay them7iuclud?nft7e oe~ ent?*^ TUe UCnCheS bei “ g 

pres.lures^ofSSo^^o'lbs^pe/^sq^hmhhthey^furii'is^h'^eiuler^riSSwilfOf* P^pPS. For 

ami from 1 hi to 4 ins internal diam ; or round pipes 1 inch ^^1!°, 7 ‘ US S, ] U! ‘ ,e externall F. 
asphaltum cement. At their ends, both the sou-ire and .t!,? , ■ b e ' C01 -‘ ed externally with 

pressures from 40 to 160 lbs per sq Inch, the round wooden .a,' d , P ?° 3 v™ banded wilb iron. For 
are spirally wrapped, by steam power, with hoop iron which f .’ ef 5 >re be >ng coated with cement, 
of coal-tar. The iron is wound so tichtlv as to he m.i^i ' ^ rst P assed through a preparation 
flush With that of the wood, The m'dsof eachoe.hfni'e lh ° lcavia « ** outer surface 
cement coating is then applied. These pipes have 8 been eJmn^'T® eXt s ra bandin g- The asphaltum 

years ago. are frequently quite sound and still'«**r,! r ,?*’ 1,0 ,W *p’ 1:11,1 1,1 Phihda 50 to f>0 
" hen tins is removed, many of these old pipes h^ave been rePud\n f° r ? *. u,er . sa P wood >-s decayed, 
around wooden pipes, excludes the contact of air and th.w "k ° r ' CS ’ * C ' C,a f wel1 Packed 
Lo^o porous soils, such as grave,. Ac. on the contra^. $? U S££X£ 10 Uwir duraW »^ 

have been used for both water and’ g'a's!*''*y a^mu^le?! , !!' epare< ? 1 ” nder b r reat pressure, 
not weigh or cost more than about half as mucV piwt il l 6 to than cast-iron, and do 
K-st stratus of 320 lbs per sq inch; equal to a wa Jr head of 507 £” H ^ tUick ’ have resisled 






WATER-PIPES. 


295 


Art. 6. Fig 38 isastantlard form of pipe-joint, Phila, 1886. The 
clear distance, d, between the spigot and the faucet, is nearly uniform for all sizes 
of pipe, varying only from T 5 B inch for 4-inch 
pipe, to t 7 5 inch for 30-inch pipe. The depth, 
m n, of the faucet varies from 3 ins in 4-iuch 
pipe, to 4 ins in 30-inch pipe. 

The small beads at s and m, s' and m' on the 
spigot end of the pipe, project about % inch; 
and are to prevent the calking material from 
entering the pipe. The calking consists of about 
1 to *2 ins in depth of well-rammed, untarred 
gasket, or rope yarn; above which is poured 
melted lead, confined from spreading by means 
of clay plastered around the joint. The lead is 
afterwards compacted by a calking hammer. 

The lead is poured through a hole left in the 
clay on the upper side of the pipe. In large 
pipes two additional holes are left in the clay, 
one at each side of the pipe, and lead is first 
poured into the side holes by two men at once, 
one man pouring into each side hole until the 
joint is half full. The side holes are then 
stopped, and, after the lead already poured has 
hardened, the two men finish the pouring by 
means of the top hole. This course is necessary, 
because the great weight of melted lead in the 



tj 


.SCALE OF INCHES, 

2 3 4 5 6 

■Fig. 3S 


8 


entire large joint would press away the clay at the lower side of the joint, and 
thus escape. 

The moisture in the clay is liable to freeze in cold weather, and to render it too 
hard to be used. It is also liable, at all times, as is also any dampness in the pipe, 
to be converted into steam by the heat of the melted lead. The steam sometimes 
breaks out, or “ blows” through the clay, allowing the lead to escape. 

Art. 7. The Watkins patent “Pipe Jointer ” avoids these difficulties by- 
dispensing with the ring of clay. It consists of a ring R, Figs 39 and 40, of square 



cross-section, and made of packing composed of alternate layers of hemp cloth and 
India rubber. This ring is encircled by one or more thin strips of spring steel, 
which are riveted to it at intervals, as shown. E E are iron-elbows riveted outside 
of the steel bands. After the gasket has been rammed into its place, the ring is 
placed around the spigot near the faucet, in the position shown in Fig 40, and is 
held loosely by the clamp, Fig 41, one point of which enters a small pit in each 
of the elbows, E E. The ring is then, by means of a hammer, driven close up against 
the end,/, of the faucet, Fig 38: the screw of the clamp is tightened somewhat, so 
as to bring the ring close to the spigot; a small dam of clay is placed in front of the 
aperture between the two elbows, E E; and the joint is ready for pouring. After 
the lead has hardened, the “jointer” is removed, and is ready for use at another 
joint. One can be used for several hundred joints. They of course dispense with 
the services of the men who prepare the clay collars, and supply them to the pour- 
ers. Upon the removal of the “jointer” the lead is found smooth, requiring no 
chipping, as it is apt to do when poured in the ordinary way. Ono of these jointers, 
for 4-inch pipe, costs, 1888. $3.50; 6-inch. $5; 8-inch. $6.50; 12-inch.$9; 16-inch, $12; 
24-inch, $1836-inch. $27; 48-inch, $32, etc. Thos. Watkins, Patentee and Solo 
Manufacturer, Johnstown, Pa. 




























296 


WATER-PIPES. 


Art. 8. As a further preventive against the escape of any of tile gas¬ 
ket into the |>i|>c, Mr Chas. G. Darrach, Hydraulic Engineer, Phila, places a 
ring of lead pipe in the joint before the gasket is inserted. This lead pipe is of 
such diameter that it can just be pushed through the space, d, Fig 38, between tlie 
spigot and the faucet; and of such length as just to encircle the water-pipe. It is 
driven sis closely as possible into the narrow annular space at o o, Fig 38. The gas¬ 
ket is then rammed in. and the lead poured, as usual. 

Art. !). John F. Ward’s tlexible joint for pipes laid across the irreg¬ 
ular bottoms of streams, is shown at Fig 4 ‘2. A portion, a o, of the inside of the 


faucet F, is bored out truly to form the middle zone 
of a sphere; and the spigot end, e n, of the other 
pipe is cast with two raised collars, n and e. The 
inner collar, o, is of such a height as barely to al¬ 
low it to pass into the faucet. The outer one, e, is 
a little lower, so as to allow melted lead (shown 
black) to be poured in at a. The outer edge or 
diarn of the spigot end at o is carefully turned so 
as to fit the turned spherical zone; so that the 
joint will admit of considerable play without dan¬ 
ger of leaking. In laying the pipes under water, 
the joints are filled with melted lead, as usual, on 
board of suitable vessels or floats. As fast as they 
are thus filled, the floats are moved forward, and 
the pipes, if small, and the water shallow, are 



passed into the water without further care. But for large pipes in deep water, suit¬ 
able apparatus is used for lowering them without undue strain on the joints. Mr 
Ward has been perfectiy successful in laying this pipe under water, in one case 40 
feet deep. One of the mains of the Philada water-works was thus laid across the 
Schuylkill River. 

In some cases preliminary dredging may be expedient, to diminish abrupt irregu¬ 
larities of the bottom. 

Art. 10. In Figs 43, A is a double braneli: which is a pipe having, in 
addition to the faucet, c, at one end, two others, s and to which pipes leading in 
opposite directions (as at cross-streets) may be attached. If either s or i be omitted, 
the pipe is a single brand*. The pipe is stronger when these extra faucets 
are near its end, than if they were at its middle. In a long line of pipes, for the sake 
ot expedition, different gangs of men are frequently laying detached portions some 
distance apart; and when two ends of different portions are brought near enough 
together to be united, as A and r, Fig C, their junction cannot be effected by the 
usual spigot-and-faucet joint. In this case a cast-iron sleeve, 11, is used, which is 
first siid upon one of the pieces of pipe; and,(after the other piece also is laid) is 
slid back into the position in the fig, so as to cover the joint. Sleeves are usually 



A 


B 


C 


D 


about a foot long; as thick as 
the pipe; and their diam is suf¬ 
ficient to allow the usual joint 
of gasket and lead. There is 
of course such a joint at each 
end of the sleeve. 


Art. 11. When h 



S 


c 


Figs. 43. 


c crack occurs in a pipe, 
a a. Fig B, already in use, it is 
repaired by means of a cast- 
iron sleeve, g g, made in two 
parts, bolted together by means 
of flanges as at n n. In other 
respects it is like the preceding 
sleeve. The intermediate white 
ring is the lead joint. If the 
crack is too long, or otherwise 













WATER-PIPES. 


297 


Cracks may at times be temporarily repaired in an emergency, by a wrapping of 
folds of canvas thoroughly saturated with white-lead paint; and tightly confined to 
the pipe by a spiral banding of thin hoop-iron or wire. 

Or. by an iron band, made in two parts, B B, Fig 44, 
and clamped together by screw-bolts, S S. Such bands 
are useful, also, for strengthening pipes that are con¬ 
sidered to be in danger of bursting. 

To attach a pipe, e, Fig 43, to one, /, 
already in use, but in which no provision has 
been made for such attachment, a piece may be cut 
out of/, as atiM), and a casting, e, furnished with 
flanges, m m, bolted over the opening, by screw-bolts 
passing through female screws tapped in the thickness of the pipe. If the new 
pipe is so large that the opening, v v, if circular, would be inconveniently wide, 
it may be made oval, with the longest diameter in the direction of the length of 
the pipe,/. In that case the casting e will be oval at its flanges; and circular 
at c c. 

Art. 12. The following table, arranged from data kindly furnished by the late 
Isaac Newton, C E, Ch Eng, Dept of Pub Wks, New York, and his Asst, Mr. James 
Duane, gives the average prices, «&c, of pipes and laying', in that 
city', for three years prior to 1885. The pipes are in lengths of 12 ft, and cost $35 per 
2000 pounds. Unpaving, digging trenches, re-filling and re-paving not included. 



44 U 44 


b( (4 44 44 


ft. 


ft. 


Yarn per joint, lbs, at 9 cts per lb.. 

“ “ “ $ . 

Hemlock lumber per length, $.. 

Coke, clay, &c, per joint, $. 

Hauling per length, $. 


ft. 


u u u « u 


ft. 


6 

12 

20 

36 

48 

430 

1000 

2000 

4860 

8250 

36 

83 

167 

405 

688 

7.52 

17.50 

35.00 

85.05 

144.38 

.63 

1.46 

2.92 

7.09 

12.03 

13 

22 

41 

80 

146 

.55 

1.10 

2.05 

4.00 

7.30 

Vs 

% 

1 

2^ 

5 

.01 

© 

• 

.09 

.23 

.45 


... 

.40 

.75 

1.04 

.02 

.04 

.08 

.15 

.25 

.40 

.80 

2.00 

3.40 

5.00 

.60 

.86 

1.27 

1.95 

3.25 

1.68 

2.82 

5.89 

10.48 

17.65 

.14 

.24 

.49 

.87 

1.47 

9.20 

20.32 

40.89 

95.53 

162.03 

.77 

1.69 

3.41 

7.96 

13.50 


Art. 13. Air valves. Air is apt to collect gradually at the high points of 
vert curves along the supply pipes; and, unless removed, produces more or less obstruction to the 
flow. This mav be prevented by air valves, 1L 

seeFig 44A,which is % of the full size of those 
once used in Philada. This simple device 
consists of a cast-iron box, ccd d. confined 
to the main pipe ift m, by screw-bolts passing 
through its flange d d. It has a cover gng, 
confined to it by screws ft; and at the top 
of which is an opening n, for the escape of 
air from withiu. In this box is a float /, 
which mav be a close tin or copper vessel, 
or of layers of cork, as supposed in the fig ; 
or &c. This float has a spindle or stem « *, 
fast to it; which passes through openings in 
the bridge-bars a a. and o; thereby allowing 
the float to rise and fall freely, but prevent¬ 
ing it from moving sideways, when the 
pipe mm is empty, the float is down; its 
base y resting on the cross-bar a a. the 
stem ss has fixed to it a valve v, which rises 
and falls with it and the float. Suppose the 
pipe mm to be empty, and consequently the 
float, and the valve v, down. Then, if water 
be admitted into the pipe, it will rise and 
fill also the box as far up as e; and in 
doing so will lift the float/, and the valve v, 
to the position in the fig; thus preventing 

S/SS «' *'<«• •»' i wdl *“ M 

purposes oi lead joints, at considerably less cost. 



Pig. 44A. 






























































298 


WATER-PIPES. 


egress to the outer air, by closing the opening at v. Now, air carried along by the 
■ft ater, will, on account of its lightness, ascend to the highest points it meets with. 

Jlence, when such air arrives under the opening aa, it will rise through it, 
and ascend to e; the closed valve preventing it from going farther. Thus 
successive portions of air ascend, and in time accumulate to such an extent 
as gradually to force much of the water downward out of the box. When 
this takes place, the float, which is held up only by the water, of course de- 
scends also; and m doing so, pulls down with it the valve v. The accumulated 
* m v? n escapes through the openings at v and n, into the atmosphere; 

fr <? o' tie 1 mined lately ascends again into the box.earrv- 

v f™ h J float; and thus agaiu closing the valve v. The valve, and the 
va ve-seat e, are faced with brass, to avoid rust, and consequent bad fit. The 
whole is protected by an iron or wooden cover, reaching to the level of the street. 

1,0 * on « er in city pipes; their place being 

? 'PP , ) - v , tlle fireplugs at average distances of about 150 yards apart. These, 
nhi"f ^ aced as uu . ,ch as Possible at the summits of undulations in the lines of 
p pes, lor convenience of washing the streets, and being frequently onened 
lor that purpose, permit also the escape of accumulated airf ^ y P 

otlier one‘n*i»»ir > *»»»« , K® >ref » Se< * air thro «8:** air valve, or 
main .K & bce,, k !' own Produce bursting of the 

water iuthfn, ? .i the .escape is instantaneous, and permits the columns of 
‘ 111 J. ? I J1 P es on both sides of the valve, to rush together with great 

forces, which arrest each other, and react against the pipes 8 8 

Air-Vessels. Motion is imparted to the water in aline of nines bv the 

strok^thiS^ th - e P iston , of a single-acting pump; but during the backward 
motlo, I stopped; and the water in the pipes comes to rest. There- 

mid thP fnrnl 1 th / 0rW ^ K stroke »» 1 u 1 the Water *' as to be again set in motion; 
wnnH kL tbat , m “ 8 * u be exerted by the pump to do this is much greatei than 

dunmr^/?H»l re r P r ey'ooslv imparted had been maintained 

during the tune of the backstroke. The addition of an air-vessel secures this 

fsh° f m °lT’ ™ d thl ! S e % cts a g^at saving of powei"^besfdes di,ni„! 
isiiing the danger of bursting the pipes at each forward stroke. It is merely a 

of the d Jne° n - 8 ?h t,ght J 1 ?, 11 box , usually cylindrical, strongly bolted on top 
tiL th u p 1 Jllst beyond the pump, and communicating frtelv with them 

DistoAen f° pening i" itS bas t- ? U ful1 ofair - The forward stroke of the 
Sf he a ir v P S e fi Wa i 0,lly a 0l, g. the P j P es > but into the lower part 
tained a R N * ,e ?P PWI , n S its base; thus compressing its con- 

the nre?sure B of t thpT g , th ® backs , troke > thts compressed air, being relieved from 
the Lines and n^ iP P ’-f^ pands - ; and ir l so dl,1 "g P resses upon the water in 
An a r vestpl 1 Jili n k f PS in . motl .°n until the next forward stroke ; and so on. 

1 air - ve ssel also acts as an air-cushion; permitting the piston to annlv its force 
to the water ,n the pipes gradually : thus preserving both the pfpes and the 

absorbed°'nnll°tIk*" sbocks ' The air in the vessel, however, becomes by degrees 
ceases b d Tn k * b 'u tbe water l and its action as a regulator then 

time hr n „ P , ve,d this, fresh air must be forced into the vessel from time to 
n y % Condenser > r air-pump. A double-acting pump does Lt so 

much need an air-vessel. There is no particular rule for the size or can* dtv of 

Lump whh L U heSht C en!, t n i a ? P f r8 ,0 Vary fr ? m about 5 to 50 times tlmt of'tlm 
Lin i i Y\ a be, g. d<l«al to two or more times the diameter A stand-nimt 
(see below) Is sometimes used instead of an air-vessel. stand-p.pe 

almv?l ta T , M^ i Pnn iS - SOmetira ? u ? d frtr ,he sa,ne Purpose as an air-vessel (see 
fts font JU tv P , P e *. 0 P®n to the air at top; and communicating freelv at 

Water-Works arefroni 125 to 70 ll. r° n , n J “ ,ed . w,t " thc Philadelphia 

boiler-iron about % inch thick near the base^aVl'nlo TiY" d u lade of riv, u ' J 



WATER-PIPES. 


299 



Art. 14. The service-pipes for snpplying single dwellings, 

are of lead ; and of % to % inch bore. They are connected with the street mains, 
n n. Fig 45, by a brass ferrule,/, here shown at 
x /$ real size. The dotted lines show its % inch bore. 

The tapering ferrule is merely hard driven into a cyl¬ 
indrical hole reamed out of the main, as ats. The lead 
j pipe, o, is attached to the other end of the ferrule; 
t overlapping it about \% ins; and the joint soldered, L 
. The extra thickness near/, is for giving proper shape 
and strength for hammering the ferrule into the main. 

The pipe and solder are shown in section. Besides the __ 

stopcocks attached to each service-pipe, and to its 45* 

branches through the house, there is an underground one by which the city authori¬ 
ties can stop off the water in case of delinquency in payment of dues; and another 
by which the plumber can stop it off when so required during indoor repail's. Gal¬ 
vanized iron tubes are being much used for service-pipes, especially for hot 
water; being less subject to contraction and expansion, which produce leaks. See 
near bottom of page 218, for such water pipes. Brass service-pipes are now much 
used in Boston. See bottom of page 218; also page 417. 

Art. 15. The so-called **• corporation stops” or “corporation cocks” 
are inserted into the pipe by a special machine, Fig 4t5. Their great advantage over 
the ferrule, Fig 45, is that they can be inserted into a pipe when the 
latter is full of water under pressure. Besides, 
inasmuch as they are screwed into the pipe, they are in no danger 
of being forced out of it by any pressure within it. As their 
name implies, they are furnished with a stop-valve, which is kept 
closed while the valve is being inserted into the pipe,and is then 
opened, and generally remains open permanently. 

I*ipe-tapping - machines, for drilling and tapping the 
pipe, and for attaching these stops, are made in a variety of forms. 

Fig 46 shows one made by 
Walter S. Payne 
Co., Fostoria, Ohio. Each 
of these machines is furnished 
with a number of malleable 
cast-iron saddles, which fit the various diams of pipe 
with which it is to be used. The saddle is not shown 
in the fig. It is fastened to the pipe by a chain slung 
around the latter. The chain is tightened by a bolt 
and nut at each end. 

The brass cylinder, C C (into which a tap-and-drill, 
T, and the stop, S, have first been inserted), is then 
screwed into the saddle by means of the thread at A. 
The stop is temporarily screwed on to a mandrel, M. 
This mandrel, and the drill-sliank, K, pass through 
stuffing-boxes cast in one with the head of the cyl. 
By means of a handle, not shown in the fig, this head 
is now revolved (while the body of the cyl remains 
stationary) so as to bring the drill,T,and stop, S, into 
the respective positions shown in the fig. When the 
cyl head comes to the proper position, it is stopped by 
n | ut r insid° of the cyl. The drill is then immediately over the center of a large 
circular opening in the base of the cyl, C C, and over a similar opening, through the 
gad die to the surface of the pipe to be tapped. It is then pushed down until it 
touches said pipe. The ratchet-wrench, W W, is then set on the square head of the 
drill shank K ; the feeder-yoke, Y, with feed screw, F, is put in position as shown ; 
and the pipe is drilled and tapped by working the wrench ; whereupon the water m 
the pipe, if under pressure, rushes out through the hole thus made, and fills the 

CJ Bv reversing the position of the switch on the ratchet in the wrench, W, and by 
working the latter, the tap is now withdrawn from the hole, hut remains in the cyl. 
The cyl head is now revolted so as to reverse the positions of S and T ; the lug in¬ 
side of the cyl stopping the head when the stop is immediately over the hole. By 
means of the ratchet-wrench, applied to the square head of the mandrel, M,the stop, 




ex cee 
#«S)» >">W 


the stop. 


































300 


WATER-PIPES. 


Tbe mandrel, M, is made in two lengths (one of which screws into the other) in 
order that the upper part may be out of the way of the wrench handle while drill¬ 
ing. It hits three or more cliff threads at its foot, to suit diff sizes of stop. Stops, 
made to suit the machine, are furnished as wanted. 

The machine can work in any direction radial to the pipe, and can therefore be j 
used lor tapping a pipe in any part of its circumference. 

After the stop is inserted, the service-pipe is attached to its outer end by a coup¬ 
ling nut passing over the thread there shown. 

The machines are guaranteed to tap under a pressure of 600 lbs per square inch. 
They are made in five sizes, weighing from 15 to 6t> pounds, and costing, 1888, from 
875 to $175. The smallest size taps holes from % inch to % inch, and the largest 
size from % inch to 2 inches, diameter. These prices are for the machines complete, 
including 3 taps and drills and 3 saddles, but exclusive of stops. 

Other forms of pipe-tapping machines are the Boston, made by Whittier Machine 
Co., office, Granite and First Sts, Boston, Mass; the Lennox, by Lennox Foundrv and 
Machine Works, Marshalltown, Iowa; Young’s by the Easton (Pa) Brass Works; 
Letzkus’, James H. Harlow, Eng’r, agent, Pittsburgh, Pa; Sperring’s, Mueller’s, and 
Hadesty’s. Any of these can be had through the larger manufacturers and dealers 
in plumbers’ supplies; as Haines, Jones & Cadbury, 1136 Ridge Ave, Phila: McCam- 
bridge & Co., 527 Cherry St, Phila; Chas Perkes, 627 Arch St, Phila. 






STOP-VALYES. 


301 


Art. 16. Stop-valves, or gates, opening vertically in grooves, are placed 
across the street pipes at intervals of from 100 to 300 yds. Their use is to shut off 
the water from any section during repairs ; the water of such sections being allowed 
to run to waste, and to soak into the ground. 

The details are much varied by dill makers. 


I 



Fig. 47. Fig. 48. 


Figs 47 and 48 show such a gate made by Chapman Valve Mfg Co, Indian 
Orchard, Mass. The valve, v, is cast in one piece. W hen down, as in the figs, it 
closes the pipe. As in other styles, it opens vert by means of a screw, D, the valve 
rising into the cast-iron case or box, B B, and leaving, when all the way up, an 
opening of the full diam of the pipe. The screw is turned by a wrench fitting on 
its square head, h. The screw, D, itself, is prevented from moving vert by the col- 
lur C 

The two principal castings which compose the box or cover are bolted together by 
means of flanges, g. The joint faces of the castings are carefully smoothed; and a 
thin strip of lead is inserted between them, as a precaution against leaks. The recess, 
R, admits small particles of foreign matter which might otherwise prevent the gate 
from closing perfectly. The valve seats are faced with Babbitt metal. At the top 
of the cover, the screw stem passes through a stuffing-box, which prevents leaking 
at that point. Very careful workmanship is required throughout. 

The following are the weights and approximate prices of these valves. 
They give a tolerable average of the wts and prices of similar gates by other first- 
class makers, among whom are Isaac S. Cassin,2d St and Germantown Ave, Phila; 
Ludlow Valve Mfg Co, 938 River St, Troy, N Y; and Whittier Mach Co, office, Granite 
and First Sts, Boston. The latter make Coffin’s patent double-disk valve, besides 
other patterns, fire hydrants, locomotive and stationary engines, dredging machines, 
iron bridges, and heavy tools and machinery in general. 





















































































302 


STOP-VALVES. 


Chapman Bell-emI Water-gates. 


Bore. 

Wt. 

List 

Bore. 

VTt. 

List 

Bore. 

Wt. 

List 

Bore. 

Wt. 

List 



price.* 



price.* 



price.* 



price.* 

ins. 

lbs. 

$ 

ins. 

lbs. 

* 

ins. 

lbs. 

$ 

ins 

lbs. 

$ 

o 

32 

10 

6 

195 

30 34 

12 

600 

82 

20 

1700 

230 

3 

55 

15 

7 

245 

36 

14 

843 

112 

24 

2750 

333 

4 

116 

19 

8 

290 

45 

16 

1080 

150 

30f 

6400 

700 

5 

135 

25 

10 

439 

62 

18 

1475 

194 

36f 

8300 

1200 


Art. 17. Fig 49 shows an arrangement with outside screw for raising 
and lowering the valve. Here the screw, D, does not revolve, but is attached to the 

valve, and rises and falls with it, being raised or low¬ 
ered by turning the wheel, YV, at the center of which is 
a nut through which the screw passes. The uut is fixed 
in the wheel, and is so coufiued that it, and the wheel, 
cannot move vertically. 

Art. 18. A lour-way stop, or four-way 
valve. Figs 50 and 51, is placed at the intersection of 
two mains; the four <*ids of which areattached, respec¬ 
tively, to the four openings, M 31 31 31. At the bottom 
is an additional opening, connecting, by means of an 
elbow, II. with pipes running to a fire-hydrant at the 
street curb. See Arts 20 and 21. Two or more of such 
bottom openings may be made, if desired, for the sup¬ 
ply of as many fire-hydrants. All of the openings are 
opened or closed at one time by raising or lowering 
the valve or plug, P, by means of a wrench or key ap¬ 
plied to the square head, S, of the screw stem. As in 
Figs 47 and 48, the screw turns, but is prevented from 
rising and falling, and the plug moves up and down on 
the screw. 

Inasmuch as all sediment escapes into the bottom 
opening which leads to the fire-hydrant, the valve is not 
liable to clogging through this cause. The fire-hy- 




49 


Fig. 50 


Fig. 51 


\ 


drant, being fed from both of the mains, obtains a fuller supply than would be possi¬ 
ble it it were fed, as usual, through only one main. 

Figs 50 and 51 represent Viney’s four-way valve, made by Keystone Valve Co, office 
l.)10 Brown 8t Philadelphia. The plug, P, is a hollow iron casting, in the shape of a 
truncated four-sided pyramid. Each of its sloping sides is faced with brass; and the 
seats are of white metal. For prices and discounts, address as above. The weights 
are as follows: for 4 inch pipes, 170 tbs ; 0 inch, 395 lbs ; 8 inch, 495 lbs; 10 inch 7o0 
lbs; 12 inch, 1000 lbs. The Co. also makes three -way valves. 


* DiNCOiinf, 18b8, about35 percent. 

1 Furnished with gearing to aid in opening and closing the valve. The 20 inch and 24 inch witea 
can a SO be furnished with such geanug when desired. This increases the weight about one eiehth 
and the cost about one fourth. ® 


















































































STOP-VALVES. 


303 


Art. 19. Whatever the style of the gate may be. it is, when attached to the pipe, 
protected by a surrounding’ box, generally of plank or cast-iron, with 
four sides, which taper so that the box is of smaller hor section at top than at bottom. 
It is open at bottom, but has a movable iron top, level with the street. This top is 
taken off when the valve is to be opened or closed, or inspected. Two of the oppo¬ 
site sides of the box of course have openings for the passage of the pipes to or from 
the valve. 

The gates, especially of large mains, must be closed very slowly. Otherwise, the 
too sudden arresting of the momentum of the flowing water would be apt to break 
either them or the covers; or burst the pipes. As a precaution against this, the 
covers for very large valves are cast with outside strengthening ribs. 

No self-acting air-valves (Fig 44 A) are now placed at street summits, to allow 
confined air to escape. The fire-plugs answer instead. The rad for hor bends in 
mains is if possible not less than about 12 times their diams; they are made as large 
as the widths of the streets will admit; usually about 50 ft. Fire-plugs, Figs 
52. &c, are placed as much as possible at summits, so as to serve also for washing 
the streets; and for the escape of accumulated air. They average about 8 in num¬ 
ber to each mile of pipe; or 1 to each block of buildings. 


! 1 











304 


FIRE-HYDRANTS. 


Art. 20. Fig 52 represents a common street fire-plug, or fire-hydrant, 

as made by Gloucester Iron Works, 



office, No 6 N 7th St, Pliila, and used 
in that city. 

The valve, v, is of layers of well-4 
hammered sole-leather; and, when 
closed, shuts against a brass ring 
seat, n , which is confined to its place 
by a lead joint. The valve is opened 
by working down the screw, s, 
which, by means of a swivel joint 
at «, can revolve without turning 
the valve-rod, y. When the valve, 
v, is closed, after the plug has been 
in use, the chamber, c, is full of 
water, which, if allowed to remain, 
would be in danger of freezing, 
and of bursting the plug. But, in 
closing the valve, we raise the 
flange, l, on the rod, and thus allow 
the water to escape through the 
opening at i, whence it runs to 
waste into the ground, through the 
open lower end of the frost- 
jstclcet, jj, which is a hollow 
cast-iron cylinder surrounding the 
working parts of the hydrant. 
Being free to slide vert it rises and 
bills when the level of the ground 
is disturbed by frost, and the hy¬ 
drant is thus protected against in¬ 
jury from this cause. 

The top, t, of the hydrant case, is 
cast iu one piece with the cham¬ 
ber, c. 

The stopper, e, screws on over 
the nozzle, n. 

Art. 21. In the Chapman 
firc-liy<lrant, Figs 53, 54, and 
55, made by the Chapman Valve Mfg 
Co, Indian Orchard. Mass, the 
valve, v v, is a sliding one. 
The stem, y, Fig 53, to which the 
screw, s, is attached, is, like that in 
Figs 47 and 48, prevented from mov¬ 
ing vert by a collar, fast to it near 
its top, and confined in a circular 
groove. When the rod and screw 
are made to revolve, by means of a 
wrench applied to the square head 
of the former, the valve slides up 
or down on the screw, admitting 
the waterto,orshutting it off from, 
the hydrant. The valve slides on 


the two guides, g g, which are cast in one piece, respectively, with the two vert sides 
of the lower part of the hydrant. Its circular face, where it comes into contact 
with the hydrant case, h , is faced with a gun-metal ring, e e, which bears against a 
similar ring, made of Babbitt metal, let into the hydrant case. 

Tile water, left in the hydrant case after closing the valve, escapes through 
a cylindrical hole, d, about % inch diam, bored through the guide, g, and case, h. 
This hole is at such a lit as to be just above the top of the loose plate, p, when the 
valve is closed, as in Fig 53. This plate lies in avert groove in the side of the valve¬ 
casting, and is pressed against the side of the case, //, by two spiral springs confined 


in cavities, cc, in the valve casting. When the valve begins to rise, the hole, d, is 


closed by the plate, p, and remains so until the valve is again entirely closed. 












































yr., ,.Z- 


305 




















































































306 


FORCE IN RIGID BODIES. 


MECHANICS. FORCE IN RIGID BODIES. 


In the following pages we endeavor to make clear a few elementary principle: 
of Mechanics. The opening articles are devoted chiefly to the subject i f matter it 
motion; for, while an acquaintance with this is perhaps not absolutely required ir 
obtaining a working knowledge of those principles of Statics which enter so largely 
into the computations of the civil engineer, yet it must be an important aid to thei. 
intelligent appreciation. 

Those, however, who wish merely for information on problems of Statics, may turr 
at once to Articles 25, etc., pp. 318 e, etc. 

(*■)• Mechanics may be defined, as that branch of science which 
tr ats of the effects of force upon matter. 

This broad definition of the word “Mechanics” includes hydrostatics, hydraulics 
pneumatics, etc., if not also electricity, optics, acoustics, and indeed all branches ol 
physics; but we i-ha!l here confine ourselves chiefly to the consideration of the action 
of extraneous forces upon bodies supposed to be rigid, or incapable of change of shape 
(b) Mechanics is divided into two branches, namely: 

Kinematics; or the study of the motions of bodies, without reference to th i 
causes of motion; and 

Dynamics, or the study of force and its effects. 

The latter is sub-divided into 

Kinetics; which treats of the relations between force and motion; and 

C o V Ch con f der « tho8e 8 P ecial . but very numerous, cases, where equal 
and opposite forces counteract each other and thus destroy each other’s motions. 

, 3 Mat , ter ’ or substance, may be defined as whatever occupies space; 
as metal, stone, wood, water, air, steam, gas, etc. P ’ , 

se,iamttdhi°/SLm y «r r /h 0n °' ™ tteT \bich * 8 either more or less completely 
!3 .7,/> f t from a11 otf l er matter, or which we take into consideration bv itsel 
and as ,f it were so separated. Thus, a stone is a body, whether it be fall Z throng 

a 'bodv w ! y ‘if R w deta ^n ed " P ° n the S round ’ or built up into a wall. Also, the walUs 
tt i d ‘r’ ? r ’ w . e W18 h, we may consider any portion of the wall as anv nartirnlnr 

r ““ taCh -r. each wh ee. „ 

-Mi sspsx , : E&r&w.&x * bodT any port “ m ° f « - ; 

i C) bodfe 8 i which u f nd°e^o no ^ "T** 7 8tated) insider chiefly rig* bodie 
or pulled apart or penetfated bv anoth" ft®’ 8u . C n as by beinK cnJshed or Btretche 
or less subject’to some such changes of shine i f n ™ e . mor 

rigid; but we may properly for oonvA^ P ’ ’ v°i ly ,s tn f act «bsolutel 

many bodies are soneSrlv rSid tw i ’ 8u i!P° se 8Uch bodies «o exist, becair 
little or no change of shaU fla K Under ( ’ rdinary circumstances they undergo 

s.dered under the distinct head of Strength "of MaSs RS d °” ° CCUr “ ay be con ‘ 

.vX^jitanuhS 1 r™“d ,o the m r< ' sar ' i '' d »“ i" c * i>abi6 " r ch "^- /»”»■ a 

tured when the whole weight of the st L 16 ° dge ” w .' , be in dan S er of being ft ac¬ 
ting. we resort to the orulhfng strength Up °V? duri , n ® the P rowsfi »pset- 

pu«hed or pulled in several directions at nor a ^I* 6 ’ P 18 ,llat a body, when 

tendency to move in one direction rather than anotw*’ “V whoIe ’ ]'. ave the slightest 
tend to move in one direction and nthprl in no .^ ler ’ yet ®°me of its particles must 
Materials,” pp. 434, etc d ° thera lQ other Actions. See “Strength of 







FORCE IN RIGID BODIES. 


307 


Art. 3 (a). Motion of a body is change of its position in relation to another 
>cdy or to some real or imaginary point, which (for convenience) we regard as fixed, 
r at rest. Thus, while a stone falls from a roof to the ground, its position, relatively 
o the roof is constantly changing, as is also that relatively to tho ground and that 
Relatively to any given point in the wall; and we say that the stone is in motion rela¬ 
tively to either of those bodies , or to any point in them, hut if two stones, A and B, 

' II from the roof at the same instant and reach the ground at the same (subsequent) 
istant, we say that although each moves, relatively to roof and ground, yet they 
iive no motion relatively to each other; or, they are at rest relatively to each other; 
/or their position in regard to each other does not change; i. e., in whatever direction 
und at whatever distance stone A may be from stone B at the time of starting, it 
remains in that same direction, and at that same distance from B during the whole 
time of the fall. Similarly, the roof, the wall and the ground are at rest relatively 
to each other, yet they are in motion relatively to a falling stone. They are also in 
motion relatively to the sun, owing to the earth’s daily rotation about its axis, and 
its annual movement around the sun. 

(to) If a train-man walks toward the rear along the top of a freight train just as 
fast as the train moves forward, he is in motion relatively to the train; but, as a 
whole, he is at rest relatively to buildings , etc. near by; for a spectator, standing at 
a little distance from the track, sees him continually opposite the same part of such 
building, etc. If the man on the train now stops walking, he comes to rest relatively 
to the train , but at the same time comes into motion relatively to the surrounding 
buildings, etc., for the spectator sees him begiu to move along with the train. 


(c) Since we know of no absolutely fixed point in space, vre cannot say, of any 
jody, what its absolute motion is. Consequently, we do not know of such a thing as 
absolute rest, and are safe in saying that all bodies are in motion. 

(a). The velocity of a moving body is its rate of motion. A body (as a 
railroad train) is said to move with uniform velocity, or constant velocity, 
when the distances moved over in equal times are equal to each other, no matter how 
small those times may be taken. 


(to) The velocity is expressed by stating the distance passed over during some 
Given time , or which would be passed over during that time if the uniform motion 
continued so long Thus, if a railroad train, moving with constant velocity, passes 
N ver 10 miles in half an hour, we may say that its velocity, during that time, is 
•i e that it moves at the rate of) 20 miles per hour, or 105,600 feet per hour, or li 60 
feet ner minute, or 29% feet per second. Or, we may, if desirable say that it moves 
>t the rate of 10 miles in half an hour, or 88 feet in three seconds, etc.; but it is 
generally more convenient to state the distance passed over in a unit ol time, as in 
\e day one hour, one second, etc. 

(c) If, of two trains, A and B, moving with constant velocity, 

c A moves 10 miles in half an hour, 

B moves 10 miles in quarter of an hour, 


sen the velocities are, 

;> A, 20 miles per hour, 

i B, 40 miles per hour. 

other words the velocity of a body (which may be defined as the distance passed 
Wer in a given time) is inversely as the time required to pass over a given distance. 

( t V\ Bv unit velocity is meant that velocity which, by common consent, is taken 
as equal to unity or one. Where English measures are used, the unit velocity gen¬ 
erally adopted in the study of Mechanics is 1 foot per second. 

When we say that a body has a velocity of 20 miles per hour, or 10 feet per 
second, etc., we do not imply that it will necessarily travel 20 miles or 10 fee , etc.; 
dr \t may not have sufficient time for that. We mean merely that it is traveling at 
the rate of 20 miles per hour, or 10 feet per second, etc ; so that if it continued to move 
at that same rate for an hour, or a second, etc., it would travel 20 miles, or 10 feet. etc. 

/f\ when velocity increases, it is said to be accelerated. When it decreases . 
it is said to be retarded. If the acceleration or retardation is in exact proportion 
to the time ; that is, when during any and every equal interval of time, the same degree 
of change takes place, it is uniformly accelerated, or retarded. When otherwise, the 
words variable and variably Pre used. 

(e) A body may have, at the same time, two or more Independent veloci¬ 
ties requiring to be considered. For instance, a bail fired vertically upward from a 



308 


FORCE IN RIGID BODIES. 


gun, and then falling again to the earth, lias, during the whole time of its rise ai 
fall, (1st) the uniform upward velocity with which it leaves the muzzle, and (2nd) tl 
continually accelerated downward velocity given to it by gravity, which acts upon 
during the whole time. Its resultant (or apparent) velocity' at auy moment is tl 

difference between these two. 

Thus, immediately after leaving the gun, the downward velocity given l 
gravity is very small, and the resultant velocity is therefore upward and ver 
nearly equal to the whole upward velocity due to the powder. But after awbil 
the downward velocity (by constantly increasing) becomes equal to the upwar 
velocity; i. e., their difference, or the resultant velocity, becomes nothing; the ba 
at that instant stands still; but its downward velocity continues to increase, an 
immediately becomes a little greater than the upward velocity; then greater an 
gieater, until the ball strikes the ground. At that instant its resultant velocity 


{ the downward velocity which it would 
have acquired by falling during the 
whole time of its rise and fall. 




the un form upward 
elocity given by the 
powder. 


M e have here neglected the resistance of the air, which of course retards bot 
the ascent and the descent of the ball. 

(Ii) As a further illustration, regard ab nc fig. 9V£ p. 320, as a raft drifting in the 
direction c a or n b. A man on the raft walks with uniform velocity from corner n tc 
corner c while the raft drifts (with a uniform velocity a little greater ihati that ol 
trie man) through the distance n h. Therefore, when the man reaches corner c, 
lat corner has moved to the point which, when he started, wits occup ed by a 
le man s resultant motion, relatively to the bed of the river op to a point on shore, 
has therefore been n a. His motion at right angles to » a, due to his walking, is 
tc, ut t lat due to the drifting of the ratt is o b. These two are equal and opp >sito. 
fence his resultant motion at right angls to n a is nothing; he does not move from 
tne line n a. His walking moves him through a distance equal to n i, in the 

1 ection n a; and the drifting through a distance equal to ia, aud the sum of these 
two is n a. ^ ’ 

(I) All the motions which we see given to bodies are but changes in their unknowr 
absolute, motions. lor convenience, we may confine our attention to some one oi 
moro of these changes, neglecting others. 

Thus in tbe case of the ball fired upward from a gun (see (g) above) we mn- 
neglect its uniform iipward motion and consider only its constantly accelerate 
down ward motion under the action of gravity ; or, as is more usual, we may conside. 
only the resultant or apparent motion, which is first upward and then downward Tr 

Ihe ea C rTh S in W spa n ce gleCt m ° tiOD9 ° f ^ baU CaUSed by the 8eTeral moti ™ 8 <> 

Art. 5 (a). Force, the cause of chance of motion. Sunnose * 
perfectly smooth ball resting upon a perfectly hard, friction less and level sin-lace 
and suppose the resistance of the air to be removed. In order to merely move the 

»■»« «Tnpo“'ii (, o r : i? 

P'-Hed >>y impressed *' be c " ni 

p'; d 3ir«c" , " ierKo ch “' e °- F ° r 

(t») I orce Is an action between two hndlra ... . 

bod^q^nfothe^in'st^n^s^f ^i^t^s^force . 60 * 168 ^ 6 b * tWeeQ th ° particles of * 
by^cau^i^g^onta^t > betm>^^it^”a*»Hier^)ody'^A! C whTch^iaa^ t° r< rt * * b " dy ^ 

toward B A force is 

some way which wo cannot understand), and this force pushes B forward (or in th( 








FORCE IN RIGID BODIES. 


309 


direction of A’s tendency to move) and pushes A backward, thus diminishing its for¬ 
ward tendency.* 

If, for instance, a stone bo laid upon the ground, it tends to move downward but 
docs not do so, because a repulsive force pushes it and the earth apart just as hard as 
A the force of gravity tends to draw' them together. 

Similarly, when we attempt to lift a moderate weight with our hand, we do so by 
giving the hand a tendency to move upward. If the hand slips from the weight, 
this tendency moves the hand rapidly upward before our will force can check it! 
But otherwise, the repulsive force, generated by contact between the hand (tending 
upward) and the weight, moves the latter upward in spite of the force of gravity 
} and pushes the hand downward, depriving it of much of the upward velocity which 
it would otherwise have. It is perhaps chiefly from the effort, of which we are 
conscious in such cases, that we derive our notions of “force.” 

When a moving billiard ball. A, strikes another one, B, at rest, the tendency 
of A to continue moving forward is resisted by a repulsive force acting between it 
find B. This fore© pushes B forward, and A backward, retarding its former velocity. 
As explained in Art. 23 (a), p. 318 d, the repulsive force does not exist in either body 
until the two meet. 

| (d) The repulsive force thus generated by contact between two bodies, continues to 

act only so long as they remain in contact, and onlv so long as they tend (from 
some extraneous cause) to come closer together. But it is generally or always 
j accompanied by an additional repulsive force, due to the compression of the particles 
of the bodies and their tendency to return to their original positions. This elastic 
repulsive force may continue to act after the tendency to compression has ceased 

(e) Force acts either as a pull or as a push. Thus, when a weight 
is suspended by a hook at the end of a rope, gravity pulls the w’eight downward, the 
weight pushes the hook, and the hook pulls the rope, each of these actions being 

j accompanied, of course, by its corresponding and opposite “reaction.” When two 
bodies collide, each pushes the other, generally for a very short time (see Art. 24 
p. 318 c.) ’ 

(f) Equality of action and reaction. A force always exerts itself equally 
upon the two bodies between which it acts. Thus, the force (or attraction) of 
gravitation, acting between the earth and a stone, draws the earth upward just as 
hard as it draws the stone downward; and the repulsive force, acting between a 

{.table and a stone resting upon it, pushes the table and the earth downward just as 
hard as it pushes the stone upward. This is the fact expressed by Newton’s 
third law of motion, that “to every action there is always an equal and 
| contrary reaction.” For measures of force, see Arts. 11, 12, 13, pp. 312 to 314. 

If a cannon ball in its flight cuts a leaf from a tree, we say that the leaf has reacled 
against the ball with precisely the same force with which the ball acted against the 
leaf. That degree of force was sufficient to cut off a leaf, but not to arrest tbe ball. 

! A ship of war, in running against a canoe, or the fist of a pugilist striking his 
i opponent in the face, receives as violent a blow as it gives; but the same blow’ that 
! ill upset or sink a canoe, will not appreciably affect the motion of a ship, and the 
blow which may seriously damage a nose, mouth, or eyes, may have no such effect 
upon hard knuckles. 

The resistance which an abutment opposes to the pressure of an arch; or a retain- 
ling-wall to the pressure of the earth behind it, is no greater than those pressures 
themselves; but the abutment and the wall are, for the sake of safety, made capable 
■ f sustaining much greater pressures, in case accidental circumstances should pro¬ 
duce such. 

(p) In most practical cases we have to consider only one of the two bodies 
between which a force acts. Hence, for convenience, we commonly speak as if the 
force w'ere divided into two equal and opposite forces one for each of the tw’o bodies, 
and confine our attention to one of the bodies and the force acting upon it, neglect- 
! ing the other. Thus we may speak of the force of steam in an engine as acting 
! upon the piston, and neglect iis equal and opposite pressure against the head of 
1 the cylinder. 

(I») That point of a body’ to which, theoretically, a force is applied, is called the 
point of application. In practice we cannot apply force to a point according 
1 to the scientific meaning of that word; but have to apply it distributed over an ap¬ 
preciable area (sometimes very large) of the surface of the body; still, as explained in 

* We ordinarily express all this by saying simply that A pushes B forward, and this 
j is sufficiently exact for practical purposes; but it is well to recognize that it is merely 
, i convenient expression and does not fully state the facts, and that every force neces¬ 
sarily consists of two equal and opposite pulls or pushes exerted b.tween two bodies. 

21 






310 


FORCE IN RIGID BODIES. 


Art 58 a n 347 l , it may, as regards its action upon the entire mass of a rigid body, bo 
considered as acting in one straight line, applied at one point. lor the present we 
shall assume that tlio line of action of the force passes through the center of giavity 
of the body and forms a right angle with the surface at the point of application, tor . 
cases where the force forms other angles with the surlace, see Art. -6, p. 818/. 


Aw * Acceleration. When an unresisted force, acting upon a body, 

s df !t in motion cH. gives it velocity) in the direction of the force, this velocity 
increases as the force continues to act; each equal interval of time (.it the foicf 


tncreutics tuc av/iw w»v.— . r 

remains constant) bringing its own equal increass ot velocity , 

Thus if a stone be let fall, the force of gravity gives to it, in the first in¬ 
conceivably short interval of time, a small velocity downward. In the next equal 
interval of tinv, it adds a second equal velocity, bo that at the end of the seconc 
interval the velocity of the stone is twice as great as at the end of the first one, andj 
s > on Wo may divide the time into as small equal intervals as we please. In each 
such interval tho constant* force of gravity gives to the stone an equal increase o!| 

Ve Such^increase of velocity is called acceleration.! When a body is thrown vertically 
upward, the downward acceleration of gravity appears ns a retardation of the upwarc 
motion. When a force thus acts against the motion under consideration, its accelera 
tion is called negative. 


Art. 8 (a). Tbe rate of acceleration! is the acceleration which taken 

place in a given lime, as one second. 

(b) The unit rate of acceleration is that which adds unit of velocity in ,'J 
unit of time; or, where English measures are used, one foot per secoud, per second 

(c) For a given rate of acceleration, the total accelerations are of course propor l 
tioual to the limes during which the velocity increases at that rate. 


Art. 9 (a). Laws of acceleration. Suppose two blocksof iron, one (which 
we will call A) twice as large as the other (a), placed each upon a perfectly frictionlea ( 
and horizontal plane, so that in moving them horizontally we are opposed by no fore* 
tending to hold them still. (See Art. 27 (d), p. 818 i). Now apply to each block 
through a spring balance, a pull such as will keep the pointer of each balance alwayt 
at the same mark, as, for iustance, constantly at 2 in both balances. We thus have > 
equal forces acting upon unequal masses.}: Here the rate of acceleration of a i. , 
double that of A; for when tile forces are equal tbe rates of accelera* . 
ration are Inversely as tbe masses. 

In other words, in one second (or in any other given time) the small block of iron j 
a, will acquire twice the increase of velocity that A (twice ns large) will acquire; 8‘ 1 
that if both blocks start at tbe same time from a state of rest, tbe smaller one, a, wil ' 
have, at the end of any given time, twice the velocity of A, which has twice its mass 

(b) Again, let the two masses, A and a, be equal, but let the force exerted upon * 
be twice that exerted upon A. Then the rate of acceleration of a will (as before) b , 
twice that of A; for, when tbe masses are equal, tbe rates of accelera ], 
ration are directly as tbe forces. 

(c) We thus arrive at th© principle that, in any case, tbe rate of acceleratloi I 
is directly proportional to tbe force and inversely proportlona 
to tbe mass. 


*We here speak of the force of gravity, exerted in a given place, ns constan k 
because it is so for all practical purposes. Strictly speaking, it increases a very littl t- 
as the stone approaches the earth. tl 

! Since the rat j . of acceleration is generally of greater consequence, in Mechanic; h 
than the total acceleration, or the “acceleration” proper, scientific 'writers (for th ( 
sake of brevity) use the term “acceleration” to denote that rate ., and the terr t 
“total acceleration” to denote the total increase or decrease of velocity occurrin i 
during any given time. Tims, the rate of acceleration of gravity (about 32.2 ft. pe ( 
second per second) is called, simply, tbe “acceleration of gravity.” As we shall nr ]j 
have to use either expression very frequently, we pliall. generally, to avoid misappnj, 
li ‘iision, give to each idea its full name; thus, “total acceleration” for the iaho\ 
change of velocity in a given case, and “rate of acceleration” for the rate of tht 
change. 

I The mass of a body is tbe quantity of matter that it contains. Seo Art. 11, p. 31: 












FORCE IN RIGID BODIES. 


311 


(cl) Hence, if we make the two forces proportional to the two masses, the rares 
of acceleration will be equal; or. for a given rate of acceleration, tlie 
forces must be directly as tbe masses. 

(e) Hence, also, a greater force is required to impart a given velocity to a given 
body in a short time than to impart the same velocity in a longer time. For instance, 
the forward coupling links of a long train of cars would snap instantly under a pull 
sufficient to give to the train in two seconds a velocity of twenty miles per hour, sup¬ 
posing a sufficiently powerful locomotive to exist. In many such cases, therefore, we 
have to be contented with a slow, instead of a rapid acceleration. 

A string may safely sustain a weight of one pound suspended from our hand. If 
[ i we wish to impart a great upward velocity to the weight in a very short time, we evi- 
dently can do so only by exerting upon it a great force; in other words, by jerking 
the string violently upward. But if the string has not tensile strength sufficient to 
| transmit this force from our hand to the weight, it will break. We might safely 
j give to the weight the desired velocity by applying a less force during a longer time. 

(f) When a stone falls, the force pulling the earth upward is (as remarked above) 
r equal to that which pulls the stone downward, but the mass of the earth is so vastly 
1 greater than that of the stone that its motion is totally imperceptible to us, aud 
,. would still be so, even if it were not counteracted by motions in other directions 

in other parts of the earth. Hence we are practically , though not absolutely, right 
when we say that the earth remains at rest while the stone falls. 

(g) But in the case of the two billiard balls (Art. 5 c p. 309), we can clearly see 
the result of the action of the force upon each of the two bodies; for the second 

» ball, B, which was at rest, dow moves forward, while the forward velocity of the 
i first one, A, is diminished or destroyed, its backward motion thus appearing as a 
retardation of its forward motion. And, (since the tame force acts upon both balls) 
mass mass rate of acceleration . rate of negative acceleration 
of A : of B :: of B • of A 


or (since the force acts for the same time upon both balls) 

mass mass forward velocity . loss of forward velocity 
of A : of B :: of B • of A 

(li) Remark. A man cannot lift a weight of 20 tons; but if it bo placed upon 
proper friction rollers, he can move it horizontally, as we see in some drawbridges, 
turntables Ac ; and if friction and the resistance of the air could be entirely removed, 
he could move it by a single breath; and it would continue to move forever after the 
force of the breath bad ceased to act upon it. It wmuld, however, move very slowly, 
because the force of the single breath would have to diffuse itself among 20 tons of 
matter He can move it, if it be placed in a suitable vessel in water, or if suspended 
from a long rope. Apowerful locomotive that may move 2000 tons,cannot lift 10 tons 
vertically. 

If we imagine two bodies, each as large and heavy ns the earth, to be precisely 
balanced in a pair of scales without friction, a single grain of sand added to either 
scale-pan, would give motion to both bodies. 

Art 10 (a). The constant force of gravity* is a uniformly accelerating force 
when it acts upon a body falling freely ; for it then increases the velocity at the uni¬ 
form rate of .322 of a foot per second during every hundredth part of a second, or 32.2 
.eet per second in every second. Also when it acts upon a body moving down an in¬ 
clined plane; although in this case the increase is not so rapid, because it is caused 
bv only a part of the gravity, wffiile another part presses the body to the plane, and a 
third part overcomes the friction. It is a uniformly retarding force, upon a body 
thrown vertically upward; for no matter what may be the velocity of the body 
when projected upward, it will be diminished .322 of a foot per second in each 
hundredth part of a second during its rise, or 32.2 feet per second during each 
entire second. At least, such would be the case were it not for the varying resistance 
of tho air at different velocities. It is a uniformly straining force when it causes a 
body at rest, to press upon another bodv ; or to pull upon a string by which it is 
suspended The foregoing expressions, like those of momentum, strain, push, pull, 
lift; work,* &c., do not indicate different linds of force ; but merely different kinds of 
effects produced by the one grand principle, force. 

(H) The above 32.2 feet per second is called the acceleration of gravitv ; and 
by scientific writers is conventionally denoted by a small g; or, more correctly speak- 


*See foot note * p 310. 










312 


FORCE IN RIGTD BODIES. 


Jne since the acceleration is not precisely the same at all parts of the earth, g 
denotes the acceleration per second, whatever it may be, at any particular place. 


Art 11 (a). Relation between force and mass. The mass of a body 
is the quantity of matter which it contains. One cubic foot of water has twice* 
as great a mass as half a cubic foot of water, but a less mass than one cubic , 
foot of iron. Thus, the size of a body is a measure ot mass between bodies!) 
of the same material, but not between bodies of different materials 


(b) When bodies are allowed to fall freely in a vacuum at a given place,, 
they are found to acquire equal velocities in any given time, ot whatever 
different materials they may be composed. From this we know (Art. J (a), 
p. 311), that the forces moving them downward, viz.: their respective weights 
at that place, must be proportional to their masses. 

Thus, in any given place, the weight of a body is a perfect measure of its mass. \ 
But the weight of a given body changes when the body is moved from one level' 
above the sea to another, or from one latitude to another; while the mass ot 
the body of course remains the same in all places. Thus, a piece of iron which 
weighs a pound at the level of the sea, will weigh less than a pound by a spring 
balance, upon the top of a mountain close by, because the attraction between 
the earth and a given mass diminishes when the latter recedes from the earth s 
center. Or if the piece of iron weighs one pound near the North or south 
Pole, it will, for the same reason, weigh less than a pound by a spring balance 
if weighed nearer to the equator and at the same level above the sea. 

The difference in the weight of a body in different localities is so slight as 
to be of no account in questions of ordinary practical Mechanics;* but 
scientific exactness requires a measure of mass which will give the same 
expression for tho quantity of matter in a given body, wherever it may 
be; and, since weighing is a verv convenient way of arriving at the quantity 
of matter in a body, it is desirable that we should still be able to express ihe 
mass in terms of the weight. Now, when a given body is carried to a higher) 
level, or to a lower latitude, its loss of weight is simply a decrease in the forcel 
with which gravity draws it downward, and this same decrease also causes 
a decrease of the velocity which the body acquires in falling during any 
given time. The change in velocity, by Art. 9 (b), p. 310, is necessarily propor¬ 
tional to the change in weight. , .. 

Therefore, if the weight of a body at any place be divided by the velocity 
which gravity imparts in one second at the same place land called or the 
acceleration of gravity for that place), the quotient will be the same at all places, 
and therefore serves as an invariable measure of the mass. 


(c) By common consent, the unit of mass, in scientific Mechanics, is said 
to be that quantity of matter to which a unit of force can give unit rate of 
acceleration. This unit rate, in countries where English measures are used 
is one foot per second, per second. It remains then to adjust the units of forct 
and of mass. Two methods (an old and a new one) are in use for doing this. 
We shall refer to them here as methods A and B respectively. 

' (d) In method A, still generally used in questions of statics, the unit 
of force is fixed as that force which is equal to the weight of one pound in a 
certain place; t. e., the force with which tho earth at that place attracts - 
certain standard piece of platinum called a pound; and the unit ol inass isj 
not this standard piece of metal, but, as stated in (c), that mass to which thi: 
unit force of one pound gives, in one second, a velocity of one foot per second. 
Now tho one pound attraction of the earth upon a mass of one pound will 
(Art. 1, p. 362) in one second give to that mass a velocity = g or about 32 feet 
per second; and (Art. 9 (a), p. 310), for a given force the masses are inversely as 
the velocities imparted in a given time. Therefore, to give in one second a 
velocity of only one foot per second (instead of g or about 32) the one pound 
unit of force would have to act upon a mass g times (or about 32 times) that 
which weighs one pound. 

This could be accomplished, with an Attwood’s machine, Art. 16 (e), p. 315,1 
by making the two equal weights each = 15 % lbs. and the third weight —* 1 lb. 


*The greatest discrepancy that can occur at various heights and latitudes.! 
by adopting weight as the measure of quantity, would not be likely to exceed 
1 in 300; or, under ordinary circumstances, 1 in 1000. 






FORCE IN RIGID BODIES. 


313 


By method A, therefore, the unit of mass is g times (or about 32 times) the 
mass of the standard piece of metal called a pound; i. e., a body containing 
one such unit of mass weighs g lbs. or about 32 lbs.; or, toy method A, 


the weight of any given body = the mass of the body, 
in lbs. J S' in nnif.a rtf mass 


Or, 


the mass of a body, in units of mass 

For instance: 

in a body weighing 
% pound 


in units of mass. 

the weight of the body, in pounds 
9 

the mass is about 

1 unit of mass 
Bf 


1 

2 

32 


41 

<< 

14 


1 

X 

T(T 

1 


41 14 

41 It 

14 14 


it 


41 


64 “ 2 

It has been suggested to call this unit of mass a “Matt.” 

(e) In method B, the mass of the standard pound piece of platinum is taken 
as the unit of mass and is called a pound; and the force which will give 
to it in one second a velocity of one foot per second is taken as the unit of force. 
This small unit of force is called a poundal. In order that it may in one 
second give to the mass of one pound a velocity of only one foot per second, it 

must (by Art. 9 ( b ), p. 310) be 1 - ^or about ^ j of the weight of said pound mas9. 

Hence, hy method B, 

the mass of any given body, in pounds 

and 


the weight of the body in poundals 

g 


the weight of a body, in poundals = g X the mass of the body in pounds. 

For instance: 

in a body weighing 
poundal = i pound 


the mass of the body is about 


1 

2 

32 

64 


44 

44 

14 

41 


1 

3T 

1 

TS 

1 


/ T pound 

l “ 




1 

2 


(f) For convenience, we sometimes disregard the scientific require¬ 
ment that the unit of force must be that which will give unit rate of accele¬ 
ration to unit mass, and take a pound of matter as our unit of mass, and a 
pound weight as our unit of force. Our unit of force will then in one second 
give a velocity of g (or about 32.2 feet per second) to our unit of mass. In 
Statics, we are not concerned with the masses of bodies, but only with the 
forces acting upon them, including their weights. 

Art 13(a). Impulse. By taking, as the unit of force, that force which, in 
one second will give to unit mass a velocity of one foot per second, we have 
(by Art. 9, p. 310), in any case of unbalanced force acting upon a mass during a 

given time : 


also 


Velocity 

Force 

Mass 

Time 

Force X time 


force X time 
mass 

velocity X mass 
time 

force X time 


velocity 

mass X velocity 
force 

mass X velocity. 


( 1 ) 

( 2 ) 

( 3 ) 

(«) 

( 5 ) 















314 


FORCE IN RIGID BODIES. 


To the product, force X time, in equation (5\ writers now give the name 
Impulse, which was formerly given to collision (now called impact). (See 
Art. 24 (a), p. 318 e). The term impulse, as now used, conveys merely the idea 
of force acting through a certain length of time. Equation (5) tells us that an j 
impulse (the product of a force by the time of its action) is numerically equal 
to the momentum* which it produces. Equation (2) tells us that any force is 
numerically equal to the momentum which it can produce in one second. In 
other words, the momentum of a body moving with a given velocity is 
numerically equal to the force which in one second can produce or destroy 
that velocity in that body; or, a force is numerically equal to the rate per 
second at which it can produce momentum. Thus, forces are proportional to 
the momentums which they can produce in a given time; or, in a given time, 
equal forces produce equal momentums. Therefore a force must always give 
equal and opposite momentums to the two bodies between which it acts. 

Art. 13 (a). The usual way of measuring a force is by ascertaining 
the amount of some other force which it can counteract. Thus we may meas¬ 
ure the weight of a body by hanging it to a spring balance. The scale of the 
balance then indicates the‘amount of tension in the spring; and we know that 
the weight of the body is equal to the tension, because the weight just pre¬ 
vents the tension from drawing the hook upward. 

Thus, forces are conveniently expressed in weights, as in pounds, 
tons, &c., and they are generally so measured in Statics, and in our following 
articles. The scientific method of measuring force, Art. 9, p. 310, is hardly 
applicable to ordinary practice. 

(b) A force may be constant or variable. When a stone rests upon 
the ground, the pull of gravity uyx>n it (t. e., its weight) remains constant, 
neither increasing nor decreasing. But when a stone is thrown upward its j 
weight decreases very slightiy as it recedes from the earth, and again increases 
as it approaches it during its fall. In this case, the force of gravity, acting 
upon the stone, decreases or increases steadily. But a force may change 
suddenly, or irregularly, or may be intermittent; as when a series of unequal 
blows are struck by a hammer, in what follows we shall have to do oqly with 
forces supposed to be constant. 

Art. 14 (a). Density. The densities of materials are proportional to the 
masses contained in a given volume, as a cubic inch; or inversely as the volume 
required to contain a given mass. Or, since the weights at a given place are 
proportional to the masses, the densities are proportional to the weights per 
unit of volume (or “specific gravities’’) of the materials. Thus, a body weigh¬ 
ing 100 lbs. per cubic foot is twice as dense as one weighing only 60 lbs. per 
cubic foot at the same place. 

Art. 15 (a). Inertia. The inability of matter to set itself in motion, or to j 
change the rate or direction of its motion, is called its inertia, or inertness. 
When we say that a certain body has twice the inertia (inertness) of a smaller 
one, we mean that twice the force is required to give it an equal rate of accele¬ 
ration ; and that, s'ince all force (Art. 5 (f), p. 309) acts equally in both direc¬ 
tions, we experience twice as great a reaction (or so-called “ resistance”) from 
the larger body as from the smaller one. The “ inertia” of a body is therefore 
a measure of the force required to produce in it a given rate of acceleration; or, 
which is the same thing, it is a measure of the mass of the body. We may 
therefore consider “inertia” and “mass” as identical. 

(b) What is called the “resistance of inertia” of a body, is simply the 
reaction, (». e., one of the two equal and opposite actions) of whatever 
force we apply to the body. Hence, its amount depends not only upon the 
mass of the body, but also upon the rate of acceleration which we choose to 


♦The momentum of a body (sometimes called its “quantity of motion”) j 
is equal to the product obtained by multiplying its mass by its velocity. If wo 
adopt the pound as the unit of mass, as in “method B,” Art. 11 (e), p. 313, the 
product, weight in pounds X velocity, is numerically either exactly or nearly 
the same as the product, tnass in pounds X velocity, depending upon whether 
or not the body is in that latitude and at that level where a mass of one pound 
is said to weigh, one pound. But the product, weight in poundals X velocity, is 
exactly g times (about 32.2 times) the product, mass in pounds X velocity; also, 
by “ method A,” iveight in pounds X velocity = g X mass in “ matts " X velocity. 
For the ideas conveyed by the term “ momentum,” see Art. 20, (a), p. 318 c. 














FORCE IN RIGID BODIES. 


315 


give to it. Therefore we cannot tell, from the mass or weight of a body alone, 
what its “resistance of inertia” in any given case will be. 

Art. 1G (a). Forces in opposite directions. When two equal and 
opposite forces act upon a body at the same time, and in the same straight liuo, 
we say that they destroy each other’s tendencies to move the body, and it remains at 
rest. If two unequal forces thus act in opposition, the smaller force and an 
equal portion of the greater one are said to counteract each other in the same way, 
but the remainder of the greater force, acting as an unbalanced or unresisted force, 
moves tiie body in its own direction, as it would do if it were the only force acting 
upon it. Caution. Seep, p. 316. 

The result, however, will be the same, as regards the motion produced, if (in either 
case) we consider both forces as causing the full motions due to them respectively, 
and these motions as counteracting each other; completely or partially, according to 
whether the forces are equal or unequal. 

Thus, when we move bodies, in practice, we encounter not only the 
“resistance of inertia” (i. e., we not only have to exert force in order to move 
inert matter), but we are also opposed by other forces , acting against us, as 
friction, the resistance of the air, and, often, all or a part of t tie weight of 
the body. By “ resistances,” in the following, we mean such resisting forces, 
and do not include in the term the “resistance of inertia." 


(b) Suppose, for convenience, that the resistance of a certain railroad 
train remains — 1 ton at all speeds (see Art. 20, p. 374e); that is, that a pull of 
1 ton on the draw-bar would in all cases balance the resistance and maintain 
the speed uniform. Suppose the train to be pulled first by an engine exerting 
a constant pull of 2 tons on the draw-bar; and then by one pulling 3 tons. 
Strictly speaking these engines impart to the train, in a given time, velocities 
which are to each other as 2 to 3, or as the respective pulling forces of the two 
engines; but in both cases the supposed uniform resistance of 1 ton w'ould, in 
the same time, cause a retardation (or counter-acceleration) of 1. Hence the net 
or resultant velocities (i. e., the observed velocities) would be — 2 — 1 and 
3 _l, or = 1 and 2; or in proportion to the net or resultant forces. 


(c) An “Atwood’s Machine,” consists essentially of a pulley, a flexible 
cord passing over the pulley, tw r o equal weights (one suspended at each end 
of the cord), and a third weight, generally much lighter than either of the 
other two. The two equal weights balance each other by means ot the pulley 
and cord. The third weight is laid upon one of the other two weights. The 
force of gravity acting upon the third weight, then sets the masses of the three 
weights in motion at a small but constantly increasing velocity. In order to 
do this it must also overcome the friction of the pulley and cord, and the 
rigidity of the latter; but, as these are made as slight as possible, they are, 
for convenience, neglected. The machine is used for illustrating the accelera¬ 
tion given to inert matter by unbalanced force, and forms an excellent exam¬ 
ple of the two distinct duties which a moving force generally has to perform, 
viz.: (1st) the balancing of resistance, and (2nd) acceleration. 

(d) In the case of a locomotive, drawing a train on a level, friction 
and the resistance of the air are the only resistances to be balanced; for the 
weight of the train here opposes no resistance. Unless the force of the steam 
is more than sufficient to balance the resistances, it cannot move the train. 
If it exceeds the resistances, the excess, how'ever slight, gives motion to the 
inert matter of the train. If, at any moment while the train is moving, the 
force of the steam becomes ,just equal to the resistances (whether by an increase 
of the latter or by diminishing the force) the train will move on at a uniform 
velocity equal to that which it had at the moment when the force and resis¬ 
tance were equalized; and, if these could always be kept equal, it would so 

m But so long V as the excess of steam pressure over the resistances continues 
to act the velocity is increased at each instant; for during each such instant 
the excess of force gives a small velocity in addition to that already existing. 

But, as the velocity increases, the steam pressure in the cylinders decreases 
(supposing the boiler pressure and the valve opening to remain the same), 
fo? tSe p“s S ton travels /aster through the cylinder and the boiler can no onger 
supply steam fast enough to fill the cylinder and maintain the original pn s- 
su?r. Thus,* after a time the force of the steam and the resistance become 


♦Some of the resistances also increase while the velocity increases. 











316 


FORCE IN RIGID BODIES. 




equal, and the train moves at a uniform velocity, simply because there i? 
now no opposing force to retard its motion, and no moving force to accelerate it 
When it becomes necessary to stop at a station some distance ahead, steam 
is shut off, so that the steam force of the engine shall no longer counterbal-J 
ance or destroy the resisting forces; and the number of the resistances them-! 
selves is increased by adding to them the friction of the brakes. The 
resistances, thus increased, are now the only forces acting upon the train, and 
their acceleration is negative , or a retardation. Hence, the train moves more 
and more slowly, and must eventually stop. 

(e) Similarly, when a horse is drawing a light carriage, and has given it a I 
very high speed, he can no longer do more than keep himself in advance of it, 
and exert a pull sufficient to balance the resistances of friction (at the axles and 
through the air, etc.)* Or, the horse may, of his own accord, or at the com¬ 
mand of his driver, intentionally abate the force of his pull to the same extent. 
\V hen this takes place, we say that the carriage is not acted upon by any force 
(for there is now no “remainder” left to act upon it), and it moves forward at 
a uniform, velocity, because it is unable, of itself, to change the rate or direction 
of its motion. 

If the pulling force is still further diminished, or the resistances increased, 
until the latter exceed the former, the “remainder” becomes a minus or negative 
quantity, and its “acceleration” of the velocity is also negative, or in the 
direction opposite to that of the motion. In other words, “ retardation ” of 
velocity takes place. 

(f) In a stationary engine the “governor” takes advantage of the increase , 
or decreaso of velocity caused by an excess of force or of resistance, respect-1 
iveiy, and by automatically decreasing or increasing the supply of steam, 
restores the equilibrium between its pressure and the resistances, and thus I 
restores also the uniformity of the velocity. 

(») When two opposite forces are in equilibrium, an addition to 

+»? e °>j ^ ^ “- >rces does not always form an unbalanced force; for in many cases 
the other force increases equally , up to a certain point. For instance, when we 
attempt to lift a weight, W, its downward resistance , R, remains constantly just 
equal to our upward pull, P, however P may vary, until P exceeds W. Thus, R 
can never exceed W but may be much less than it. Indeed, when we stop 
pulling, R ceases, although W (the attraction between the earth and the weight) r 
of course remains unchanged throughout. Such variation of resisting force to 
meet varying demands, occurs in all those innumerable cases where structures 
p^To' 1 var ^ in £ oa ^ s within their ultimate strength. See also Friction, Art. 4, 


Worlt * Force, when it moves a body,* is said to do “work” 
upon it. The whole work done by the force in moving the body through any dis¬ 
tance is measured by multiplying the force by the distance; or : Work = Force X 
distance If the force is taken in pounds, and the distance in feet, the product (or 
the work done) will be J n foot-pounds; if the force is in tons and the distance in 
Indies, the product will bo in inch-tons; and so on.f 

Thus, if a force of moves a body through we have work =» 

1 pound 10,000 feet 10,000 foot-pounds 

100 pounds 100 “ 10.000 “ 

. I 0 ’ 0 ™ “ 1 foot $000 « 

? r ’ ,n “ n F ca , 80 ’ j* toTCe F pounds, the whole work done by it in moving a 
body through s feet, is F * foot-pounds. s 

fout ‘ ton \ tho inch-pound, the inch-ton. etc., etc., are called 
units of work.) We may adopt any unit we please, just as wo may state a 
distance in feet, or m miles, etc., at pleasure; but it is always convenient to take for 
our unit of work, the product of that unit of force, and that unit of distance, which 
wo are employing at the time. ’ 


A man who is standing still is not considered to be working, any more than 
is a post or a rope when sustaining a heavy load; although ho may bo supporting 
an oppressive burden, or bolding a car-brake with all his strength; for his force 
moves nothing in either case. ** 

the head Levers, it will be seen that the tendencies (called moments'* 
which the power and the weight respectively have to commence motion about tho 
fulcrum as a center are also measured iu such terms. Work and moments, are 
howevor, effects of force so different from each other, that confusion is no ^nore 

th ° B “ m “ meMuro ’ OD ° ^0o ‘• 10 materlal3 ** difforeu ‘ 




1 













FORCE IN RIGID BODIES. 


317 


For practical purposes, in this country, forces are most frequently stated in 
• pounds, and the distances (through whicti they act) in feet. Hence the foot- 

I pound is here the ordinary unit of work. 

(c) In most cases, a portion at least of the worlt done by a force is expended In 
overcoming resistances. Thus, when a locomotive begins to move a train, a 
ji portion of its force works against, and balances, the resistances of friction or of an 
up-grade, while the remainder, acting as unbalanced force upon the inert mass of 
the train, increases its velocity. 

i For, in order to move a body previously at rest, we must apply to it a force greater 
than any resistances which tend to keep it at rest. Otherwise there will bo no unbal- 

II anced excess of foice to give motion to the unresisting mass of the body. An upward 
pull of exactly one pound will not raise a one pound weight, but will merely balance 
the downward force of gravity. If we increase the upward pull from one pound 
(= 16 ounces) to 17 ounces, the ounce so added, being unbalanced force, will give 
motion to the mass, and will accelerate its upward velocity as long as it continues 

i to act. If we now reduce the upward pull to 1 pound, thus making it just equal 
to the downward pull of gravity, the body will move on upward with a uniform 

velocity; but if wo reduce the upward force' to 15 ounces (= yf pound), then there 
will be an unbalanced downward force of 1 ounce acting upon the body, and this 
downward force will generate in the body a downward or negative acceleration or 
1 retardation, and will destroy the upward velocity in the same time as the upward 
excess of 1 ounce required to produce it. If now we wish to prevent the body from 
falling, we must again increase our upward force to 1 pound. 

It is plain that during any time, while the 17 ounces upward “force” were acting 
against the 1C ounces downward ‘ resistance,” the product of total upward force 
X distance must be greater than that of resistance X distance. The excess is tho 
work dono in accelerating the velocity, by virtue of which the body has acquired 
kinetic energy or capacity for doing work in coming to rest. See Art. 19, p. 318 a. 

On the other hand, while the upward velocity was being retarded, the product of 
total upward force X distance was less than that of resistance X distance, the 
difference being the work done by the kinetic energy against the resistance of 
gravity. 

But the total work dono from the time when the force was applied to that when 
the weight again came to rest, is equal to the product, mean force X distance. 

In practice, the term “ work ” is usually restricted to that portion of the work 
which a force performs in balancing tho resistances which act against it; in other 
words, to the work done by so much of the force as is equal to the resistance. 

j With this restriction, we have 

Work = force X distance, = resistance X distance. 

Thus, if the resistance be a friction of 4 lbs., overcome at every point along a dis¬ 
tance of 3 feet; or if it be a weight of 4 lbs., lifted 3 feet high, then the work done 
amounts to 4 X 3 = 12 foot-lbs. The lifting of weights; and the friction encoun¬ 
tered in merely moving them by sliding or rolling, constitute the principal sources 
of resistance, and of work, in practice. 

(d) In cases where the velocity is uniform , as in a steadily running 
i machine, the force is necessarily equal to the resistance; and where the velocities at 

tho beginning and end of any work are equal (as where the machine starts from rest 
and comes to rest again) the mean force is equal to the mean resistance. In such 
i cases, therefore, the two products, mean force X distance and mean resistance X dis¬ 
tance are equal, aud we have, as before, 

Work = force X distance = resistance X distance. 

(e) The work done by a horse against resistances, in drawing a heavy load on a 
level road, consists entirely in overcoming the friction at the axles and rims of 
the wheels; but in drawing it wp hill, he partly lifts the load and himself also. 
In the first case, his work is not the product of the weight of tho load multiplied by 
the length of the haul (for here the weight in itself is not the resistance), but 
so many pounds of rolling and axle friction multiplied by that length. In going 
up u hill , his work consists of friction overcome through one distance; namely, the 
length of the hill; and of the weight of the load, vehicle and himself, lifted through 
another distunce, namely the vertical height of tho hill. 






318 


FORCE IN RIGID BODIES. 


(f) In calculating the work done by machinery, etc., allowance must be made for 
this expenditure ot a portion of the work in overcoming lesistances. Thus, in pump¬ 
ing water, part of the applied force is required to balance the friction of the different 
parts of the pump; so that a steam or water ‘’power,” exerting a force of 100 lbs., 
and moving t> feet per second, cannot raise 100 lbs. of water to a height of G feet 
per second. Therefore machines, so far from gaining power, according to the popular 
idea, actually lose it, in one sense of the word. In the practical a, plication of all 
machinery, the object is two-fold; namely, to enable us conveniently to apply force 
(1st) to balance, leact against, or destroy, the resisting forces of friction, and the 
cohesive forces of the material which is to be operated on; and (2d) to give motion 
to the unresisting matter of the machine, and of the material operated on, after tho 
resisting forces which had acted upon them have thus been rendered ineffective. 

(§) That portion of the work of a machine, etc., which is expended against fric¬ 
tion, is sometimes called “lost work,” or “prejudicial work,” while only 
that portion is called “ useful work ” which renders visible and tangible service 
in the shape of output, etc. Thus, in pumping water, the work done in overcoming 
tho friction ot the pump and of the water, is said to be lost or prejudicial, while the 
useful work would be represented by the product, weight of water delivered X 
height to which it is lifted. 

r lhe distinction, although artificial, and somewhat arbitrary, is often a very con¬ 
venient one ; but the work is ot course not actually “lost,” and still less is it ‘‘pre¬ 
judicial ; for the water could not be delivered with< ut first overcoming the 
resistances. A merchant might as well call that portion of his money lost which he 
expends for clerk-hire, etc. 


(k) For a given force and distance, the work done is independent of the 

time ; for the product, force X distance, then remains the same, whatever the time 
may be. But the distance through which a given force will work at a given velocity, 
is ot course proportional to the time during which it is allowed to work. Thus, in 
order to lift 50 pounds 100 feet, a man must do the same woik, (= 5000 foot-pouuds) 
whether ho do it in one hour or in ten; but, if he exerts constantly the same force,■, 
he will lift 50 lbs. ten times as high in ten hours as in one, and thus will do ten times 
the work. Thus, for a given force, the work is proportional to the time. 

Art. 18 (a). Power. The quantity of any work may evidently be considered, 
without regard to tho time required to perform it; but we often require to know 
tho rate at which work can be done; that is, how much can be done within a 
certain time. 

The rate at which a machine, etc., can work is called its power. Thus, in selecting 
a steam-engine, it is important to know how much it can do per minute, hour, or 
day. We therefore stipulate that it shall be of so many horse powers ,* which means 
nothing more than that it shall be capable of overcoming resisting forces, at the 
rate of so many times 33.000 foot-pounds per minute wheD running at a uniform 
velocity or so that force X distance = resistance X distance. 

(b) The horse-power, 33,000 foot-ponnds per minute, is the unit of power, 
or of rate of work, commonly used in connection with engines. See p. 377. 
In the study nf Mechanics the foot-pound per second is used. 


(c) Up to the time vvhen the velocity becomes uniform, the power, or rale 
of work, of the train, in Art. 16 ( d ), p. 315, is variable, being gradually accele¬ 
rated. lor in each second it overcomes its resistances (and moves its point 
of application) through a greater distance than during the preceding second. 
Also, alter the steam is shut off, tho rate of work is variable, being gradually 
retarded. When the force ol the steam just balances the resistances, the rate 
of work is uniform. 

All these remarks, as well as those of Art. 17, apply alike to all kinds of 
heavy machmery; no matter by what kind of force it is driven; the machinery 
takes the place of the train just spoken of; and the friction of the cog-wheels 
gudgeons pivots the grinding, sawing, or whatever the work may be, takes the 
pfoce of the grades curves, and friction of the train. All alike are simply 
cases of force at work. 1 y 


(d) Power == force X velocity. Since the rate of work is the work 
done in a given time, as so many foot-pounds per second, we may find it bv divid¬ 
ing tho work in foot-pounds done during any given time, by the number of 
seconds in that time. Thus , J 


Power ==» rate of work = f orce in pounds X d istance in feet 

time in seconds 










FORCE IN RIGID BODIES. 


318a 


But this is equivalent to 

Power =* rate of work — force in pounds X ( ^ 9 *' ance * n fe e ^ 

time in seconds 

*= force in lbs. X velocity in feet per second. 

Or, if we treat only of the work of that force which overcomes resistances: or in 
cases where the velocity is either uniform throughout or the same at the 
beginning and end of the work; 

Power, = rate of work resistance, v velocity, 
in ft-lbs. per sec. in ft-lbs. per sec. = in lbs. x in ft per sec. 

Thus, if the resistance is 3300 lbs. and is overcome through a distance of 10 
feet in every minute; or if the resistance is 33 lbs. and is overcome through 
a distance of 1000 feet per minute, the rate of the work is in each case 
the same, namely, 33,000 foot-pounds per minute, or one horse-power; for 

lbs. vel. lbs. vel. 

3300 X 10 = 33 X 1000 = 33,000 foot-pounds per minute. 

(e) The same “power” which will overcome a given resistance through a 
given distance, in a given time, will also overcome any other resistance through 
any other distance, in that same time, provided the resistance and distance 
when multiplied together give the same amount as in the first case. Thus, 
the power that will lift 50 pounds through 10 feet in a second, will in a second 
lift 5oo pounds, 1 foot; or 25 pounds, 20 feet; or 5000 pounds ^ of a foot. 
In practice, the adjustment of the speed to suit different resistances, is usually 
effected by the medium of cog-wheels, belts, or levers. By means of 
these the engine, water-wheel, horse, or other motive power, exerting a given 
lorce and running at a given velocity, may be made to overcome small resist¬ 
ances rapidly, or great ones slowly, as desired. 

Art. 19 (a). The work which a body can do by virtue of its 
motion $ or (which is the same thing) the work inquired, to hrinsr 
the body to rest. Kinetic energy, vis viva, or **living force. ,> 

remarked, a force equal to the weight of any body, at any place, 
will, in one second, give to the mass or matter of the body a velocity = a, or 
(on the earth s surface) about 32.2 feet per second. Or if a body be thrown 
upward with a velocity = g, its weight will stop it in one second. 

Since, in the latter case, the velocity at the beginning and at the end of the 
second are, respectively, = g feet per second, and = 0, the mean velocity of the 

body is _iL feet per second. Therefore, during the second it will rise JL- feet, 

. 2 
or about 16 feet. In other words, the work which any body can do, by virtue 
of being thrown vertically upward with an initial velocity (velocity at the 
start) of g feet per second, is equal to the product of its weight multiplied by 

~2~ feet. Or, 

work in foot-pounds = weight X 9 

2 

Notice that in this case (since the initial velocity v is equal to g), v = 1. 

Suppose now that the same body; be thrown upward with double the former 
velocity; i. e. f with.an initial velocity equal to 2 g (or about 64 feet per second), 
uince gravity requires (Art. 8 (c), p. 310), two seconds to impart or destroy this 
velocity', the body will now move upward during two seconds, or twice as long 
a time as before.. But its mean velocity now is g, or twice as great as before. 

1 herefore, moving for double the time and with double the velocity, it will 
travel four times as far, overcoming the same resistance as before (viz.: its 
own weight) through four times the distance. 


Thus, by making its initial velocity v = 2g, i. e., by doubling its v making 


it ■=* 2, we have enabled the body to do four times the work which it could 
do when its was 1; so that the work in the second case is equal to the 


9 














3186 


FORCE IN RIGID BODIES. 


product of that in the first case multiplied by the square of 


Work 
in ft-lbs. 


weight 


X 



«= weight X 
= weight X 


9 

2 
v 2 
^9 




And it is plain that this would be !the case for any other velocity. Now the 
total amount of the work which the body can do, is independent of the 
amount of the resistance against which it is done; for if we increase the 
resistance we diminish the distance in the same proportion, so that their 
product, or the amount of work, remains the same. The above formula 
therefore, applies to all cases; i. e ., the total amount of work, in foot 
pounds, which any body will do, against any resistance, by virtue of its motion 
alone, in coming to rest, is 

"Work = weight of moving body, in lbs. X S( l uare its velocity in ft per sec.’d 

*9 

= weight of moving body, in lbs. X fall in ft required to give the velocity 
= weightof moving bod y, in lbs. v square of its velocity in ft per second 
~~9 ’ 2 " 

In these equations, the weight is that which the body has in any given place 
and y is the acceleration of gravity at that same place. 

(b) Since the Weight C 'J a body is its mass (Art. 11, p. 312), the last formula 
becomes, by “method A/’ Art. 11 (d), 

work _ mass of moving body v square of its velocity in ft per second 
in foot-pounds in “matts ” ^ ' jg-- — 

and by “method B,” Art. 11 (e), 

work _ mass of moving body v square of its velocitv in ft per second 
in foot -poundals in pounds - % --- 

(c) In the above equations the left hand side represents the work (or resis¬ 

tance overcome through a distance) in any given case, while the riqht hand 
side represents the kinetic energy of the body, by which it is enabled to do 
that"work. Some writers call this energy “vis viva,” or “ living force” a 
namo formerly given (for convenience) to a quantity just double the enercv 
or = mass X velocity 2 . 6-M 

(d) As an illustration of the foregoing, take a train weighing 1 120 000 
pounds, and moving at the rate of 22 feet per second. The kinetic ene’rev 
of such a train is 

energy = weight X - y . eI ° clty2 ; Q r, 

'2g 

1,120,000 lbs. X — = 8,400,000 ft.-lbs. 

64.4 

That is, if steam be shut off, the train will perform a work of 8,400,000 ft -lbs 
in coming to rest. Thus, if the sum of all the resistances (of friction air 
grades, curves, etc.) remained constantly =~ 5000 lbs.,* the train would travel ’ 


8,400,000 ft.-lbs. 
5000 lbs. 


= 1680 ft. 


(*) We thus see that the total quantity of work which a body can do by virtue 
of its motion alone, and without assistance from extraneous forces, is in pro¬ 
portion to the weight of the body and to the square of its velocity when it 
begins to do the work. For example, suppose that a train, at the moment 
when steam is shut off, has a velocity of 10 miles an hour and that the kinetic 
energy, which that velocity gives it, will by itself carry the train against the 


* In practice, this would not be the case. See pp. 374, etc. 























FORCE IN RIGID BODIES. 


318c 


resistances of the road, etc., for a distance of one quarter of a mile before it 
stops. Then, if steam be shut off while the train is moving at 5, 20 , 30 or 40 
miles per hour (i. e. with Y, 2, 3 or 4 times 10 miles per hour) the train will 
travel i , 1, 2 hi or 4 miles (or 4, 9 or 16 times mile) before coming to 

rest.* 

But the rate of work done is proportional simply to the resistance and the 
velocity (Art. 18 d, p. 318). Therefore, the locomotive whose steam is shut off 
at 20, 30 or 40 miles per hour, will require, for running its 4, 9 or 16 quarters 
of a mile, but 2,3 or 4 times as many seconds as it required at 10 miles per hour. 

The same principle applies to all cases of acceleration or of retardation.! 

For instance, in the case of a falling body, the distance through which it 
must fall in order to acquire any given velocity is as the square of that 
velocity, but the time required is simply as the velocity. Also, if a body is 
thrown vertically upward with any given velocity, the height to which it will 
rise by the .time gravity destroys that velocity, will be as the square of the 
velocity, but the time will be simply as the velocity. See Caution, p. 362. 

Art. 30 (a). The momentum of a moving body (or the product of its 
mass by its'velocity) is the rate, in foot-pounds per second, at which it will 
begin to work against a resisting force equal to its own weight , as in the case of 
a body thrown vertically upward. At the instant when it comes to rest, its 
momentum, or rate of work, is of course = nothing. Therefore its mean rate 
of work, or mean momentum, is one-half of that which it has at the moment 
of starting. 

Thus, suppose such a body to weigh 5 lbs. Then, whatever its velocity may 
be, 5 pounds is the resisting force, against which it must work while coming 
to rest. Let the initial velocity be 96 feet per second. Then its 

momentum = mass X velocity = 5 X % = 480 foot-pounds per second; 
and, while coming to rest, its 

mean momentum = mass X Y . eloclt Y = 240 foot-pounds per second. 

& 

Now, in falling, the weight of the body (5 lbs.), would give it a velocity of 96 
feet per second in about three seconds. Consequently, in rising, it will destroy its 

velocity in the same time. In other words, the time =* ae< !eieration * = * vel ^ cifc Y 

= 96 = 3 . Three seconds, therefore, is the time during which it can work. 

Now, if the mean rate of work in foot-pounds per second (at which a body 
can work against a resistance) be multiplied by the time during which it can 
continue so to work, the product must be the total work done. Or, in this case, 


240 X 3 = 720 foot-pounds. 


720 ft-pounds. 


work _ mean rate of work v time, 
in ft-lbs. in ft-lbs. per sec. ^ or No. of secs. 

_ weight X Te . l0 2 cit ? X 

=. weight X Tel ° city - i as io Art. 19 (a), >=* 5 X 'j-r’-- 
2 cj 

(h) We may notice also that since, in the case of a falling body, or of one 

thrown upward, Y eloci - Y is the time during which it must fall in order to 
g 

acquire a given velocity, or during which it must rise in order to lose it, 
therefore, 

velocity v velocity 

2 g 

so that 


mean velocity X time = distance traversed; 


weight X vel ° - 1 ^ 2 “ weight X 


velocity ^ velocity 


2 g 


weight X distance traversed = the work. 


g 


*This supposes, for convenience, that the resistances remain uniform 
throughout, and are the same in all the cases, which, however, would not hold 
good in practice. See Art. 11, p. 374. 

f Retardation is merely acceleration in a direction opposite to that of the 
motion which we happen to be considering. 




















318c? 


FORCE IN RIGID BODIES. 


Art. 21 (a). Energy Is indestructible. Energy, expended in work, is 
not destroyed. It is either transferred to other bodies, or else stored up in the 
body itself; or part maybe thus transferred, and the rest thus stored. But, 
although energy cannot bo destroyed, it may be rendered useless to us. Thus, 
a moving train, in coming to rest on a level track, transfers its kinetic energy 
into other kinetic energy; namely, the useless heat due to friction at the rails, 
brakes and journals; ami this heat, although none of Mis destroyed, is dissipated 
in the earth and air so as t) be practically beyond our recovery. 

Art. 22 (a). Potential energy, or possible energy, may be defined as 
stored-up energy. We lift a one-pound body one-foot hy expending upon it 
one foot-pound of energy. But this foot-pound is stored up in the “system” 
(composed of the earth and the body) as an addition to its stock of potential 
energy. For, while the stone falls through one foot, the system will acquire 
akinetic energy of one foot-pound, and will part with one foot-pound of its 
potential energy. 

(b) The potential energy of a “system” of bodies (such as the earth and a 
weight raised above it, or the atoms of a mass of powder, or those of 
a bent spring) depends upon the relative positions of those bodies, and 
upon their tendencies to change those positions. The kinetic energy of a 
system (such as the earth and a moving train of cars) depends upon the masses 
of its bodies and upon their motion relatively to each other. 

Familiar instances of potential energy are—the weight or spring of a clock 
when fully or partly wound up, and whether moving or not; the pent-up water 
in a reservoir; the steam pressure in a boiler; and the explosive energy of 
powder. We have mechanical energy in the case of the weight or springs or 
water; heat energy in the case of the steam, and chemical energy in that 
of the powder. 

(c) In many eases we may conveniently estimate the total potential energy 
of a system. Thus (neglecting the resistance of the air) the explosive energy 
of a pound of powder is = the weight of any given cannon ball X the height 
to which the force of that powder could throw it, = the weight of the ball X 
(the square of the initial velocity given to it by the explosion) -f- 2 g. But in 
other cases we care to find only a certain definite portion of the total potential 
energy. Thus, the total potential energy of a clock-weight* would not be 
exhausted until the weight reached the center of the earth; but we generally 
deal only with that portion which was stored in it by winding-up, and which 
it will give out. again as kinetic energy in running down. This portion is = the 
weight X the height which it has to run down = the weight X (the square of 
the velocity which it would acquire in falling/reeh/ through that height) -r- 2 a. 

(d) There are many cases of energy in which wb mav hesitate as to whether 
the term “kinetic” or “potential” is the more appropriate. Thus, the pres¬ 
sure of steam in a boiler is believed to be due to the violent motion of the 
particles of steam, which bombard the inner surface of the boiler-shell - so 
that, from this point of view, we should call the energy of steam kinetic But 
on the other hand, the shell itself remains stationary; and, until the steam is 
permitted to escape from the boiler, there is no outward evidence of energy 
in the shape of work. The energy remains stored up in the boiler ready for 
use. From this point of view, we may call the energy of steam potential energy 

(e) It seems reasonable to suppose that further know ledge as to the nature 
of other forms of energy, apparently potential (as is that of steam), might 
reveal the fact that all energy is ultimately kinetic. 

Art. 23 (a). There is much confusion of Ideas in regard to the actions to 
which, in Mechanics, we give the names “ force,” « energy,” “ power ” etc. 
This aiises from the fact that in every-day language these terms are used^ndis- 
cruninately to express the same ideas. 

Thus, we commonly speak of the “force ” of a cannon ball when flying through the 
air ; meaning, however the repulsive force which would he ex- rted between the bail 
and a building etc., with which it might come into contact. This force would tend 
to move a part of the building along in the direction of the flight of the ball, and 
would move the ball backward; (t. e., would retard its forward motion). But 

forlovrwA rCPU 8 t' V0 d ° eS , n0t exist until ,he bal1 strikes the building. 

ev / n o,’ ? 0m t ,' tt velocit - v an<1 weight of the ball what the amount 
..ilintil of m th r S deppuds , up ' m the strength, etc., of the building. If the 

r!\. f f 8 ’ * ! e . 1 f ’ r . c , e ! nay be 80 sll tfht as s a reel y to retard the motion of 

th b. 1 1 perceptibly, while, if the building is an earthwork embankment, the force 

* ( !‘’ or convenience we may thus speak of the energy of a system of bodies /'tbe 
earth and the clock-weight) as residing in only one of tiie bodies. 











FORCE IN RIGID BODIES. 


3186 


will bo much greater, an 1 may retard the motion of the ball so rapidly as to entirely 
stop it before it has gone a foot farther. 

The moving ball has great (kinetic) energy; but the only force that it exerts 
during its flight is the comparatively very slight one required to push aside the 
particles of air. 

The energy of the ball, and therefore the total work which it can do, is independent 
of the nature of the obstruction which it meets ; but since the work is the product of 
the resistance offered and the distance through which it can be overcome, the dis¬ 
tance must be inversely .‘is the resistance offered; or (which is the same thing) iu- 
j versely as the force required of, and exerted by, the ball in balancing that resistance. 

Since wor ^ j n ft-Iba = force, in lbs. X distance traversed, in feet, 


we have 


force, in lbs. = 


work, in ft-Ibs. 
distance traversed, in feet 


In other words, we may d fine force as th.e rate per foot of doing 
work. 

Art. (a). An impact, blow, stroke or collision takes place when a moving 
j body encounters another body.* The peculiarity of such cases is that the time of 
I action of the repulsive force due to the collision, is so short that generally it is im- 
i possible to measure it, and we therefore cannot calculate the force from the 
i m mentum produced by it iu either of the two bodies ; but since both bodies undergo 
a great chauge of velocby (i. e„ a great accelerat on) during this short time, we 
know that the repulsive force acting between them must be very great. 

We shall consider only cases of direct impact, or impact whero the centers 
of gravity of the two bodies approach each other iu one straight lino aud wlieie the 
nature of the surfaces of contact (Art. 25, p. 318/) is such that the repulsive force 
c iused by the impact also acts through those centers and in their line of approach. 

(b) This force, acting equally upon the two bodies (Art. 5 (/), p. 309) for the same 
| length of time (namely, the time duriug which they are in contact), necessarily 

pro luces equal and opposite changes in their momentums (Art. 12, p. 314). Hence, 

J the total momentum (or product, mass X velocity) #f the two bodies is always the 
s ini * after impact as it was before. 

(c) But the relative behavior of the two bodies, after collision, depends upon their 
elasticity. If th ',y could be perfectly inelastic, their velocities, after impact, would 
be equal. In other words, they would move on together. If they could be per¬ 
fectly elastic, they would separate from each other, after collision, with the same 
vel »city with which they approached each other before collision. 

(d) Between these two extremes, neither of which is ever perfectly realized in 
practice, there are all possible degrees of elasticity, with corresponding differences in 
the behavior of the bodies. The subject, especially that of indirect impact, is a very com¬ 
plex one, but seldom comes up in practical civil engineering. We therefore refer the 
reader, for full treatment of it. to such works as those of liankiue and Weisbacli. 

(e) ** In sonn careful experiments made at Portsmouth dock-yard, England, a man 
of medium strength, and striking with a maul weighing 18 lbs., the handle of which 
was 44 inches long, barely started a bolt about }/$ of an inch at each blow ; and it 
required a quiet pressure of 107 tons to press the bolt down the same quantity; but 
a small additional weight pressed it completely home.” 



Art. 25 (ft). “Applied” and “imparted” forces. When force is 
applied by contact (Art. 5 (c) p. 308), the repulsive force generated, and which 

* The term “ impulse ” ivas formerly applied to cases of collision. It is now r used 

to signify the action of a force during a certain time, as explained in Art. 12 (a), 
p.314. 













318/ 


FORCE IN RIGID BODIES. 


tends to push the two bodies asunder, acts (theoretically) always at right angles 
to the surface of contact, no matter what angle the applied force may form with 
that surface. (For apparent exceptions, see below). We shall confine our¬ 
selves to cases where the two bodies tend to approach each other in one 
straight line; and, for convenience, we shall give the name applied force 
to the tendency of the body a to aproach the other body (B), and the name 
Imparted force to the action of the repulsive force upon B. 

(1>) If the two rigid bodies tend to approach each other in a direction at right 
angles to the surface of contact, as at a. Fig. 1, or at right angles to a tangent 
to that surface, as at c, d and e, then the imparted force is equal to the applied 

But if the applied force, F g, Fig. 1 a, is oblique to the surface s t of contact, 
then (theoretically) the imparted force v g, is less than the applied force. 

To find the amount of the theoretical imparted force in such cases; let iq 
represent by scale the amount of the applied force (in pounds, etc.), and draw 
t s at right angles to s t. Then i s will represent, by the same scale, the 
imparted, force, acting upon B in the direction v g; and s g will represent the 
lorce with which a now tends to slide up the plane s t. As it thus slides, it will 
(it it still retains its original tendency in the direction F g), continue to exert 
the pressuro v g (= i s ) against B at each point of its path, s t. 


(c) But in practice we invariably find apparent violation of tilts 
principle. Thus, in many cases, B, Fig. 1 a, would move in the direction F q of 
the applied force, while a, instead of sliding along s t, would remain in contact 
with B at g and continue to move onward with it in the direction F q, but with 
diminished velocity because of the retardation (or negative acceleration) 
caused by the repulsive force acting in the direction g F. This shows that 
Pft r p, ( or ? e h ere a< ^ 9 m the line F g, or obliquely to the surface , s t, of contact. 

1 his, however, is merely because even the most highly polished fiat surface, 
as s t, is not (as it appears to the eye) a plane, but is, in fact, a more or less 





jagged surface, Fig. 2, as would appear under a sufficiently powerful micro¬ 
scope, so that the force F g, instead of forming the apparent angle Fes with 
one smooth surface st of application, really becomes a series of mrallel 
forces, as c, d and e, which form other angles with a number of surfaces m m 
n n etc., of application, inclined (often in different directions) to the gen era! 
surfaces f, as shown. Among these surfaces maybe some, as m m, at right 
angles to the applied force; and, the force c will be imparted to them in its 
original directum, although applied obliquely to the apparent surface s t or 
in the case ot the two forces d and e, applied to the surfaces, n n and o o if 
the sliding tendencies along the two surfaces are equal and act in opposUion 
to each other, the combined resistance of the two surfaces, n n and o V is 

aSesUE 1 forces f ° rCeS ’ “ W ° Uld be that of a ^ngle surface at right 

(d) It is of course entirely out of the question to ascertain the exact 
resistance of each such microscopic projection in any given case. Instead of 
this, we find by experiment the combined resistance which all of the nroiec- 
turns in a given case offer to the sliding force, s g Fig. 1 a, and give to this 

See Arts. 01 to G4/pp. §53 to 356; S3 ‘‘ Frlc- 










FORCE IN RIGID BODIES. 


318<? 



Fi<?. 3 


w4 


(e) If it were not for friction, a body, Fig. 64, p. 355, would slide down an 
inclined plane w x, no matter how slight its inclination might be; but we 
know that friction often prevents such sliding, even when the plane forms a 
considerable angle, y x w, with a horizontal lineyx. When this angle becomes 
so great that the body is just on the point of starting to slide down, it is 
called the angle of friction; and in Fig. 1 a, if the force F g, does not form 
with the perpendicular v g an angle v g F, greater than this angle of friction, 
then friction will oppose all the sliding force s g, no matter how great it may 
be; so that« g also will be imparted, in its own direction, to B; and the com- 
bined effect, or resultant of s g and v g, is t g. In other words, the entire 
applied force i g will be imparted to B at g , and in its original direction b g. 
This is merely another way of explaining the action of the forces, c, d and e, 
Fig. 2. 

(f) This remark is particularly applicable to the case of the masonry 
joints ill tlie abutments of stone arches; especially those of large 

span, with small rise. The pressure which such an arch 
exerts upon its abutments is very great; and its line 
of direction changes at each joint, as at o, n and m, Fig.3. 

It therefore becomes necessary first to find the posi¬ 
tion of this line (see Art. 72, p. 359), so as to know how to 
draw the varying inclinations of the joints nearly at 
right angles to it; otherwise, the upper courses 
are liable to slide outward upon the lower ones, as 
shown by the arrow. In small arches of considerable 
rise, the sliding portion of this force may be safely 
resisted by good mortar or cement, if sufficient time be first allowed for it to 
harden properly; but in large ones, the direction of the joints must be relied 
on, unless we increase the expense by making the abutments unduly thick. 

The angle of friction of masonry on masonry (see table, p. 373) is about 32°. 
Therefore, if at any bed-joint of masonry, as in the abutment of Fig. 3, the 
resultant that cuts said joint at the line of pressures, o nm , does not differ 
more than 32° from a perpendicular to said joint, there will be no unresisted 
tendency to slide at that joint. 

(g) Fig. 4 is added merely to illustrate more strikingly the necessity for 
clearly distinguishing between applied 
and imparted forces. Here the great 
force a o is applied to the body B B at 
the point o; but all of it that is theo¬ 
retically imparted to the body, or pro¬ 
duces any kind of effect upon it, is the 
very small amount represented by c o, 
at right angles to the surface of B B at o, but we have seen that in practice 
friction increases it, and makes its direction coincide more nearly with that 
of a o. 

t All this will be better understood after studying Composition and Resolution 
of Forces, Arts. 28, etc., pp. 319, etc. 

(h) If force f be imparted to any rigid body, as N, Fig. 5, at any point 

c; and if fo represent the direction 
in which it was imparted, whether 
as a pull or as a push, then the force 
will produce the same effect upon 
the body considered as an entire 
mass, as if it had been imparted as 
either a pull, or a push, in the same 
direction, at any other point of the 
body in said line; as at i. t , s, o, &c. 

Under Composition and Resolution 
of Forces, it will be seen to be some¬ 
times necessary to consider a push 
f c, to be changed to a pull o h, and 
vice versa, when we wish to ascertain the joint effect or resultant of a pull 
and push imparted to a body at the same time. See Remark 1, Art. 29, p. 320. 

(I) The foregoing important principle holds good, no matter how many differ¬ 
ent forces may be acting upon the body at the same time, in different directions; 

22 


B Fig. 4= B 



Jig 5 














318 A 


FORCE IN RIGID BODIES. 


or how much the direction of their joint effect, or resultant, may differ from 
that of any one of them; the action of each force, considered separately , may 
be regarded as just stated. The tendencies of several forces, acting at the „ 
same moment, may therefore frequently be first investigated one by one;^ 
and these tendencies then combined into one ; or the forces themselves may 
first be combined into one or more resultants, as directed under Composition I 
and Resolution of Forces, and the effect of these resultants considered. The 
engineer has generally to divide all the forces acting upon his structures into \ 
two classes; namely, those whose tendency Is to secure the stability he 1 
requires; and those which tend to impair that stability. He, therefore, first 
finds the resultant, or joint effort, of each class separately; and then compares 
these two resultants with each other. 

(J) It is plain that if, instead of regarding the body as rigid, we considered it as 
elastic, or as breakable, an entirely different course would be necessary, as 
the question would then become one on the strength of materials: for the force 
f , applied at c or t , as a push, might break off the pieces c and t; and so with 
the same force as a pull at s or o. Although masonry, iron, timber and other 
building materials are by no means absolutely rigid, yet generally they may 
be assumed to be so when we are investigating the effect of force to over¬ 
throw or derange the structure as a whole. 

Art. 26 (a). When different forces act upon a body, it is absolutely essen¬ 
tial, in considering their effects upon it, to know whether they all act in ilie 
same plane) for if they do not, their effects become totally different. 


c 


o 

o 


A flat piece of paper is a plane, and if on it we draw any number of straight Hues, 
in any direction whatever, they will represent so many forces all acting in that same 
plane; that is, the same flat surface coincides with the directions of all of them It 
will evidently do the same in whatever position this plane surface may bo placed 
whether horizontal, vertical, or inclined. Straight lines drawn on the floor of a room’ 
will represent forces in that same plane; lines on the ceiling, forces in that same’ 
plane; which of course is not the same plane as that of the floor; so with lines on the 
sides of the room. All the lines o t, i t, a t, c t. Fig. 6, are in the same plane toic. 
Although i t is in the plane toic; and t e at the same time in the plane t eg c, and 
in the plane tone; and s e in the piano sn eg; and t s in the plane i tes, still all of 
these, namely, it,te, se , and t s, are evidently in the same plane ites. Any two 
lines which meet, or would meet or intersect each other if sufficiently extrnd^d in 
either direction, are in the same plane; as o t, i t; or s t, i t; or t s, c s. Still two 
lines may be in the same plane, and yet not meet if extended; as for instance, the 
parallel lines c t, g e, in the plane tege. The lines at and g e, being in parallel 
planes, oict and ns eg, cannot meet if extended. Kor parallel forces, see p. 347. 

Remark. Wo must not confound acting in, with acting on, upon or against 
tho same plane. The floor of a room is a plane, and upon or against that same 
plane, forces in a thousand different planes may act. The distinction is so 
self-evident, that a bare allusion to it will prevent mistake. See Fig. 1 , p. 347. 

Art. 27 (a). Stress or strain takes place when two forces act upon a 
body or particle, either in opposite directions or in directions meeting a£ an 
angle * This occurs when force is applied to a body in such a way that at a 


x 



* Modern writers on the Strength of Materials call this action of opposite 
forces "stress;" and apply the term “ strain ” to the change of shape which a 
body undergoes when thus acted upon. We use the word “strain” in 
its usual sense, viz.: as denoting stress, or the action of the opposing forces 
Under Strength of Materials, pp. 4 54, etc., we use the word “stretch” for 
change of shape, as better expressing that idea than does the word “strain ” 












FORCE IN RIGID BODIES. 


318l 


gi von instant, it acts unequally (or in different directions) upon different parts 
of it; in other words, when force is communicated through matter. Different 
particles of the body then tend to move with different velocities, or in different 
lirections, and thus to separate from each other. This is prevented, if at all, 
by the inherent cohesive forces of the material, which tend to hold the 
jparticles together or in their original positions. These forces thus transfer 
the extraneous force from one particle to another, and compel it to give a 
small acceleration to the entire mass of the body instead of a greater accelera¬ 
tion to the small part lying near the point of application. 

n Thus, if we attempt to move a heavy weight suspended by a cord, however 
.ong, by pulling upon a string attached to one side of the weight, the pull 
{tends first to set in motion the particles near its point of application, while the 
other particles of the body have not yet been acted upon by it. This tendency 
is resisted by the inherent cohesive force thus called into action between 
these particles and those next beyond them, which force, while it pulls the 
first particles in a direction opposite to the tension of the string, pulls the next 
ones in the direction of that tension, thus (as it were) transmitting the pull 
of the string to them. The force, or stress, or strain, is thus rapidly trans¬ 
mitted from particle to particle throughout the body, each particle being 
subjected to a strain by the two forces which act upon it in opposite directions. 


! (b) The inherent cohesive force of matter, by which its particles are held 

e in close union, frequently causes the matter itself to appear to resist straining 
Iforce; thus, a cake of ice may sustain a great pressure; but if we destroy its 
{cohesive force by converting it into water, it will yield readily, So with the 
metals and stones if reduced to dust. It is not the material that resists being 
{broken; but the inherent cohesive force which holds the particles of the ma¬ 
terial together. If they are not sufficient to resist the separating tendency 
of the extraneous forces, the body will be broken. 


This view of the subject is treated of under “Strength of Materials,” pp. 4:H, 
etc. We here confine ourselves to the consideration of the effect produced upon the 
rigid body, as a whole , by two external forces acting upon it either in opposite direc* 
tions or in directions meeting at an angle. 

j (c) Stresss, or Strain, as thus considered, is the action of equal forces, or of 
{equal parts or components of unequal forces, acting upon a body in opposite directions 
’ in the same straight line. See foot-note *, p. 318 h. 

t : (1) If the imparted forces, as ca and b a , Fig. 9 l /£, p. 320, form an angle with each 

other, then equal components, c i and bo, of the two forces, react against each other 
{ at the point a as strain, while the other two components, (iaof c a, and oaof ba, 
which may or may not be equal) form a resultant, na, which acts at a as unbalanced 
i (force, giving motion to the body unless prevented by some third force. 

(2) If unequal forces act upon a body in opposite directions in the same straight 
lino, the smaller force, and a portion of the greater force equal to the smaller one, 
act against each other as strain, and the remainder of the greater force (i. e. the 
resultant), gives motion to the body in its own direction. 

(3) If two equal and opposite forces act upon a body, they have no resultant. 

In all these three cases the two equal and opposite forces, or components, destroy 
jeach other’s effects by acting against each other as strain, and thus give the body no 
tendency to move in either direction. 

i Strictly speaking, the two equal forces, straining against each other, do not even 
jkeep a body at rest; but the body rests merely because the two forces balance each 
{other, and therefore cannot prevent it from resting. As a matter of convenience only, 
we may, however, say they keep it at rest. 

(d) The mere fact that a body is subjected to great strains from equal forces acting 
upon it in opposite directions, does not of itself render the body more difficult to 
move than if it were free from strains. . 

Thus, let 15, Fig. 7, be a block resting on a horizontal support, and acted upr>n by a 
downward f ace d of 100 tons, produced by an immense block of granite resting upon 
B. Now it is plain that this 100 tons downward force will be met and balanced by a 
| ioi tons upward force w, being the resistance of the horizontal support. These two 
equal reacting forces produce in the body B a strain of 100 tons; but evidently do 
not impart to it as a whole any tendency to move in any direction whatever; nor do 
they tend to prevent it from being moved in any direction. The body therefore 
remains as before a mere inert mass incapable of resisting the slightest moving force. 







FORCE IN RIGID B0DIE3. 


318; 


Now suppose no friction to exist at either the base or the top of B. Then th 
slightest horizontal force h, a mere breath, would slide B along the horizontal sup 
port, moving it from under the 100 ton block on its top. No matter how heavy 1 
might be, the same smallest force would slide it, the only difference being that th 
heavier it was the greater must bo its mass, and the less would be the velocity in 
parted in a given time. If perfectly hard and frictionle68 rollers were interpose 
between 11 and its support, B would slide over the rollers, under the action of force ? 
without moving the rollers in the least, or making them revolve, or even tending X 
so move them or make them revolve. 

The heaviest bodies resting upon the surface of the earth, as well as 
ourselves, would be swept along by the slightest breeze if it were 
not lor friction. 

If the screw of a vise be worked until it produces a great strain in 
the jaws of the vise, the vise is not thereby rendered more difficult 
to move. 




5 


a 


TTi cr '7 

Again,if astrainof thousandsof tons were produced bythejawsof a •=** 
vise in a body, weighing an ounce, this immense strain would not prevent, no 
even tend in the smallest degree to prevent, the ounce body from falling down fror 
the jaws of the vise. It is prevented by friction, which is simply the upward resistin 
force of the roughnesses or projections on the faces of the jaws. The two forces of thoi! 
ands of tons each, which produce the strain of the vise, are entirely destroyed , a 
regards their action upon the body as a whole. Hence they could not prevent th 
one ounce from producing motion in it ; nor could they affect it as a whole in an 
way; for all their action is against each other. It is on this principle alone thu 
strains do not interfere with motions. 

If a body H, Fig. 8, of 10 tons weight, is suspended from a long rope, its reactioi 
against the equal opposing force at the other end of the rope, produces a continuou 
strain among all the particles w hich compose the rope ; but this does not in the leas 
affect the rope considered as a whole, inasmuch as it does not tend to move it in an, 
direction. Now, in this case, there is practically no friction 
to be overcome; and we know from daily experience that it 
is therefore easy to move the unresisting body a little distance, 
by applying a very small horizontal force/. We cannot move 
it far, as, for instance, to m, because we then have not only 
to move, but to lift it through the vertical height v c. In doing 
this, it is true our force dues not have to sustain the entire 
weight of the body ; because most of it is sustained by the 
lope. Still, if we move it at all, we have to overcome some 
of its weight; otherwise, a mere breath w r ould move it, 
although very slowly. If we attempt to move it by an 
upward force u, we shall have still more of its weight to re¬ 
sist us; and if by a downward one d, we shall be resisted by 
the cohesive force of the rope. Therefore, in this case, we can move it more readil’ 
by the horizontal force f. 





(e) Since two equal opposing forces, or equal portions of unequal ones, thusbrinj 
each other to a stand-still, or equilibrate each other, they are called Static; from tin 
Latin “&<o, I stand;” and that branch of the science of force which treats only o 
cases in which all the applied forces keep each other at rest, is called “ Statics ” o 
“ Equilibrium.” It is with this branch of the science of Mechanics that th 
civil engineer is chiefly concerned. 


(f) Strain, like the force which produces it, is conveniently measured oi 
ex pressed in weights, as in pounds, tons. Ac. Its amount or quantity is equa 
to that of only one of the two equal opposing forces. Thus, if two men pul! agains 
each other at two ends of a rope, each with a force of 30 lbs , the strain on the ropi 
is but 3') lbs, as is made manifest if one of the men applies his pull through a snrini 
ba ance attached to his end of the rope. If the other man also does the same, hi 
balance also w ill show a strain of 30 pounds; but this does not indicate that the tota 
strain is 60 lbs.; for if ten such balances were inserted along the rope each woult 
show about* 30 lbs, but they would, of course, not increase the strain. If a rop 
passes over a pulley, and equal weights bo suspended at each end of it, then the tw< 
equal forces of gravity of the two weights strain against each other, and also Btraii 
the rope, to an amount equal to one of them. 


i 6 8 at "W ^/.than at the ends. See Suspension Bridget 
p. cm If toe rope had no weight or if its weight were supported upon a horizoiff, 
table in lino with the pull, the strain would be uniform (3 j lbs.) throughout the rope 

















FORCE IN RIGID BODIES. 


319 


e Art. 28. Composition and resolution of forces. If two forces, 
a o and b o , Figs. 0, whether equal or unequal, are imparted at the same time 
to an unresisting rigid body o, in directions either converging toward, or diverging 
from, the same point o, at any angle whatever; then the body o canuot possibly 



d 


be kept at rest by them ; or in other words, equilibrium cannot exist between them; 
H or they cannot balance, or completely react against each other; the body must move. 
8 Equal parts of each of the two forces will mutually destroy each other as strain 
f among the particles of the body ; while the remaining portions will unite to constitute 
i a single force r o, which will move the whole body in a direction o d, in the line r o 
extended ; and which direction o d will always be somewhere between those in which 
) the separate forces would have moved it. 

If we lay off c o and t o by any convenient scale, to represent respectively the 
t amounts of the forces a o and b o, and then complete the parallelogram o cr t; the 
i diagonal r o, measured by the same scale, will represent both the direction and the 
amount, of the singlo resultant force. 








320 


FORCE IN RIGID BODIES. 




The same process will answer also for forces which instead of motion, produce strain, not only in 
the particles of the body, but iu the body itself considered as a whole ; or, iu other words, a tendency 
to press or pull the eulire body iu a certain direction. Thus, suppose that two men were either pull¬ 
ing or pushing with the forces’ co and to; trying iu vaiu to detach a piece o of rock, from s cliff of. 
which it forms a portiou ; aud which, by its iuhereut force of cohesion to the cliff, defies their efforts^ 
Here we have a case of extraneous forces, resisted, or reacted against, or balanced, by strength op 
material. 

As in the case of motion, the two forces partly destroy each other as strain among the particles of 
the body; and the remainders combine to form the single force r o, which tends to move the whole 
body toward d. The rock resists this siugle force, by a cohesive force precisely equal, aud diametri¬ 
cally opposite to it; aud so long as it does so, there is strain but no motion. The piece of rock may 
have strength enough to oppose a much greater resistance; but cauuot actually exert it unless ti 
men also exert more force. 

In the matter of comp and res of forces, it must be remembered that when force is applied to a 
body in order to produce motion, care must be taken that there is no other force to prevent it; but 
when the force is intended to produce strain, it is equally necessary that other force should be present 
to oppose it; for strain is the opposition of forces. 

The fig oert, Figs 9, is called the parallelogram of forces. The two 
original forces co, to, are called the components of the force ro; which results from 
their joint action ; and the force r o is called the resultant of the original ones which 
compose it. The principle of the parallelogram of forces, than which there is none 
more important in the whole range of mechanical science, may be expressed thus: 
If any two forces, (both motions, or both strains,) whose directions either converge 
toward, or diverge from, the same point, be represented both in quantity and in di¬ 
rection by two adjacent sides of a parallelogram : then will their resultant be simi¬ 
larly represented by the diag of the parallelogram,* 





Rem. 1. If one of the forces, as c, upper Fig 9%, is 
n pull, ami Ibe other a push, then to find their result¬ 
ant o t we must, before drawing the parallelogram of forces, move (or imagine 
to be moved; oue of the forces to the opposite side of the point o, so as to change 
it from a pull to a push, or vice versa, so that both shall be pulls, or both pushes, 
as shown by the two lower figs. Otherwise we should obtain a wrong resultant 
n o of the top fig. Hither a push or a pull equal to ot, if applied at o, would be 
equal in effect to the push a and the pull c. The remark is of frequent use when 
finding strains in bowstring and crescent trusses ; as in many other cases. 

Rein. 2. When any three forces as a, b, c, form¬ 
ing only two angles axb and bxc, balance each other 
at any point x, then a straight line as oe can be 
drawn through that point so that all three forces 


shall be on one side from it; then also a parallelogram xn can be 
drawn on the three lines a, b c, having the middle line b for its 
diagonal; and this diagonal will be of a different character from 
the two outer forces a and c; that is ; if they are pulls, it will be 
a push, and vice versa. But if as in the three balancing forces 
t, i, s, three angles as s x t, txi, sxi, are formed, neither such a line, 
nor such a parallelogram can be drawn ; and the three forces will 
all be alike, all pulls or all pushes. All this is evident from the two 
figures. 



Rem. 3. We have alluded to equal parts of each component as being lost, or de¬ 
stroyed, by reacting against each other; thus producing within the body a straining 
of its particles; and therefore having no tendency to move, push, or pull, the body 
as a whole, in any direction. 

Let b a and c a be any two components, and na their resultant. From 
the two angles b and c, opposite to the diagoual, draw bo and ci at right 
angles to the diagonal; or to the diagonal extended, if necessary, as in 
Figfi.H- These two lines, bo, ic. will always be equal to one another; 
whatever may be the lengths and direction’s of the components b a, ca. 
When two forces, as b a, c a, are imparted at a, there occurs a loss of force 
equal to what would result from the reactiou of two forces equal to Ito and 
c i. It is lost by becoming strain against the cohesive forces of the parti¬ 
cles which compose the body a. In anticipation of what is said in Art 31, 
we will state that the force b a may be regarded as made up of the forces 
bo, oa; and the force co, of ci, ia; which act also in those directions, 
when b a and c o converge toward a. as in Fig 9^$ ; or iu the directions 
ao, ob, and ai,ic, when the forces diverge from a, as in Fig 9^. In either case, however, these 
forces, bo, ao, c i, i o, Ac, must be considered as being imparted at a. This being supposed, It be¬ 
comes plain that when 6 a and co meet at o, inasmuch as 6 o and ci destroy each other as strain 

against the internal cohesive forces of the body, there remains nothing to act upon the body consid¬ 
ered as a whole, except oa ami la; which, being together equal to na, (as seeu in the fig,) are. in 
other words, equal to, or actually compose, the resultant n a of the two components ba,ca. See Remo. 



* Components anil Resultants may be ealenlated by the form¬ 
ulas in Art 45, when a diagram is not considered sufficiently accurate. 

























FORCE IN RIGID BODIES, 


321 


We conceive that each of the original forces endeavors as it were to compel the other to leave its 
own course, aud follow that of its autagouist; aud the struggle continues until they have succeeded 
in forcing each other into the same direction. This is of course effected hy their reactions against 
each other; aud, as occurs in all cases or reaction, they expend equal parts of their forces on each 
other. When the two forces act in diametrically opposite directions, where there is no neutral diag 
direction thatcau lie adopted, there is no alternative but for the larger force to react against or de¬ 
stroy the smaller one entirely ; thereby losing an equal amount of its own force. Its remains totter 
on slowly in their former unchanged direction. The writer can see no difference of principle between 
i the reaction of opposite forces; that of oblique ones; and that of those at right angles to each other. 

Hem. 5. When the direction ab. Fig 9%, of one of the forces, forms an angle ban , 
greater than 90°, with the diagonal, the shape of the parallelogram ol forces becomes 
j) such that the two equal lines bo and ci, cannot be drawn at right angles to the diag 
a n itself; or within the parallelogram; in which case the diag must be extended 
1 each way, as to o and i ; and the lines bo , ci, must be drawn at right angles to the 
extensions. 

When this occurs, the component forces a o, at , cannot as in Fig 
9J-6 be measured on the diag a n of the parallelogram ; because they 
will be greater than it; but must, like bo, ci, be measured outside 
of the fig. And here it must be remembered that ao and a i no 
I longer measure forces acting (like those in Fi g 9 hi) in the same di- 
I reclion. Thus the strain along a b may be considered (see Comp 
; and Res of Forces, Art 31) to be made tip of two forces imparted 
at a; namely, a bor force equal to o 6, and a vert oue equal to ao, 
j acting upward. And the strain along ac, ns made up of one hor 
j force equal to tc, and a vert one ai. (greater than the whole diag,) 

I acting downward; both of them imparted at a. Hence, the re- 
I sultant a a we find is equal to the diff between the two vert compo- 
j nents a o and a i. Thus it is seen that this shape of the parallelo- 
j grata in no way affects the principle laid down in Remark 3. 



Art. 29. According to Art. 2 5h, the force w e, Fig 10, may be considered as im¬ 
parted to the rigid body B at any point whatever in its line of direction wc; also, 
the force act, at any point in its direction x d; conse- w 
quently, both of them may be considered as imparted at 
the same point a; inasmuch as it is situated in both these 
lines. Hence, it is immaterial, so far as regards the effect 
of those two converging forces upon the body considered 
as one entire rigid mass, whether they are actually im¬ 
parted like z o and yo, at the same point o; or like w e aud 
xi, at diff points i and e. For in either case their result¬ 
ant, or joint effect upon the body as a whole, is precisely 
the same; namely, a tendency to move the body in the 
same line of direction oat. This tendency will actually 
produce motion if no opposing force prevents; otherwise 
it will produce strain in the body. 

Rem. 1. Hence the resultant R, of two converging forces F/, Fig 10^; or of two 
diverging ones F/, Fig 10)4, acting in the same plane, hut imparted at diff point* 















322 


FOKUJri liN RIGID bodies, 


of a rigid body W, may be found as readily as when imparted at the same point; as 
at o, Figs 9, or Fig 10. 

Thus, produce their liues of direction, either forward as in Fig ; or backward as in Fig ; l j 
a* the case may require, until they meet, as at b. Make bn by any scale, equal to the force/; and r 
b c equal to the force F. From a aud c, draw lines respectively parallel to be aud ba ; thus complet-'! 1 
ing the parallelogram of forces, b n i c. The diag b i of this parallelogram, measured by the same i 
scale, will represent the reqd resultant R both in quantity, aud in direction. It is thus seen that it | 
is not necessary that the point b shall be in the body itself. 

Rkm. 2. It is perhaps almost useless to agaiu remind the young student that the bodies are all along i 
assumed to be rigid; or iuelastic, and incapable of being broken or bent by the imparled forces. For , 
otherwise the force/, in Fig 10>6, might split off the top of the body ; or F might crush to dust its . ] 
toe t ; or both might penetrate it. ltut, assuming that the material is sufficiently strong to resist ' i 
such splitting, crushing, and penetration, we at present coniine ourselves to the effect of the forces, * 
whether as motion, push, or pull, upon the hotly os a whole. The splitting, crushing, &c, is a mat¬ 
ter that must be considered under the head of Strength of Mnten'ials. It is of course quite as neces- J i 
sary in practice to pay attention to these effects as to the others, but it must be done by a separate 
process. 

Art. 30. Since the effect produced upon a rigid hotly (con¬ 
sidered as a whole) by the resultant (a c, Fig 11) of any two forces , 

( bc,dc ) tending to or from the same point, is the same as the joint 
effect of those two forces themselves, it follows that if we oppose 
to those two forces a third one (n c) equal to the resultant (a c), 
and diametrically opposite to it, that this third force will com¬ 
pletely react against, balance, or destroy said two forces; or rather 
their remains. It is frequently necessary to consider such a third I 
force, ( n c,) equal and opposite to a resultant (o c) ; and inasmuch 
as we do not know that any specific name has been applied to it, 
although one is needed, we suggest anti-resultant. Re¬ 
sultant (ac) may be defined to be a single force which will pro¬ 
duce upon a body considered as a whole, the same result that its 
components (6c, dc) produce. Or as a force which, if its direction 
were reversed, (thus making an anti-resultant,) would balance its components. 

Ip the preceding Figs, the arrows represent pressures; if all the arrows be reversed, thus indi- j 
eating pulls, the principle and processes remain precisely the sumo ; for force is still only force ; and 
its effect upon a rigid body, considered as u whole, is the same whether it act as n pull, or as a push. I 

When the forces diverge from the same point, their strain is a pull, or a tension; when they con¬ 
verge toward it, a push, or pres, or compression. 

Art. 31. T5y a process the reverse of that in Art 2S, any single force, od, Fig 12, 
may he resolved into two component ones, nd,md, one on each side of it, and in 

the same plane with it; 
which would produce the 
same effect as it upon a 
rigid body, d, (considered 
as a whole.) by merely 
drawing from d, 2 lines 
d g,dt, showing the di¬ 
rections of the two forces; 
and then, drawing from 
o two other lines o n, o m, 
respectively parallel to dg, dt ; thus completing the parallelogram (dnom) of forces, 
upon nd tut its diag. Then measure dn, and dm, by the same scale as od; and they 
will give the amount of each of those forces. 

It is plain that an infinite nnmber of differently proportioned parallelograms, such as dnom, dsoa, 
Jtc, may be drawn upon any line od as a diag; and iD »dj one of them, two adjacent sides will rep¬ 
resent components equal in effect to the single force od. represented by the diag. Thus the forces 
nd. md, are equal to o d, us regards their effect upon a rigid body d, as a whole. So are also the 
forces a d and a d ; consequently the effect of * d, and a d, is eqnal to that of n d , and m d. It will bs 
observed that the longer any two components on the same diag are, (as nd, m d, longer than sd, a d.) 
the more nearly in a straight line, and more directly opi>osed to each other, do they become: and 
consequently the more nearly do they mutually destroy each other; leaving smaller portions of eoch 
to act upon the body. Thus the portion of the great forces nd. m d, left to act upon the body d, is 
u# greater than that of the small forces sd, a d; this remainder being in both cases represented by 
the resultant o d. 

Rem. Tlenee, if we have two forces, as the two pulls 
ah, ac, Fig 12J6, whose amounts and directions both are 
given ; and which are counteracted, or held in equilibrium, 
by two other forces such as the two pulls af.ae, whose 
directions alone are known, it becomes easy to fiDd the 
amounts a d and a o of these last, thus : Complete the 
parallelogram bnct; and draw its diag at. Make ai 
equal to at, and in a line with it. Complete the paral¬ 
lelogram a dio-, then plainly ad will he the amount of 
the force in the direction a/; and oo that in the direc¬ 
tion u e. 












FORCE IN RIGID BODIES. 


323 


s Art. 32. It follows from the foregoing articles, that a single force cannot he 
resolved into two components, one of which only is in the same direction as that 
, force itself; for if a line representing that force be taken as a diag, it is self-evident 
l that no parallelogram can be drawn upon it which shall have any of its sides par- 
- allel to said diag. 

Therefore a rope, as a b. Fig 15, sustaining a wt w, so long as it remains perfectly vert, that is, pre¬ 
cisely in the direction of the force of gravity of the wt, will receive no assistance in upholding the 
wt by having added to it a single rope as ob, or by; or one extending front the wt itself in any in- 
i cliued direction. In other words, a perfectly vert rope cannot sustain one part of a load, and one in- 
1 cliued rope another part. All this, indeed, is a result of the fact stated in Art 15; that any force, 
1 however great, (as the vert force of an immense suspended weight w, Fig 15.) will be turned out of 
1 ! its direction by any other force, however small, (as a slight pull from a rope ob, or by,) unless there 
i be some third force to prevent it. In the present instance, this third force might be a third rope; 
1 for the rope a b will be relieved, and still remain vert, if we employ two oblique oues to assist it. pro¬ 
vided they be exactly opposite each other; or, in other words, that all three ropes, or forces, be in 
one plane. 



So also in the case of a vert post sustaining a load; the pres from the load cannot pass vert through 
the axis or the post, if the load at the same time is partly sustaiued by a single oblique brace pressing 
against the post. Indeed, such a brace, by turning away the direction of the strain from the axis 
of the post mav very materially diminish the power of the latter to sustain the load; for « will he 
found under Strength of Materials, that if the strain along a post or column does not Pass directly 
through its axis, the column may in some cases lose two-thirds of its strength. The principle of 
course applies to force in any other direction, as well as vert. 

A resultant may be greater or less than either one of its two oblique components; 
but it can never be greater, or even quite equal, to both of them: on the plain prin¬ 
ciple that any two sides of a triangle are greater than the third side. It the com¬ 
ponents are equal, and inclined to each other at an angle of 120 , the resultant will 
he equal to one of them; therefore, the same weight that would break a single vert 
rope, or post, would break two ropes each of the same strength as the single one, or 
two posts, inclined 120° to each other. If the angle o a b, or y a 5, which either of 
the forces form with the diag a b, exceeds 90°, see Rems 5, of pp 3-1, 337. 

Art. 33. The principle of the parallelogram of forces is of constant applica¬ 
tion in*constructions of every kind; for instance, bridges, centers, roofs, retannng- 
valls Ac Figs 13, 14, 15,16, show a few- of the most simple cases of force (the load 
W ) applied to produce strain ; by reacting against opposing forces ya oa, presented 
bv the walls. In all these, the load w, applied at a, is a single force of gramty; and 
consequently acts in a vert direction downward. It is to he resolved into two com¬ 
ponent forces in the direction a m, an, in order that we may find the strains which 
it produces (according to the ordinary phraseology) along the pieces a m, a n, so that 
we may proportion their dimensions to resist those strains ; which strains are in 
fact produced by the reactions of the three forces, of the load, and the two w alls. To 
do this in all the figs, from a draw a vert line a b, to represent the direction of grav, 
or of the force in the load w. On this line, lay off by any convenient scale, the dist 
a b to represent the amount in !bs, tons, Ac, of the load w Also, trom a draw the tw o 
lines a. vi, a v, in the directions of the reqd component forces. Then complete the 
parallelogram of forces, by drawing lines bo, by, from b, respectively parallel to 
\m an Then will a o, measd by the same scale as a b, give the amount of strain, 
whether push or pull, which the load w produces along the piece a m; and in like 
manner will a y give the amount which it produces along the piece a n. 

It must be especially borne in mind, that we here speak only of the amounts and directions of the 
» i y v t y, p pxtrnneou* load w alone; without reference to those produced by the Tveigbt 
'SS SiSC»t » i. not bin cwi r . then th. o, „ i. 

must of P course be drawn oblique; but if the force at a is gravity or wt, it mitsl be vert. 














324 


FORCE IN RIGID BODIES, 


Caution. See foot-note, p 556. 


Fig 16|4 shows that the strains e t, e s, are really due to the action and reaction 
of the wt and the ivalls ; although we often speak of them as due to the load l alone , 
which is represented by the diag e i. We have said that a force cannot produce strain 
unless there is opposing force to strain against. Now, when we place the force of the 
load l at the point e, it is evident that it is upheld by the walls at A and B; or in 
other words, that it reacts against these walls; and the walls against it. The wall A 
furnishes the force indicated by the arrow A ; and which may be considered as the 
resultant of the hor force c; and of the vert one o. So also the force B; as the re¬ 
sultant of m aud n. •; 


Now these forces 
A and B are ap- j 
plied at the point 
e, just as well as 
the load l is; for 
they pass up as 
pushes, along the 
rafters; as the 
force of l passes up 
as a pull, along the 
rope. The rafters 
and rope are mere¬ 
ly the mediums 
through which the 
three forces reach 1 

«; and the forces in passing through them from end to end, of course produce in 
them strains respectively proportionate to the forces. Now, the forces e t and e s, 
which are usually said to be produced by the load, are nothing more or less than the 
two forces A and B, produced by reaction of the walls; and which, for convenience 
of drawing the parallelogram of forces in practice, are laid off each way from e. We 
have then three forces t e, s e, and e t, all acting at e, to produce strain alone; and 
this they must do by straining against each other. 



The following is the manner in which they do so. The two hor components m and c, (which wilt 
always be equal to each other; no mutter how different the slopes of the two rafters may be,) being dia¬ 
metrically opposite iu direction, react or straiu against, or balance, each other ; thereby producing 
a hor strain, equal to one of them, throughout every part of each rafter. The two vert components o 
and n, (however uuequal they may be.) will together be equal to the load l; or to its representative 
e i; and having a direction exactly opposed to it, they react agaiust, or balance it; thereby producing 
in every part of the rafter ei.a vert strain equal to «; and iu the rafter e t, one equal to o. There¬ 
fore, since », is here greater thau o, the rafter e s bears more of the load l, than the rafter e t does; 
aud in the same proportion. 

Thus, we see that every part of each of the three forces « f, e t, e *, produces strain, by balancing 
an equal part of one of the others.. The walls really oppose to the load no force greater than its own ; 
namely, o and n, against e t. With the hor components t/i aud c, the walls react only against each 
other. J ^ 

As it is difficult, however, to introduce a new phraseology, in place of one which, although errone- 
ou.s, is in universal use, we also shall speak of component strains like el, es, as if they were real lv pro- 
duced by the resultaut, or load, e i. And in alluding to resultant motion, we shall probably often say 
they are the effects of component *, instead of effects of their remainder s, after the components have 
partially destroyed each other's moving forces by straining against each other to produce chance of 
direction. r b 


IIem 2. The truth of such examples as Fig 14, with a rope or string, may easily be 
shown by means of two spring balances, to which the ends m and n of the string may 
be fastened. Suspend a weight w from the string, and the balances will stuAv the 
strains along a m and a n. The balances must be held in inclined positions. 


The student should try all such experiments. This one will show that in proportion as the two parts 
a m, a n, of the rope, approach nearer to one straight line, the greater will be the strain produced 

upon them by any given load, or force u>; and so great will 
this be, that if the weight w he only- one pound, two of the 
C strongest men cannot strain the rope perfectly straight be¬ 

tween them. Or if they stretch the rope alone to as nearly 

gd "‘‘i- Matt * straight line as possible, and if then a weight of a few lbs 

_5{_ VWV “e suspended from it. this small weight will puli the men 

closer together. Or if the rope be stretched nearly straight 
between the two spikes so firmly driven as to require a great 
force to draw them, it will be found that a much smaller 
force applied as at w, will draw them readily. In other 
words, a rope so situated, and with force, or power ui, applied 
to it in this manner, between its ends, and oblique to its di¬ 
rection, becomes a machine; for by it power may, (to use tb« 


!’ 


% 

'4 




Iiq IfiJ 


0 



















FORCE IN RIGID BODIES. 


325 


ordinary incorrect expression,) be gained. Itiscalled the funicular lJIHCtlln© * or some¬ 
times simply the COl’tl. Fig shows tlie principle on which this machine is frequently employed 
for overoomiug a great resistance, r, through a short distance, by a small power p. One end c, ot a 
rope c d r. is firmly fixed. The rope passes over a pulley d ; and its other end is tied to the resist¬ 
ance, or load r. By applying a small downward force p. at the center of the rope, drawing it down 
to #, the load r is thereby raised a short dist; for the same great strain w hich the small force p pro¬ 
duces from s to d, extends also down the rope, from d to r ; except a slight loss produced by the 
friction of the pulley. Thus, the strain along the back-stays of a suspension bridge, is equal to that 
on the main chains just inside of the suspension piers; supposing the cables to rest upon rollers, in 
the theoretical consideration of ropes and chains, they are in most cases assumed not to stretch; to 
5>e perfectly flexible; without weight; and infinitely thin. 

In such a machine the two parts s c, s d, Fig are to be considered as two entirely distinct ties ; 
in the same manner as a m and o n, Figs 13 aud lti, are two distinct struts Each of these ties may 
have to sustain a different amouut of strain, depending on their respective inclinations to s p. Thus 
if the loadp. Fig 16>£, be suspended from a perfectly frictionless pulley or slip knot resting on the 
perfectly flexible cord c s dr, and if this pulley or knot be at first placed near c or d. it, with its load p 
will descend by gravity along the cord until it comes to rest at s. which is the lowest point, that the 
cord admits of its attaining and at which alone the angles of Inclination of s c and s d to s 
p become equal; and the strains on the two parts will then ue equal. But if as iu Fig 1+ the 
short string which sustains W is tied fast to the cord (so as not to move as the pulley did) at any 
point a, such that the angles of inclination of am and a n to the diagonal a b shall be different, then 
the strains or pulls along a m and a n will also he different. 

It Is immaterial whether m and n. Fig 14, or c and d, Fig 16M» arc at the same 
height or not. 

For more on the funicular machine see p 344. 

Let the end g of the rope g c o n be fixed ; a power of 9 tons at n ; the rope passing over a pulley 
at P; aud bent out of line at c by a fixed pin. Make c g and c o by scale each equal to the power 9 
atn; and complete the parallelogram ; the diagonal c x of which is then found to be. say 6; oraresult- 
ant of 6 tons. Now, in this case, theoretically the strain lengthwise of the rope is everywhere equal t» 
the power n, or 9 tons; aud we have found that it produces also a strain c x, against the pin at c, of IS 
tons. It also produces’a pushing strain on 
the pulley P. Its amount may be found in the 
same way, by measuring 9 tons by scale each 
way from o toward c and n; completing the 
parallelogram; and measuring its diagonal 
resultant. But now let us use this rope as a 
funicular machine ; and apply a power x c of 
6 tons at c. We find that this 6 tons produces 
a strain c g or c o, of 9 tons along the rope; 
and this strain along co will pass along to n; 
and thus the power of 6 balances a resistance 
of 9 tons acting at n iu the direction n o. 

The diagonal c x or any other will plainly be vertical only when the angles of 
inclination of c g and c o, with the horizon are equal. If they differ, both the di¬ 
rection and the length of the diagonal will change. 

All will remain the same if the end g instead of being fixed, is passed over a 
pulley as at P, and a load or a pull equal to that at the other end is applied to it. 

Rem. 3. The surfaces of contact of pieces used in construction, are called 
Joints. When a piece is intended to resist compression, or push, it is called a 
strut; or if inclined, it is often called a brace; or if vertical, a post, pillar, 
or column. When to resist tension or pull, a tic. When to resist both tension 
and pull alternately, a tie-strut, or a strut-tie. A strut should be still or 
inflexible; but a rope, chain, or thin rod, may answer for a tie. 


OC 



Rem 4. To distinguish a tie from a strut at a glance is sometimes 
difficult; but it may he done thus. From the point a, Figs 16%, at which the force 


acts, draw a line a c, in the direction 
away from that point. On any part, 
a o, of that line as a diag, draw a paral¬ 
lelogram of forces. Through the point 
a draw a line » t, parallel to the other 
diag 11. Then all the pieces which 
are on the same side of that line, that 
a c is, are struts; while those on the 
opposite side, are ties. We may also 
frequently determine, by imagining 
the piece to be a rope or chain, in¬ 
stead of a beam ; and seeing whether i 
a tie ; if not, a strut. 


iu which the force, it at liberty, would move 



would then bear the strain. If it would it is 


When a piece of material is used to resist forces which tend to bend or break it crosswise, or trans¬ 
versely of its length, as in Figs 47, 48, 49, 50, it is called a beam; such as joists, girders, &c. The 
same piece however, frequently acts at once, as a beam, and as a tie, &c. Its own weight strains a 
beam transversely; but in our present illustrations of comp and res of forces, this strain, although 
frequently the most important one, could not he well considered at the same time. 




326 


FORCE IN RIGID BODIES, 


Art. 34. Since any single force may be resolved into two oblique ones in the 
same plane with it, and which shall produce the same effect upon a rigid body con¬ 
sidered as a whole, it follows that the single strain along any piece a m or a n, of the 
four figs on p 323, may bo thus resolved. In practice, it is frequently necessary to do 
this; and especially so for finding components at right angles to each other, in hor 
and vert directions. 

For instance, the joint o d, Fig 17, at the foot of the beam A, if made 
at right angles to the resultant r r of all the pressures along the beam, 
of course receives the whole of these pressures; which consequently are 
all imparted to the abutment; leaving no portion unresisted, so as to pro 
duce sliding; or even a tendency to slide along the joint o d. Conse¬ 
quently. this joint is perfectly adapted to its duty. But a joint 

of the form of b i c, which is equally effective, is sometimes reqd for re¬ 
ceiving a single strain (like that along A) along a piece E; and in order 
to properly proportion the vert and hor faces 6 t, and c i, of the joint, we 
must find the proportion existing between the vert, and the hor compo¬ 
nents equal to the single strain r r along E. To do this is very easy; for 
we have only to lay off by scale, auy length e n along r r, to represent 
the single strain in that direction; and on it as a diag, from e and n 
draw vert and hor lines e t, n t, meeting in f. Then e f measured by the 
same scale, will give the vert strain ; while n f will give the hor one. The 
parts b i, i c of the joint, must consequently have the same proportion as 
these two components have to each other; bearing in mind, however, that 
since joints should be at right angles to the forces they have to sustain, 
the vert part b i must bear the hor strain; and the hor part i c, the 
vert one. 



When, by Art 33, we are finding, by 
means of the parallelogram of forces 
on y g, Fig lb, the total strains o n, o g, 
which an extraneous load F produces 
along two beams. FR. F g, itiseasy at the 
same time to find the vert and hor compo¬ 
nents also; by drawing the two hor lines 
n t, gj, and measuring them by the same 
scale used for the diag oy. Likewise 
measure o t. and oj, for the correspond¬ 
ing vert forces at the joints; because 
when nt and gj may be drawn inside 
of the parallelogram, (which is not 
always the case; as see Fig 18^.) the 
component forces in the direction of any 
diag, whether vert or not, are measured 
respectively from tho poipt o, where the 
extraneous force F Is imparted to the beams; to those points t and j, where the diag is met by the 
equal lines n t, gj. 

Rem. 1. It is an important fact that however diff may be either the inclinations, 
or the lengths of tho two beams; or how diff tho total strains in the directions of 
their respective lengths; tho /tor strains, caused both by tho extraneous load and 
by the weights of the beams themselves, will always bo equal on both of them. 
1 bus, in I ig 18, n t is equal to gj ; and in Figs 13 to 10, if hor lines bo drawn from o 
and y , to a b, those iu any one fig will be equal to each other. 

Rem. 2. It is plain that each beam may lie considered as receiving from the load 
F, either one force or its two components. 



The vert component oj, of the triangle o gj, being longer than of, of the triangle of n, shows that 
the beam og bears morelof the vert force or weight of the load F, than oR does; and in the same 
° L i ,T he , tW0 components on the diagonal, (when inside of the parallelogram,) 
ml It ill t0 F u h or eq u al t h e leu g t h of the diag, or the weight F. But as n f and g j are of the same 

k\vw 1S ' 1 ' e ^ lndlcate tliat 1,01,1 beams are pressed sideways, or hor, to the same extent. 

" t reat °'Vl USS r u l we sha)1 find lhis method of obtaining vert components, by 

.7 us l ful - The len Kths of the beams o It, og, do not affect tho amount of 
their lengths ^ th ,° load . 1 ’' at their summits ; but as their own wts must increase with 

wta into consideration ^ them must inorease also ! but we have not yet taken their own 

wts into consideration, neither are we yet prepared to do so. 



Rem. 3. If, as in Fig 18%, one of the beams, as n o, ig 
hor, it is plain that all of the diag oy, that is, all of the 
weight 1 or W, is borne by the other beam og; and n o 
sustains hor strain only. The l>eam og of course bears 
an equal hor strain also, as shown by y g, equal to n o. 

Rem. 4. It is immaterial (Art 18) whether tho load 
rests on top, as F; or is suspended below, as W; for in 
either case it is simply vert force imparted at o." 















FORCE IN RIGID BODIES, 


327 



Rem. 5. When one of the forces, as 
ti o, makes an angle n oy, greater than 
90°, with the diag o y, the positions of 
the beams on, off , become as in Fig 18*^, 
and we have a case like Fig 9%; that ii, 
the hor lines nb, ga, from the angles n 
and ff, and at right angles to the diag, 
cannot be drawn inside, of the parallelo¬ 
gram. Therefore we must extend the 
diag both ways, to a and b. If we wish 
to consider each of the forces on and off 
as made up of two components; then 
for those of o n , we have b n, and o b ; 
and for those on og, we have ag and 
o a. Hence when the angle no a ex¬ 
ceeds 90, the vert strain on o g is greater than the loan w, 

which (according to the ordinary phraseology) produces it. But the part ay of the vert o a, has 
no reference to the load; but represents an upward vert force produced by the wall M, to balance a 
downward vert one equal to 6 o from the wall P. This excess ay over the diag, occurs only when 
one of the beams forms an angle greater than 90° with the diag. We call the attention of the student 
to this case, because we do not remember to have met with it in any book. It will perhaps be a new 
idea to many, that the vert pres on the wall M, can be greater than the entire load. 

Art. 35. As a simple practical example of very common occurrence, of the ap¬ 
plication of the foregoing principle of finding the resultant of two forces in the same 
plane, and tending to one point; let S, Fig 19, represent a block of stone weighing 
3 tons; and standing on a hor base mn; but not attached to it in any way by ce¬ 
ment, &c., but with a stop at», merely to prevent sliding toward b. 

In this case there will be no force acting upon the body in such 
a way as to prevent its being overturned around its toe n as a 
turning point, except its wt, or force of gravity, which always 
acts vert downward. By Arts 56, 57, all this force may be con¬ 
sidered to be concentrated at the cen of gray f of the stone; and 
as acting at any point whatever in its vert line of direction l g. 

Now suppose a pres /ft, of 2 tons, (which may be either one 
simple force, or the resultant of many forces,) to be imparted to 
the stone, in the same plane with the force of gravity ; which it 
will evidently be only if its direction /c meets the direction of 
gravity Ig, at some point; as, for instance, at a; because then a 
plane surface would coincide with both directions. The 

question is, which of these two forces will prevail; the two tons 
of fh. to overturn the stone ; or the 3 tons of gravity to prevent 
its being overturned? By Art 29, both forces may be considered 
to be imparted to the rigid stone, at the point a, where their lines 
of direction meet; and we may make a c equal to 2 inches, ft, &c, 
to represent the amount and the direction of the 2 tons of pres of the 
force/* ’ and a t>, by the same scale, for the direction, and 3 tons 
of pres of the weight of the stone. From c draw a line cd paral¬ 
lel to a v ; and from v draw v d parallel to a c ; these will meet at 

a d ofTh?”parallelogramsenile^will give about 2^ tons for the single resultant 
force which would by itself produce upon the rigid stone the same effect as gravity and /ft com¬ 
bined This force may be considered as imparted to the stone at any point in the lineof 

iU direction /^ o) through the stone ; as a push at w, as shown by the arrow *«<; or as a at o, 
bv means*of a rope 6 o, fastened to the stone at o. Since this resultant is supposed to take he p ace 
h^th of trravitv and of fh, the two last must of course be considered as annihilated , so that the stone 
^cornel’asit werean unresisting body of matter without weight; and acted upon by the force ad, 
oTg w which must of course move it, and thus compel it to overturn around n as a pivot. 

Rem 1 Had the direction of the resultant a d struck the base of the stone at n, 
instead of striking outside of the base as at b, the stone would barely have stood; 
because then the resultant, on leaving the body at «, would have encountered the 
resisting force of the ground on which the stone stood, acting upon the bodj at that 
point Had the direction struck within n, that is, between « and m, the stone would 
o+ond etill more firmlv and t ^ ie more fi rm ly in proportion as 

it strikes nearer g, where the direction Iff of the gravity of the stone meets the base. 
The direction of a resultant may strike within the base ; and the body remain fiim, 
so far as regards overturning ; but yet may slide. See Art 63; very important. 

m, • ill oi=n the necessity Tor assuming at times that bodies are rigid, or unbreakable. 

This exrample shows ali3 o'the ia . »y ^ application of the great force of the resultant so 

iSSdiE ?Z £2 -Kh on 














328 


FORCE IN RIGID BODIES, 


■which it stood. A knowledge of the direction of the resultants of forces acting on bridge abutments, 
retaining-walls, &c, is therefore of use also by enabling us to guard against such accidents, by select¬ 
ing the strongest stones for the most strained parts of the structure; as well as by adopting extra 
precautions in preparing those portions of the earth foundations, upon which those parts rest. 

It must be remembered, however, in such cases as the foregoing, that with the exception of the point 
o. at which the resultant leaves the body, ad is not the direction which the resultant actually follows 
in the body ; but is one which we may assume it to have, so long only as we assume the body to be 
practically rigid: that is, that it canuot. be in any way broken, bent, or have its form changed, by 
the forces actually imparted. Frequently we cannot safely assume a mass of masonry to be thus 
rigid; for it may be composed of many separate pieces merely placed in contact with each other 
without mortar, as in dry masonry ; or even if mortar or cement be used to unite these pieces, it may 
not have time to set or harden properly, before the deranging forces are brought to bear upon it. In 
that case, although the resultant might fall eutirely within the body, and within the base, thus de¬ 
noting perfect security to a rigid body; yet the structure might be completely destroyed by the 
sliding or other derangement of its parts among one another, under a force much less that: would be 
required to overturn it. On this account, if we wish to obtain security at the least expense, we must 
frequently trace the actual curved direction of the resultant through its entire course; so that we 
may at every point of it place the joints of our masonry at right angles to it, or adopt 

other precautions to prevent the parts of the structure front separating. See Art 72. 

Kkm. 2. If iu Fig IS we suppose strong mortar or cemeut to exist between the base of the stone 
and a rigid foundation of masonry or rock, upon which we may assume it to stand, then it mav not 
be overthrown, although the direction of the resultant a d falls outside of the base m n. For then a 
third force, namely, the cohesive strength of the mortar, is brought to act upon the stone ; and the 
resultant of all three forces may fall within the base. In all cases where a body remains at rest, not¬ 
withstanding that the resultant of the forces falls outside of its base, (whether the base be hor, ve't, 
or inclined.! we may be certain that it is because some other force, which we have neglected, is acting 
upon it at the same time ; for when the direction of all the forces passes beyond its base, and is con¬ 
sequently force unresisted, the body must move. See Rem, Art 65; also see Art 72. When one body 
is tints cemented to another, the two beconte in fact one body, so long as the cement docs not give 
way under the imparted forces: so that a problem which is one in Statics, if there is no cement, may 
become one in Strength of Materials, when there is cement. The cement takes the place of natural 
cohesive force between the bodies which it unites. 



Art. 36. When the number of 
forces in the same plane, whether 
tending; to or from the same point 
or not, is greater than two, their result* 
ant may he loiiml in the manner 
already given for two. Thus, with the 
three forces b a, ca,oa, Fig 1% first find the 
resultant of any two of them; as, for instance, 
the resultant n a, of o a, and c a. Then consider 
oa and c a as removed and na as taking their 
place; and then find the resultant m a, of n a and 
fp. |A l then is via the single resultant that will 

1IC1 IwA, produce upon the rigid body W, the same effect 

u J A as the three forces oa, c a, ba. If the three 

forces are imparted at diflf parts of the body, proceed as in Figs 10V£ and 1014 If 
the number ot forces be greater than 3, the process is precisely the same; find 1st 
the resultant of two of them ; then the resultant of the 1st resultant and 3d force- 
then the resultant of the 2d resultant and 4th force; and so on to the end. 


X 



Fitf 1 &14 Illustrates a ease in 
whieli three forces a, b , and c, in 
the same plane, do NOT tend to¬ 
wards the same point. We may be¬ 
gin with any two of the forces at pleasure. We 
will take b and c; and, as at Fig 10b£, 
prolong them backwards to h, and find their 
resultant hi. Then prolonging hi, and the 
third force a to meet at n, we lay off from n 
two sides of the parallelogram equal respec¬ 
tively to a and to h i, and complete the paral¬ 
lelogram no. Then the diagonal on is the re¬ 
quired resultant, to be applied to the body 
J at e, as shown by xe. This resultant, would 
then by itself produce upon the rigid body 
considered as a whole, the same effect as would 
the three forces a, b, c. Or if its direction were 
inverted, so as to pull, instead of push at e, it 
would become an antiresultant to the three 
forces, and thus hold them in equilibrium. 

The same process applies to any number of 
forces. 









FORCE IN RIGID BODIES. 


329 


Art. 37. It sometimes happens, after having 
found the resultant of all the forces except the last 
one, that said resultant and remaining force are in 
the same straight line. Thus, with the forces u, r, w, 

Fig 20; the resultant r of u and w, is in the same 
straight line with the last remaining force v ; and of 
course no parallelogram of forces can be drawn which 
shall have v and r for two of its sides. When this 
happens, if v and r are of diff lengths, we have the 
case given in Art 16, of two unequal forces meeting 
in the same straight line; but in opposite directions 
along it. Consequently, the small one, and an equal part of the large one, mutually 
destroy each other; and the remainder of the large one is the resultant of the two. 

But if v and r are of the same length, then we have the case of two equal opposing forces, which 
mutually destroy each other entirely ; and the body remains at rest. Consequently, there is no result¬ 
ant in this case; for no single force can have the effect of keeping a body at rest; but will always 
move it. In other words u, v, and w, are then in equilibrium. 

Art 38. The polygon of forces. The resultant of any number of 



forces in the same plane; may be 
thus: Let a, b, and c, be three 
such forces ; whose resultant R is to 
be found. Begin with any one of 
them, as a, and draw a', parallel, and 
equal to it; and place an arrow-head 
at the proper end of it, to show its 
direction. From this arrow-head, 
draw b', equal and parallel to i>; 
piacing an arrow-head at its end. 
From this second arrow-head, draw 
c', equal and parallel to c; and so on 


found by means of the polygon of forces, 



Bo 21 


Finally, from the arrow-head d of 


with any number of forces ; taken in any order, 
the last of the forces, draw a line A to the butt-end, «, of the first one ; thus closing 
the figure; and place an arrow-head as on the others. Now, this closing line A, or 
dn, with its arrow, represents both in quantity and direction the autiresultant of 
the three given forces; or if its arrow be reversed, it will represent their resultant. 
Consequently, we have only to draw from o a line, o R, parallel to A; and to make 
R equal to A; but pointing in the opposite direction. Then is R the resultant. 
This process will give different figures, according to which force 
we begin with; or whether we take the forces in right, or left-hand order; still, A 
will always come out the same in all of them. If the three given forces (or any 
greater number, as the case may be) had been in equilibrium with each other, that 
is had mutually destroyed each other’s tendency to cause motion, they of course 
could have no resultant, or single force that would produce an equal effect; because 
a single force, if the only one acting on a body, must produce motion. W hen this 
is the case the forces will of themselves form a closed polygon. In either case some 
of the lines may cross each other as do a' and c' at A, or not, as at N below. If the 
forces do not all act through the same point, see Art. 40, p 332. 

When any number of forces are in equilibrium, 
any one of them is the anti-resultant of all the 
rest, because it keeps them all in equilibrium; 

Also, any number of the forces balance all the 
rest. 

If any number of forces, as a'b'c'd' Fig 22, 

(whether they act through one point, as s, or y 
not) are in equilibrium; and if we know the 
directions of all of them, and the amounts of all 
but tvjo; then the polygon N will give us the 
amounts of those two. Suppose we know the 
amounts of a ' and b' and require those of c' and T7’n 99 

d'. First draw by scale the known ones a and 
b (see polygon N) parallel tp their' actual direc- ' 

tions, and then complete the polygon with lines c and d, respectively parallel to e 
and d'. Then the lengths of c and d will give the amounts of c' and d' respectively. 

Auy diagonal across a polygon of forces, represents the resultant 
of all the forces on either side of it. 

If tlie forces are not all in one plane, the polygon is not in one 

plane, and cannot be drawn on a flat surface, but is twisted, or “gauche,” as is 
oct y, Fig 36, p 333. 

















330 


FORCE IN RIGID BODIES, 


Rem. 1. It must not be inferred because forces balance each other, that therefore they balance the 
body to which they are imparted; for the body might move under the iutiueuce of other forces. Thus, I 
the forces a', b', c', d‘, may be supposed to be the balancing forces of several persous holding a body J 
at rest in a railroad car moving with great speed. Their forces prevent each other from giviug motion 
to the body ; but do not prevent the steam force of the engine from doing so. It is ouly when all the 1 
forces imparted to a body, including its own weight, are in equilibrium, that the body itselj is also at 
rest. Or, if we hold a book, ruler, &c, vert betwecu our thumb and forefinger; the opposite and 
equal pressures of the thumb and finger, hold each other in equilibrium, so that they cannot move 
the book either to the right or left, and the friction between them and the book, holds the gravity or 
weight of the book in equilibrium, so that it does not fall. So thatso far as these forces are concerned, 
they produoe no motion in the book; but we can move it vertically up and down, by introducing 
the third force of our wrist; or hor by stretching out our arm; or by walking; none of which will 
interfere with the equilibrium of the other forces; they only prevent eacA other from producing any 
motion. i 


Rem. 2. A triangle being a polygon of 3 sides, if any 3 forces which form a trianglo, I 
be applied in one plane, to a body, and in directions parallel to the sides of the tri¬ 
angle ; and tending either to or from one point; they will hold each other in equili¬ 
brium. And, vice versa, when we see a body kept at rest solely by the action of 3 
forces which are not parallel to each other, we may be sure that those forces are pro¬ 
portional to the sides of a triangle drawn parallel to them; that they are in one j 
plane; that they all tend either to or from one point; and that any one of them 
acts in the direction of an antiresultant to the other two. Moreover, each of the j 
forces is proportionate to the sine of the angle included between the other two; so 
that if we know one of the forces, we can readily find the others if we have the 
angles. This is very often of use in practice; as in finding by calculation alone, the 
lino of pressures through an arch ; the pres of earth against retaining-walls, Ac. It 
must be remembered that the wt of a body usually constitutes one of the forces to | 
be considered as acting upon it. This rein is very important. 



Kx. 1. Let a c, Fig 22J£, be a beam ; Its foot rest¬ 
ing onot; and its head c merely leaning against a 
6mooth vert wall; aud whether a c be unloaded ; or 
w hether it supports a load placed in auy manner 
upon it, or suspended from it; let the vert line 
which passes through the cen of grav of the beam 
and its load (both of which are R'ipposed to be 
known) be represented by p g. The beam audita 
load may be regarded as a single body, acted upon, 
and kept at rest, by three forces ; namely, its own 
gravity or wt; the force A, at c; and the force/, at 
a. No other forces act on it. Now, gravity acts vert 
only : and in the case before us it may all be re¬ 
garded as acting in the line p g. The force at c can 
act ouly at right angles to the surf or joint at that 
place, and since the joint is vert, the 

force A must be hor, or along h p. The question 
now is, howto find the direction of the third force/. 
To do this we must avail ourselves of the principle 
that wheu three forces, uot parallel to ea-h other, 
hold a body at rest, or in equilibrium, as tbe.se 
three forces hold the beam a c, their directions all tend to or from one point; which is either at the 
cen of grav of the body ; or in a vert line passing through said cen. Hence, since the vert direction 
P !J of the force of gravity of the body; and the direction A p of the force A, meet atp, therefore, the 
direction / p, of the force /, must also meet there. Hence we have only to draw a line fp, in order 
to find the reqd direction. A post intended to support the end a of the beam, should have the position 
fa-, and the joiut o i should be at right augles to fa; and not to a c, as might at first be supposed 
from Figs 13 and 16, in which the wtof the beams is not considered, and in which the extra¬ 

neous weight w is applied ouly at the joint, a. 






Having found the directions of the three forces in Fig 22^, it only remains to find their amounts. 
To do this, we already have one of them given, namely, gravity, or the wt of the beam aud its load ; 
and we know that they must be in proportion to the sides of the triangle drawn parallel to their di¬ 
rections. Consequently, if on the vert direction p g, we lay off by scale any portion whatever, as p d 
to represent the force of gravity, then will the hor side of the triangle, p d b, represent by the same 
scale the hor pres at c; and the side b p, the oblique pres at a. The hor pres at the foot is equal to 
that at the head of the beam. It is of course included in the oblique pres /; which is compounded 
of said hor force, and of the vert force at o. The vert force is equal to the weight of the beam aud 
its load; none of which is sustained at c; nor can be, so long as the joint at the head and wall la 
vert. See Fig 5£, p 552. 

Ex. 2. This is very similar to the preceding. Let abij, Fig 22%, he the half of 
any arch bridge, loaded or unloaded equally throughout; and of which wo know in 
either case the total wt; and that the cen of grav of said wt is somewhere in the vert 
lin egg- Now this half bridge is, like the preceding beam, kept in equilibrium, or 
at rest, by three forces only; namely, the wt; a hor pres h, at the crown, arising 
from the other half of the arch; and an oblique pres o, at the springing, or skew- 
hack/ i. To find the directions and amounts of these forces, from a draw a hor line, 
meeting the vert gg at c. From c draw a line to the center of j t; this is the direc¬ 
tion of the oblique force o. From c measure down by scale any dist cs, on the vert 
direction p g, to represent the weight; and from s, draw s t hor. Then s t , measd by 
the same scale, will be the hor pres h ; and ct, the oblique one o. The joint ji at 
the spring of the arch, bears all of cs; that is, all the wt of the half arch, and half 
















FORCE IN RIGID BODIES 


331 


load. No vert pres or wt is sustained at the center In of the arch ; nothing but the 
hor pres. Buty i also sustains this hor pres, for c t is composed of c s and s t. 

The oblique force c t constitutes the total thrust exerted by the entire arch, 
against each of its two abuts; and the line ct shows the direction in which this 
thrust enters the abut at the skewback j i. After entering at that point, it begins to 
curve downward, on the principle explained in Art 72. Since ct is the hypotlienuse 
ot a right-angled triangle, of which the leg cs represents the half wt; and st the 
hor pres of the arch, it follows that the total thrust ct of an arch may be found thus: 
add together the square o f half its wt; and the square of the kor pres ; and take the sq 
rt of the sum. This applies also to arches of iron or wood. 

The joints of any arch which is a portion of a circle, are 
usually drawn toward the center of the circle; and, practi¬ 
cally, this answers every purpose; but it is plain that strict 
theory would require the joint ij to be at right angles to co. 

Bo also the other joints of the archstones would be reqd to 
be pcrp to the pres which they have to sustain. 

Moreover, since the pres ct, upon the joint ji, is much 
greater than the pres st, upon the joint In, theory would 
require the joint ji to be proportionally deeper than In; 
whereas in practice they are usually made the same, except 
in very large arches. See Stone Bridges. 

The last few lines of Ex 1, respecting the hor pres at the 
foot, and at the head, apply equally here. The young stu¬ 
dent should familiarize himself thoroughly with the princi¬ 
ple illustrated by these two examples, as it is one of very 
frequent application in practice; as in retaining-walls, abut¬ 
ments, &c. 

Here, as in the preceding example of the beam, we do not 
consider the strains produced along the length of the arch 
Inji itself; but merely the two forces which, acting at its center In, and at its foot ji, keep each 
Other, and the wt of the half arch and its load, in equilibrium. For the others, see Stone Bridges. 

Art. 39. A third mode of finding' the result:mt R, Figs 23, 
of any number of forces E, F, G, in one plane; and actiug; through one 
point, x. 



Draw two lines, H H, and V V, at right angles to each other. From their point of intersection o, 
draw lines by any convenient scale, to represent the directions and amounts of the forces. By Art 
| 34, resolve each of these forces iuto two component ones parallel to II H ar.d V V. Thus F o is re¬ 
sol ved into u o and e o; Go, into m o and to; E o, into t o and no. Then measure by the scale, and 
add together, those components io and to, parallel to H H, and which tend to move the point o 



toward the left hand. Also add together those (in this case only one) u o, which tend to move o 
toward the right hand. Subtract the least sum from the greatest; their diff, equal to s o, will be the 
resultant of the two sets of forces which respectively tend to move o to the right, and to the left. la 
this instance, this so must evidently be placed to the right of o, because the components on that side 
give the greatest sum. 

Next, add together those components eo, no, parallel to W, which tend to move the point o up¬ 
ward. In like manner add together those (in this case only one) components mo, parallel to V V, 
which tend to move o downward. Subtract, as before, the least sum from the greatest; their difT, 
equal to a o. will be the resultant of the two sets of forces which respectively tend to move o upward 
and downward. In this instance, no must be measd off below o, because the upward tendency is the 
greatest. By this process, then, we first reduce all the original forces to two components, so and ao. 
This being done, we have only to complete the parallelogram of forces osca, and draw its diag co; 
which will be the final single resultant of all the original force*. From a; draw xy parallel to co, 
and make b y equal to e o; then is by, or R, the reqd resultant; and b the point for imparting it to 
the body P, so that its effect may be equal to that of the three original forces combined. 


23 


























332 


FORCE IN RIGID BODIES. 


Art. 40. Even when any number of forces In the same 
plane «lo NOT tend to or from the same point, the principle of 

the polygon of forces, or of Art 39, may be used in precisely the same manner as at 
Figs 21 and 23,for finding the length and direction of their resultant. Or if they 
are in equilibrium, and hence can have no resultant, they will still form a closed 
figure as A Fig 21, or N Fig 22, as well as if they acted through one point. There 
will however be this ditrerence, that when all the forces, as a, b, and 
c, Fig 21, tend to or from one point o, we know' that their resultant, as R, must 1>6 
applied parallel to A, and must tend to or from that same point n. In other words, 
we know where its point of application must be. And so with 
the resultant co. Art 39. But when the forces do not tend to or from one point, and 
we find their resultant by Art 38 or 39, we know only its amount and direction ; 
but do not know where to apply it. In such cases we may use Art 
36, Fig 1934- If the lines, representing any number of forces in one 

plane, form a polygon, then we know that those forces, if they all act 
t/trough one point, are in equilibrium; and we know that if they do not all act 
through one point they are either in equilibrium, and hence have no resultant; 
or else they have two resultants, equal and parallel to each other, and acting in 
opposite directions; forming a “couple” and causing the body to 

revolve around a point midway between them; while 6aid point, and the body us a 
whole, remain stationary. 








Art. 41. Forces in different planes; bnt tending’ to or from 
the same point. Such forces cannot, like those in one plane, be correctly rep¬ 
resented together on one flat surf, such as a sheet of paper. 
Thus, let Fig 27 be a cube; and tx,cx , ix , three forces 
acting in the directions of its edges; and all tending to 
the same point x. It is plain that the relative positions 
of these forces are not correctly represented; for txc,txi, 
and cxi, are in reality right angles; whereas, in the fig, 
txc appears to be an acute one; cxi, a right angle; and 
txi an obtuse one. 



On this account the resultant of such forces cannot be had! by 
measurement from a drawing. Recourse must therefore be had to 
calculation ; which, however, will be facilitated by a drawing. The 
theoretical principle is very simple; boiug, in fact' the same as when 
the forces are all in one plane; namely, first find by Art 28 the re¬ 
sultant of any two of them, (for any two are really in one plane;) 
then find the resultant of this resultant and the third force; and 
so on to the end. It is easy to find the first resultant; bnt the 
others are more troublesome. Instances are comparatively rare, in 

v rCit o o re rond t a Ka 6am? ♦ K a nil r.n i b.vt f 41. /v ^ M .. ■ « A n _ ? 


which the resnltants of such forces are reqd to be found ; the attention of the engineer being gen¬ 
erally confined to those in one plane; as when proportioning bridges, roofs, retaining-walls, &c. 







FORCE IN RIGID BODIES 


333 


Art. 42. To find the resultant of forces in different planes, 
but all tending through one point. 

In cases where mathematical accuracy is not necessary, and the number of forces 
only three, or four, the writer will venture to propose a method by models; which, 
if open to the objection of empiricism, has the advantage of requiring less time than 
I other processes; is sufficiently correct for most practical purposes; and shows the 
i resultant in its actual position, which is done by no method of calculation. 

Let a 060 , e o, Fig 30, be the three forces, meeting at 0 ; their angles with each other, a oh, hoc, 
1 co a, (.which alone are necessary in this method,) being of course known. Draw on pasteboard the 



three forces ao,bo,co, as in Fig 31, with their actual angles a oh, hoc, coa. By Art 28, draw 
the parallelogram or forces for the middle pair bo, co; and draw its diag wo, which will be the re¬ 
sultant of those two; leaving the resultant of it, and ao, yet to be found. Cut away neatly the 
whole fig, a o a c w b a. Make deep knife-scratches along oh, o c, so that the two outer triangles may 
be more readily turned at angles to the middle one. Turn them until the two edges o a, o a, meet- 
and then paste a piece of thin paper along the meeting joint, to keep them in place. Stand the model 
upon its side oiutcaaa base; and we shall have the slipper shape a oh tv, Fig 32; ow being the sole, 
and aob the hollow foot. 

We now have the first resultant w o, and the third remaining force a 0 , in their actual relative, po¬ 
sitions. Now, to find their resultant, also in its actual position, cut a separate triangular piece of paste¬ 
board of the size and shape of w a o. Find the center i, of the edge to a, and draw a line i 0 on each 
side of it. Finally, by means of the edges ao, wo, paste this piece to the inside of the model, along 
its center-line too. This done, io represents one half of the reqd resultant, in its actual position. 

The reason why it represents but one-half of it is plain ; for, as be¬ 
fore stated, we now have ao and too in their actual positions in the 
model; consequently, if we complete the parallelogram of forces 
too an, and draw its diagonal no, this last will be their resultant. 

But since the two diags of every parallelogram divide each other into 
two equal parts, the diag aw thus divides the resultant; consequently 
j' o is one-half the resultant. 

If there be four forces, as an, bn, cn, dn, Fig 34, draw them as in 
the fig, with their actual angles a n b, bnc, &c. Draw also the re¬ 
sultants no, of a n and bn ; and n w, of n c and n d. Then cut out •, 
the entire fig, as before; and paste together the tw r o edges an, an. y 
Hold the model in such a way that two of its planes (as an h and 
bnc) form the same angle as do the two corresponding planes be¬ 
tween the forces. Then we have the two resultants a v, a w, Fig 35, 
forming two simple forces, in their actual relative positions ; and we 
have only to measure their distance apart from v to w; and thence 
find their resultant a r, which will evidently be that of the four ori- 
gimal forces. 

Or, as in the preceding case, cut out a separate piece of pasteboard, 
avw. Fig 35, and having drawn on each side of it a line from a to the 
center o of v w, paste it inside of the model. Then will a o represent 
one-half of the resultant of the four forces, in its actual position. 

Should the model be exposed to hard usage by workmen, it should be 
made of wood; the triangles anh, bn c, kc, being cut out separately ; 
the joining edges bevelled ; and then glued together. See also Art 43. 



IW 34 


AY 



Ko 35 


Art. 43. The parallelepiped of forces. If any three forces, ao,bo, 
co. Figs 36, in diff planes, meet at one point o, whether they all be strains, or all 
motions, their resultant or joint ef¬ 
fect will be represented, both in c \ _~i 

quantity and in direction, by the 
diag y o, of a parallelepiped a ctb i, 
of which three converging edges 
may be assumed to represent the 
three converging forces. 

This suggests another mode of showing t- A 

the resultant of three such forces by a *- 

model; for it is only necessary to prepare 
a box A or B, as the case may be; and y 0 
will represent the reqd resultant. 



0 > f\ 

J 

rc 1 i . 

"■-i 


;r\ 




























334 


FORCE IN RIGID BODIES 


No three forces In different planes can be In eqniHbrinm. 

Art. 44. Forces in different planes; and not tending 1 to or j 
from one point. It is but rarely that such forces have a resultant, or anti- i 
resultant; that is, no single force can usually be found either to produce an equal 
effect, or to balance them. It is so seldom that they present themselves to the engi- j 
neer's' attention, and their solution is so tedious, except in very simple cases, that 
we shall confine ourselves to one of that kind. As in Art 41, the resultants cannot 

be had by measurement from a drawing. 

Let ao, a o, a o, Fig 37, be three such forces; and suppose 
them all to act against the same plane 

p pp \ and against the same side, or surf of it; that is, uoue j 
of them pointing upward against the under side of the plane ! 
in the fig. Having the points o of application, and the rela j 
tive positions of the forces themselves, as well as the angle *• 
aoc which they form with the plane ppp, resolve each of th | 
forces into two components; one of which, co, coincides wit! j 
the plane ; while the other (parallel and equal to a c, but meet !i 
ing c o at o) is at right angles to c o, or to the plane. We then, ‘ 
have two sets of forces ; one set in the plane, and the other at 
right angles to it. Since those in the plane do not tend to or 
from one point, their resultant must be found by Fig 19}$> 
while that of the several parallel components (equal to a 
e, but applied at o) may be obtained by Arts 56 and 59. 
These two resultants will rarely be in the same plane with 
each other, and consequently can have no joint resultant. If 
they should chance, however, to fall in the same plane, use 
Art 28 for finding their resultant. In simple cases, where the 
forces act against one plane, as in onr fig, pieces of wire, cut to lengths to represent the forces ; and 
stuck into a piece of smooth board, in their proper relative positions, will greatly facilitate the find¬ 
ing of the resultant approximately enough for most practical cases. 

The same general process must be used, no matter how great may be the number and directions of 
the forces. A plane must be assumed to pass somewhere through the system; and the directions of 
all the forces must be conceived to be so extended as to terminate at points of application in said 
plane. K.ach force must then be resolved into two, as in the foregoing example; aud the resultants 1 
of the two sets of forces, as well as their joint resultant, if they have one, must be found as before, j 
If any of the forces should be parallel to the assumed plane, but not in it, it evidently cannot be re- : 
solved into two, one of which shall be in the plane; for (Art 32) no force can have one of its compo- , 
Dents parallel to itself. Hence, in such a case, the resultant cannot be found by this process. 



Art. 45. It is comparatively seldom that strict mathematical accuracy is req; < 
in finding the resultants of forces in engineering practice; therefore, the foregoing, J 
easy methods by measurement from a drawing, or model, will usually answer every ‘ 
purpose. Moreover, they appeal to the eye; and are therefore much less liable to 
serious errors than methods involving numerous calculations. But when more cor-• 
rect results are needed, they may be had by means of a table of nat sines, tangents, 
kc. Thus, in the case of two components and their resultant, calling the components, 
Fig 38, C and c; and the resultant It, then 



If the angle m b n 
nents C, c is 90°, R 


between the compo- 

will = /C 2 + c2; and C = 


|/R 2 —c 2 ; and c = fR2- C 2 . Or R will = C -r- cosiL 1 
of a b n = c -r cosine of a b in. And C will = Rxc' 3 
sine of a b n ; and c = It X cosine of a b m. ( 

_n whether the angle between the con 

oo P 0 "**" 1 * ® »«<* « *>e 90°, or more, or less, i 

Fig. 38. m I lg 39, 

P \/ c? nn r r Tl v / • * 


C will = 


R X sine of v 
sine of C be ’ 


and c — 


R X s ine of x p 
sine of C b c ’ 



Observe that v is used for finding C ; and x for fine" 1 

ing c. 

And R will = c * s * ne of ? c X sine of Cbc 1 
sine of x or sine of o 

If the angle Cbc, or either of the others exceeds 
90°, subtract it from 180°, aud use the sine of the 
remainder. 


i 



























FORCE IN RIGID BODIES, 


335 


I 

Art. 46. Moments. Leverage. If a b, 

r Fig 40, represent any force acting in any direction 
'■ whatever; aud if o, or t, be any point whatever, 

1 ‘ whether in or out of the body on which the force 
1 ‘ is acting; and if from said point a line o s, or i c, 

1 be drawn at right angles to said direction of the 
force, then said line a s, or i c, is called the arm, 
or leverage of the force a b, about said point. 

; And if the amount of the force in lbs, &c, be mult 
! by the length of the arm or leverage in feet, &c , 
the prod in ft-Ibs, &c, is called the moment of 
he force about that point. Thus, if the force a b 
8 lbs, or tons; and the line os, (5 feet, then the 
noment of a b about o is 8 X 6 = 48 ft-lbs; or ft- 

ions. A force has of course no moment about any point through which it passes. 

i This moment represents the total tendency of the force to produce motion about the given point. 

| We cannot hold hor. between the ends of a thumb and forefinger, a piece of stick a foot long, which 
r has a 3 ft wt at the other end of it; because the tendency of the wt to produce motion is too great for 
the force of our fingers to resist; but we can in that manner hold a stick two feet long, with a 3 ft wt 
a at each end, if we take it at the center. For although in this case there is twice as much moment as 
'■ before exerted at our fingers ; yet it is not now exerted against them ; because we now have two equal 
■ moments in opposite directions, reacting against each other ; and leaving nothing for the fingers to 
“ react against, except the mere vert tut of 6 fts. 

!1 Siuee the moment about o tends to produce motion at that point in the direction in which the hands 

* of a watch move, or from the left hand, toward the right, it is called a right-hand, moment. But the 

* moment of the same force, about the point i. tends to produce motion at that point, from right to left, 
1 as shown by the arrow-head on the small circle ; hence it is called a left-hand moment. The moment 

of the force d y, with its leverage y i, about the point t, is a left-hand one; as is also that of x w 
< with its leverage e i. 

>f When the arm o s, or i c, instead of being merely an imagined dist, is a rigid bar, at one end of 
id which, as s or c, the force is imparted ; thus giving the bar a tendency to move around the point o or 
1* i as a fixed center, it is frequently called a levee; and the point o or i, the fulcrum of the lever. 

If the lever, instead of being like c i, at right angles to the direction a to of the 
!• force, should be oblique to it, as in i to, or i a ; or should be curved, or bent in any 
way, as ig t; this in noway affects the leverage, or moment of the force; for the 

everage is always tlie perp (list, or in other words , tlie shortest (list 
( from the fulcrum, to the direction of the force: and is entirely 
y independent of the length of the lever itself. This is a grand funda* 
o mental principle of all levers, and leverages; and the young 
student should carefully impress it upon his memory, inasmuch as it is of constant 
!, application in practice. 

i, The fulcrum is not always at one end of the lever, but may be between the two 
ends; so that there are two arms. Cog-wheels are merely continuous circular levers, 
with the fulcrum at the center. 

Art. 47. As a further illustration,let 
: a fb, Fig 41, be a bent lever, turning on 
is fulcrum/; and m and n two wts sus- 
•jnded from its ends, constituting two 
c rces acting in the vert directions a n, 
c t. Now ,fc, at right angles to the di- 
i Action be; and/a, at right angles to 
tae direction a n, are the arms, or lever- 
ges of the forces to, and n, about the 
1 oint f. 

„':.ln the fig these leverages are equal, say each 
j 6 ft; and let each wt, m and n, be 100 fts : then 
| the right-hand moment of m, and the left-hand 
one of n, about /, are each 6X100=000 ft-fts ; and 
lince both the forces, and /. are all in the same 
plane, it is evident that the two opposite mo¬ 
ments, balance each other ; although the lever 
fb, is much longer thau/a. 

Now suppose the wt m to be removed ; and that instead of it. a person pulls in the direction b s, by 
means of a string fastened at b. With what force must he pull in order to balance the wt n? First 
measure the leverage ft, from the fulcrum, and at right angles to the direction b s of the new force; 
and suppose it is found to be 9 ft. Now we have already found the moment of n about / to be 600 ft- 
fts ; and we require the same moment on the opposite side; so that all that is reqd is to find what 

number the 9 feet leverage must be mult by, in order to make 600. This is plainly —— = 66.66 fts, 

for the pull which the person must exert; because 9 X 66.66 — 600. 

So it is seen that with the same length of lever, fb, we can have diff powers, (so called,) or lever - 

















336 


FORCE IN RIGID BODIES 


ages, according to the direction in which we apply our force to the lever. This, however, evidentl; I 
has its limit; for the greatest power is gaiued (to use the popular expression) when we apply oui 
force in the direction by, at right angles to a line fb. drawn from the fulcrum to the outer end of tht 
lever. If we apply it in the direction b d, we get only the leverage fh. 



On the same principle as in the foregoing example, it o t and g a, Fig 
42, be two beams of equal scantling, but of diff lengths; with one end i 
of each tirmly fixed in a vert wall, and both sustaining equal suspended 
wts tv, x; the moments of the wts about the points o and g will be equal, 
because the arms or leverages oe.ga, are equal. Therefore the wt w 
will have no more tendency to break off the long beam at o, than x has 
to break the short one at g. The wts of the beams themselves are Trot 
here takea into consideration; and this is always the case in speaking 
of levers, unless otherwise expressed. In very many cases, the wt of 
levers of two arms does not affect the result aimed at, provided the arms 
are so proportioned as to balance each, other when unloaded; no matter 
what their comparative lengths or wts may be. 

If, in Fig 42, we apply pulls to the beams, in the parallel directions i 
tm, c n, at right angles to o t, then the leverages become changed from 
o e aud g a, to ot and g c ; and siuce o t measures 6 times the leugth of 
g c, it follows that the beam ot would be broken off at o, by I part as 
much force in this new direction, as g a would; for the leverage being i 
6 times as great, must be mult by only *. as much wt, in order to have i 
an equal breaking moment. 


Art. 48. In ordinary phraseology, the load, or resistance of any kind, which 
we wish to move, overcome, or balance, by means of a lever, is called the weight; 
while the force of whatever kind which we apply to accomplish this, is called the 
power.* Usually, but not always, the power is applied to the longer arm. 
Equilibrium, or balance, or equal moments in opposite directions, will then plainly 
take place, when the long leverage (not lever) lias the same proportion to the short 
one, that the wt has to the power; because then only can the long leverage mult by 
the small power, have the same moment as the short leverage mult by the great wt. 


This is seen in the common steelyard, Fig 43; which is merely au iron lever, turniug on a fulcrum 
/1 and having the wts of its two arms f a, fb, so proportioned as to balance each other wheu un¬ 
loaded. Here the power, PI, at the dist of two divisions from the 
fulcrum; balances the wt, W 2, at the dist of one division. If the 
wt W 2 were suspended at y, only half a division from /, it would 
balance the power P 1, suspended where W2 is in the fig, at a whole 
division from /. If the power is reqd to move, or overcome the wt, it 




X 


Fi°43 


7 


FI 


is plain that either the power itself, or the length of its arm, must be 
greater than when mere equilibrium is to be effected; in other words, 
besides the two straining forces, which by their mutual action balance 
or equilibrate each other, we need some unresisted force to impart 
motion to the inert matter. 

In the two levers of the same length, Fig 44, the leverage fw of the 
wt tv, is of the same length in both; namely, one division; but the 


leverage fp, of the power p, is but two divisions long in the upper oue, 
and three divisions in the lower. Therefore a power of 1 H> will balance 
only a wt of 2 ibs in the upper one, aud of three lbs in the lower. In the 
upper one, the power will move twice as fast as the wt; in the lower oue, 
three times as fast. When the fulcrum is between the wt and the power, 
as in the upper one, the lever is said to be of the first class; when the ful¬ 
crum is at one end, and the power at the other, second class; fulcrum at 
one end, and wt at the other, third class. In all cases it is assumed that 
the fulcrum is in the same plaiie with the directions of both the wf 

aud the power; otherwise the principles do not apply. When two weight! f 
balance each other on two arms of a lever, as a steelyard, or com mot 
weighing scales, &c, their directions are the vert lines passing throug.' 
... ,, l ,leir centers of grav ; aud the same imaginary vert plane which coincides 

with those directions, coincides with, or passes through, the fulcrum also. When this is not the 
case, no equilibrium can exist. This may be readily proved by experiment; for we cannot balance a 
bow-shaped piece of stick or wire, so long as the bow is hor; for it will turn on the fulcrum of it* 
own accord, uutil the bow becomes vert; so that the same vert plane that passes through the fulcrum 
shall pass also through the cen of grav of each half of the bow. If all the forces acting on the lever 
are hor, or oblique, the imaginary plane must be so too. 



Rkm. From what has been said respecting the lower Fig 44, it follows that when a load w is home 
at any point of a .beam fp supported at both ends, then the portion of the load supported bweach of 
these points/ and p, is in the same proportion to the whole load that the respectiveportions fw and 
p w o( the beam, are to the whole span fp of the beam ; but inversely; that is the P smal"est portion 
or the load is borne by the support at p ; and the largest portion bv /. In the fig fw is one third 
and p w is two thirds of the clear length or span of the beam ; henci, p supports* of the load t* 1 
and / supports %. Or aa/p \ w ; \fw ; load atp. And as/p t w j i p w i load at/. 


1 b - v , meai i, 8 of leverage a small power can be made to move a great wt, is in commoi 
parlance styled a gain of power. In a scientific sense the expression is absurd, yet in practice t 
universal use become very oonveuient, and we shall therefore employ it. Vhen the leve. 
° f raere y balancing the power and the weight, has to be put into motion, it is plain that the; 
must be some excess of force applied at the power eud, to produce the motion. 





















FORCE IN RIGID BODIES, 


337 



Art. 49. This example is the same as in Fig 19, 
where the question is solved independently of leverage and 
moments. 

Let S be a stone of 3 tons; standing on a hard hor base n 
to ; let i be its cen of grav; and let the dist ng from the toe 
n, and at right angles to the vert direction, ig, of the gravity 
of the stone, be 2 feet. Also let f h be a force of 2 tons, im¬ 
parted to the stone at h; and let the dist no from the toen, 
and at right angles to the direction fa of the force fh be 5 
ft. Will the foree/A upset the stone arouud the toe n, as a 
turning point? 

Here n g, or 2 ft. is the leverage of the force (3 tons) of gravity; consequently, the moment of the 
stone about the poiut », is equal to 2 ft X 3 tons = 6 foot-tons; and this moment (which is called the 
moment of stability of the stone, about w) alone tends to prevent the stone from overturning about 
n. Again, n o, or 5 ft, is the leverage of the force, (2 tous,) of /ft; consequently, the moment of /A 
about the poiut n, is equal to 5 ft X 2 tons — 10 foot-tons; and this moment alone tends to overturn 
the stone about «. Since the overturning moment is the greatest, the stone will of course upset. 
The foregoing case resembles that of an abutment resisting the thrust of an aroh ; or that of a re¬ 
taining - wall, sustaining the thrust of earth against its back; said thrust being supposed to be con¬ 
centrated at its c enter of pressure. (Art 57.) It is analogous to a 
simple bent lever efo, Pig 25%, supported at its fulcrum/; 
around which it may revolve. The short arm fe, of 2 ft, is acted, 
upon by the 3 tons wt of the stone; moment = 2X3 = 6 ft-tons. 

The long arm fo, of 5 ft, is acted upon by the 2 ton force ft; mo¬ 
ment 5 X 2=10 ft-tons; which, being greater than the 6 ft-tons 
of the stone, equilibrium cannot exist; aud motion must ensue. 

The wt of the body S, in Fig 25, constitutes one of the forces act¬ 
ing upon it; while its inert matter constitutes the two lever arms 
at Fig 25%. It frequently thus happens that a body is at the 
same time the resistance to be overcome; and the lever with 
which to overcome it. It is plain, that in the same manner as 
above, we may find separately the moments of any number of 
forces acting in the same plane, upon a body ; and may afterward 
ascertain their united effects, by adding into one sum those which 

tend to overturn it; and into another sum, those which tend to prevent its overturning; the diff 
between these sums will be their joint effect. 

Rem. In Fig 25 we have supposed the body S to turn about a single point, n; 
but in practical cases they turn about edges , as d y, Fig 25%, the assumed turning 
edge of the body B ; which may be re- 


Fig. 25% 



Fig. 25%. 



garded as a retaining-wall, or abutment, 

&c. In making calculations for the 
strength of such structures, it is usual 
to restrict ourselves, for convenience, to 
a supposed vert slice, one foot thick, 
like the shaded end of the fig; in which 
case the turning edge is one foot long 
instead of extending along the toe, d y, 
of the entire structure; (see Art 70.) 

This, however, causes no change in the 
calculations, which remain the same as 
for Fig 25; for we suppose all the wt of 
the slice to be concentrated at its cen 
of grav ; and the forces to be imparted 
in the same vert plane with the direction of the gravity ; so that it amounts virtually 
to the same thing as if we assumed our one-foot slice to be infinitely thin ; but still 
to have the same wt as if it were one foot thick. We of course restrict ourselves, 
also, to the forces acting upon such 1 ft slice. 

But in fact, it is not absolutely necessary in such cases, to suppose our applied forces to be in the 
same vert Diane with the gravity of the wall, provided that both our structure, and the base against 
which the edge d y bears, and revolves, may be regarded as practically rigid , or unyielding, under the 
action of those forces. For, if there be no yielding whatever along the edge dy. then it is immaterial, 
so far as regards overturning, whether the force be applied at o, or at t ; for d y then becomes analo¬ 
gous to the rigid axle c i, Fig 25%. with two lever-arms s e, and c a. The wt r, may be supposed to be 
that of the structure; and from that common machine, the wheel and axle, we know that it is imma¬ 
terial whether the applied force, and other lever-arm, are attached to the axle at c ; or at any other 
point along its length ; so long as the axle is equally unyielding at every point. Art 53 will perhaps 
make this more clear. 

Caution. Although it is immaterial, as just explained, as regards overturning, 
whether the force, Fig 25%, he applied at t or at o ; it is plainly not immaterial as 
regards a tendency to swing the body around horizontally, or as regards pressures 
(and consequent friction) at the ends of the stick, Fig 25%, or in the bearing //, 
Fig. 45. 


































338 


FORCE IN RIGID BODIES 


Art. 50. Eqnilibrinm of forces, and of moments. The stu¬ 
dent must distinguish clearly between those cases in which forces hold each 
other in equilibrium, and those in which the moments of forces do so. Thus, if 


in 


two equal forces n and R, Fig Y, act against each 
other in the same straight line n R, but in op¬ 
posite directions, we correctly say that these two 
forces are in equilibrium, or balance each other, 
or prevent each other from giving motion to the 
body. Rut also in the case of a lever, one arm of 
which is say 10 ft long, and the other only 2 ft, w« 
usually say that 2 lbs of force at the end of the 
long arm, will balance or hold in equilibrium 10 
ibs at the end of the short one. But this is not 
scientifically correct; for a force of 2 Ibs cannot 
possibly balance one of 10 Ibs. It is actually the 
moment of the 2 Ibs that balances the mo« 
That is, 2 ft»s X 10 ft leverage = 20 ft-Ibs, the moment of 
the 2 Ibs, balances the 10 lbs X 2 ft leverage = 20 l't-Ibs, the moment of the 10 Ibs. 
As to the two forces 2 and 10, they are balanced by the upward 12 ft) reaction of the 
fulcrum. 

Rem. When any number of forces 



it 


Fi 9 Y. 

merit of the 10 Ibs. 


any number of forces as m, n, n, p, Fig Y, in the same plane, 

whether acting through one point or not, hold each other in equilibrium, then any one, of them, as n, 
is equal to the resultant R of all the rest; and will be in the same straight line with it, but will act 
in the opposite direction. And this is the most ready method that suggests it¬ 
self to the writer for determining whether several given forces 
-equilibrium or not. Art 30 may be used for finding the 


are in 

required resultant R. See Art 40. 

Art. 51. Equality of moments. This principle consists in the follow¬ 
ing: If any number of forces as a, b, 

~L':~ o a - n • _ i 7 Z 



c, d. Fig 26, all in the same plane, and 
acting upon a body N, in any directions 
whatever in that plane, hold each other in 
equilibrium, then if any point i be taken 
in that same plane, whether within the 
body, or out of it, the moments of 
the forces will hold each other in equi¬ 
librium around that point; that is, the 
moments of all those forces (fc and d) 
which tend to turn the body N in a right 
hand direction, (or like the hands of a 
watch,) around the point i. will together 
lef/h^Vd" th ®. moment8 of aI1 those (a and c) which tend to turn it around i in a 




If the four forces a, b, c,d, mutually prevent each other from imparting to the 

teU<lenCy to l m ° v . e * “ 3 a whol f> in au y straight direction whatever, they are themselves 
in equilibrium, and this being the case, their right and left hand moments 


Forces. 

Arms. 

Right-hand Moments. 

a—6 

3.9 


c = 3 

5.8 


b-4 

4.6 

18.4 

d = 7.2 

3.111 

22.4 



40.8 Total. 


Left-hand Moments. 


23.4 

17.4 


40.8 Total. 


But moments around a point may hold each other in coni- 
lihrinm even when the forces themselves do not balance 
each other; as in the case of the lever in Art 50. balance 

Rem. If any number of forces as a, b, c, d, Fig 26 in the 

Inline o'", a,';*,.,' SRme dlreC,, °" S on? 

But it dees not follow, that because forces mar balance each other ™h.e . 

will do so if applied to a body N at any point, whatever, in the same plane. P * ^ P ° lDt ^ the * 


























FORCE IN RIGID BODIES. 


339 


»o^ r Vi 52 * . t y irt,,al velocities. Whenever the power and the wt balance 
each other, either in a single lever, or in a connected system of levers, or leverages 
ol any kind whatever, then it we suppose them to be put into motion about the full 
cium, their respective vels will be in the same proportion or ratio as their leverages; 
that is, it the leverage ot the power is 2, 5, or 60 times as long as that of the wt the 
power will move 2, o, or oO times as fast as the wt. Therefore, by observing these 
vels, we may determine the ratio of the leverages. The wt and the power are to 
each other, therefore, inversely as their vels, as well as inversely as their leverages- 
and this is based upon the principle of virtual velocities; and is very important. ’ 

Art. 53. Neither the amount, nor the effect of leverage is changed, if the arms 
of the lever, (whether straight or crooked, or in whatever relative positions they may 
be,) instead of being in one piece, and supported by a single point or edge as a fui- 
licruni, as in Fig 44, p 336; should consist of two separate pieces m, n, Fig 45, firmly 
muted to a straight rigid axle a x, 
of any length; (usually placed at TL 
right angles to the levers m, n,) and 
supported at two or more points 
f, f. The moments are then about 
the axis, or longitudinal center-line 
of the axle; any point of which 
may be regarded as the fulcrum. It 
is immaterial at what points along 
the axle, the lever-arms m, w, may 
be attached to it; both may be be- 
tween/,/; or both outside; or 43 

one in each position. But see 

Caution, p 337. This is illustrated by the common wheel and axle. 



The 


” r * ** utvu kj j vuiuiou «y ■ ■ J 11CJ 

rad of the wheel, and that of the axle, measd from their common axis, constitute 
two continuous levers. Also by series of cog-wheels, which we see placed indiffer¬ 
ently at any points along extended shafts, or axles, whether vert, hor, or inclined. 

If the levers m, n, are not at right angles to the length of the axle, then their lever-' 
ages, and not their actual lengths , (measd from the ceqter line of the axle,) must evi¬ 
dently be used in calculating the moments of forces acting upon them. See Remark, 
Art 49. 

Rem. The assumption of an imaginary axle, or axis, as in Rem to Art 49, enables to solve such 
cases as this. The entire load being Known which is sustained by three vert supports; also the posi¬ 
tion of its cen of grav; to find how much of it is 
sustained by each of the supports: Draw by scale, 
a triangle ab c, Fig 46. showing in plan the correct 
position of the supports; and of the cen of grav g, 
of the sustained load. Assume any side, as a c, to 
be an axis ; and from it draw the two perps: ib, to 
the opposite angle b ; and e g. Now, it is plain that 
e g, and i b, may be considered as two leverages 
from the supposed axle a c. At g is placed the en¬ 
tire load, whose moment about the axis of the axle, 
is equal to the load, (say 5 tons,) mult by the meas¬ 
ured (by scale) length of e g, say 8 ft. And this 
moment (5 X 8 := 40 ft-tons) is balanced by that 
of an upward force at b : which upward force must 
be equal to that share of the load which rests upon 
b, inasmuch as the two are in equilibrium. To find 
the amount of the force at b, we have only to div 
the moment (40 ft-tons) of g, by the leverage i b, of the reqd force b. Suppose i b is found by the scale 

be 25 feet; then the upward force at b, is ^ = 1.6 tons; and we have its moment about the axis 
— 1.6 X 25 — 40 ft-tons, the same as that of g. Therefore b supports 1.6 tons of the load. In pre¬ 
cisely the same manner we may assume each side in turn to be an axis ; and find how much is sus¬ 
tained at the opposite angle. The pressure on each of four legs cannot be calculated. 

The Fig T, in which a represents one end of the axis ; a g the leverage e g ot the load g ; and a s 
that (t b) of the force b, will make the principle more apparent. 



Flo 46 


Q 


7 


a 


Art. 54. Ex, 1. The condition of a beam 
a b, Fig- 47, may often be examined on the 

principle of a lever. Suppose it to be of uniform 

depth and thickness; its length a b, in the clear between its supports, 

20 ft; its wt 600 fts ; and its position hor. In this case we know that 
one-half its wt. or 300 lbs, is borue by each support. To prove this, we 
may consider its entire weight to be concentrated at its cen of grav, 
which in this case will be at its center t. See Arts 56, 57. Then we suppose 
one of the supports, as o, to be removed ; and an upward force/to be ... , .. . . 

acting upon a lever a b. 20 ft lone, without wt; and sustaining a load of 600 lbs at 10 ft from the ful¬ 
crum a. Since the force and the load both act vert, ami the beam is hor or at right- to tlieir 

directions, therefore the dist a t and a b from the fulcrum, are the true levorages o ' , . 

load. Now, the moment of the load, about the fulcrum, is 600 fts mult by 10 ft — 6000 ft-lbs, and to 


* 1 # 
Fio47 f 




































340 


FORCE IN RIGID BODIES 


6000 


balance this the upward force/ must be equal to — 300 lbs, or one-lalf of the wtof the beam; 


the proof reqd. 

If, in addition to its own wt, the beam had actually sustained a load w at its center, we must add 
this load to the wt of the beam; and then proceed as before. So also if it sustains a load uniformly 
distributed over its length. 



Ex. 2. If the cen of gravof the beam be at any point 

y, Fig 48, not at its center, we use the leverages a y and a s, instead of 
a t and a It , and if iu addition it sustains a load z at any point n what¬ 
ever, we first find as before the force reqd at F for the beam alone; and 
afterward, by usiug a n, and a s as leverages, we find the force reqd by 
the load; and add the two forces together for the total F. 


Item. 


To find the portions of z borne by a and by s 

say, as the whole span s a is to the whole load z, so is n a to the load , 
on s ; and as s a is to z so is n s to the load on a. 


Ex. 15. If the beam sustains several loads at diff points, 

as in Fig 49, calculate for each of them separately, usiug the leverages 
a i, a c, a o, &c ; and add all together for total F. For portious of each 
of these loads borne by a aud v see above Kem. 


Ex. 4. If the beam in any such case, is inclined, as in Fig 50, the 
hor dist a o, a g, &c, must be taken as measured from the fulcrum a, 
instead of at, a i, &e; because, since all the forces are vert in direction, 
only a hor line can be at right angles to them , and serve to measure 
their leverages from the fulcrum a. if the beam be rigid, and its 
ends cut hor, as shown in this fig, it will have no teudeucy to slide; 
because all the forces which through it are applied to the bodies m 
and p are vert; and since the joints are at right angles to those 
bodies at those points, the entire forces will be imparted also; no por¬ 
tion of them remaining unresisted, to act as motion, so long as the 
beam remains rigid, and consequently straight. But if it bends un¬ 
der either its owu wt, or that of its load, new forces come into action, 
which will tend to push the supports outward from each other; so 
also iu the foregoing cases 


It is only where we may practically regard a beam as rigid, or unchangeable under the forces, that j 
the foregoiug concentration of entire weights or forces at the cen of grav, can be safely assumed. 
It will not apply when we are investigating the strength, and deflections of beams; see Art 58. 

After having thus obtained F, in any of these cases, or in other words, having found how much 
of the entire wt of beam and load bears upon one support, we have only to subtract it from the entire ( 
to obtaiu that on the other support. It is plainly immaterial which end of the beam is assumed 


wt 


Fi g\s 51 


to be the fulcrum in any of these cases. 

Ex. 5. Let a o, Figs 51, be a hor beam 10 ft long, projecting from a vert wall a c ; and resting at one 

eud on a step a ; the other end being sustained by either 
a strut, or a tie o c, 12)$ ft long. The beam, and'its uni- j 
form load, weighing together 3 tons, what will be the push- j 
ing strain aloug the direction of the strut; or the pulling 
strain along the tie? Draw the Fig to scale; and meas- 
ure ai (which will be found to be 6 ft) at right angles 
toco. Now, the weight of a rigid body, when considered 
only with regard to its effect in moving the entire un¬ 
altered body, or in straining it bodily against another ; 
body, acts the same as if it were all concentrated at its 
cen of grav; aud since we are now about to consider it 
in that light, and not as tending to bend or break the 
beam a o (in which case only half its uniform load, and. 
wt must be assumed to be concentrated at its cen of grav;) 
we consider the 3 tons wt to act at g, 5 ft, or half the 
leugth of the beam from a. Now, the 3 tons, being a forcr 
of grav, will act in a vert direction; and since the beam 
is hor, a g is at right angles to this direction of the force 
, . exerted by the beam aud its load. Consequently, if v 

assume the beam to be a lever, movable about a, as a fulcrum, a g is the leverage of that force of 



f™’ ' l “ d thu . n ">ment of that force about a, consequently, is 3 X 5 = 15 foot-tons. But this moment 
is reacted against by that of another force in the direction c o; which acts at the point o of the level 
o o, to uphold the beam aud its load. The leverage a i of this force, that is, the dist from the fulcrum 
o, and at right angles to the direction c o or the force, has already been found to be 6 ft; consequently, 
tne torce ItseB, in order to have a moment of 15 ft-tons about a (as the beam aud its load have) must 

vvidently be — = 2.5 tons, the reqd strain along the strut, or along the tie, o c; for 2.5 X 6 = 15. 





























FORCE IN RIGID BODIES, 


341 


. A load resting- on two props either at its ends or otherwise. When a 
load c ot any shape whatever, rests in any position upon two props x and z the 
portions of its wt borne by the respective props will be to each 
other inversely as the horizontal distances ox, oz, from the 
cen of grav ,c of the load, to the props. Thus if o z is two, three 
or four times as great as o x, then will x bear two, three, or four 
times as much of the entire wt of the load c as z does. There¬ 
fore to find how much each prop bears, first find 
the cen of grav c of the entire load; and its hor dist (say o x) from 
either one of the props, (say x.) 

Then as entire hor dist x z . Entire wt . * . , Wt borne by the 

between the two props * of load • • 0 x • other prop z. 

And this wt taken from the entire load leaves the wt borne by x. 

This all amounts to the same as if we consider xz to be the clear span 
of a beam without wt, and supporting a load equal to c, concentrated at the cen 
of grav of c. 

Conversely, to place two props x and z so that each may bear a given por¬ 
tion of the entire load c, take any two hor dists o x and o z from the cen of grav 
c, inversely as the two portions of the load to be borne by each. Thus if a; is 
to bear two-thirds of the wt, make o z equal to two-thirds of x z. 




Ex. 6. The following is very important in its application 
to arches of any material. 

Let cndr j, Fig 52, represent one half of a bridge arch. If this half were not prevented by the 
hor pres, ha, of the opposite half, it would evidently fall, as in the shaded hg, by turning about the 
point r as a fulcrum. (See Rem 1.) Let us find what this hor pres amounts to in any case. To do 
this, we may consider the half bridge cndrj to be a lever. Suppose its wt to be 80 tons , and tv 
be concentrated at its cen of grav g ; the vert line g s being of course the direction in which it »oud 
act. And let its leverage r t y about r, be 6 ft; r t of course being at right angles to the direction 








342 


FORCE IN RIGID BODIES, 


0 a. Then is its moment about r equal to 80 X 6 = 480 ft-tons. (Art 46.) Now, whatever may be 

the amouut of the hor force A a, which acts at the 

end a of this lever, to counteract this moment of q n 

480 ft-tous, its leverage (Art 46) is plainly equal to re, 
measured from the fulcrum r, and at right angles to 
the direction h i of said force. Suppose we hud by mea¬ 
surement from the drawing that re is 8 feet. Then the 

480 

force itself must necessarily be —60 tons; which 

8 

is the hor pres which the opposite half of the bridge 
exerts against the keystoue a, of the arch; for 
60 X 8 = 480 ft-tons of moment. 

Rkm. 1. But so long as an arch is not deranged, 
but remains firmly in position, the half arch, in¬ 
stead of teudiug to revolve about the point r, presses 
equally over the entire surf r d of its skewtjack. 

Therefore, the leverage with which the hor force h 
acts upon the skewback, is actually y o, measured 
from the center of r d, and in practice it must be 
used instead of r e. In the same manner, y m be¬ 
comes the leverage for the wt, instead of r t. 

Rem. 2. The cen of grav of file 
half arch, can be found by making 
a drawing c n d rj, about 4 to fi, ins long, on pasteboard, or on a stiff drawing-paper, to a scale. 
Cut out the fig; and balance it flatways on a sharp straight-edge, or over the edge of a table, in 
two directious or positions. \\ here these two directions intersect each other is the cen of grav. It 
is not indeed this ceh itself that is needed, but the liue q s, of its direction ; which mar be found 
at once by taking care that the straight edge is parallel to the back n d, while balancing the fig. 

Rkm. 3. Under the head leverage, may be classed the tread-wheel ; windlass and lever: capstan 
and lever; and all axles turned by a winch or by a crank ; such as the drum and winch with which a 
water-bucket is raised trom a well, Ac. They are all merely continuous simple levers, of which the 
axis is the fulcrum ; the rad of the circle described by the power is one arm, and the rad of that de¬ 
scribed by the shaft, drum, &c, is the other. 

Rem. 4. Compound levers, a a, bb, cc, Fig 52^, maybe used where 



there is not space for the arms of a single lever of sufficient power.' 
extend in one line; but maybe placed 
one over the other; or in such other po¬ 
sitions as may be convenient. Their 
effect is much greater than the combined 
effects of the three simple levers, and is 
found thus: As the product of the weight- 
arms, 2 X 1 X 3 = 6; is to that of the C, 


They need not 


o- 




10 


% 52 J- 




W <J3i 


power-arms, 10 X 8 X 7 = 560; so is the 
power to the weight; or, as 6 : 560 :: 

I’l : W93J^. These arms are measd in 
all cases from the fulcrum; which is 
sometimes at the end of one or more of p j 
the levers, when compounded ; see Fig 44. A 
The combined effects of the three simple 
levers would be but 5 + 8 + 2% = 15^; or P 1, W 15^. 

A series or train of toothed pinions and wheels, working into each 

other, is merely a series of continuous compound levers. These are generally set in motion by a 
w-inch-handle, the rad of which is the first leverage of the series; while at the other end of the train 
the wt is usually suspended from a drum, the rad of which is the last leverage. To find the effect, 
mult into one prod the radii of the winch and of the wheels, and into another prod the radii of the 
pinions and of the drum ; then, as the last of these prods is to the first, so is the power to the wt. 
as in the preceding case. "M 

l u the foregoing cases of compound leverage, as in all other cases whatever of leverage, tho 
vel of the wt^is to that of the power, as the power itself is to the wt; thus, in Fig 52)4, the wt will 

move only part as fast as the power; or the power must descend 93)4 inches, in order to raise 

the weight 1 inch ; on the principle of virtual vels. See Art 52. 

is a combination of leverage, with an inclined plane ; a spiral inclined 
plane being rormed by the threads of the screw. While the power applied to the lever which turns 
moves around an entire circle, the body moves only the dist between the centers of two 
, S, ? CC ‘ r - Hl ' mechanical contrivances, the wt is to the power as the vel of the power is 

to that of the wt. so in this case, theoretically, the wt is to the power, as the entire circumf of the i 

circle described by the power, is to the dist between the centers of two threads; but in practice the 
friction ot the screw (which under heavy loads becomes very great; has also to be overcome bv the 
power; and this fact makes the calculations of but little use. overcome D> the 

« also, when a fixed one, is referable to leverage. In the fixed pulley A Fie 
f- t * lere ls oo gain of power; for here the diam a b is a lever of two equal arms revolving around 
its lulcrum at the center of the pulley. Consequently, the wt and the power have equal leverages ; 

























FORCE IN RIGID BODIES. 


343 



each equal to the rad of the 
circle ; and iu order to balance 
a wt W of say 1 ton, the power 
P must also be 1 ton; for if one 
of them moves, the other must 
plainly move with the same 
vel. To raise the wt, the power 
must exceed the wt; because 
some unresisted force is re¬ 
quired to give motion, as well 
as to overcome the friction of 
the axle around which the pul¬ 
ley revolves, and the friction 
of the rope in the groove 
around its circumf. These fric¬ 
tions become so great w hen 
many pulleys are combined, 
that theoretical calculations of 
the power are of little value. 
Although a fixed pulley gives 
no gain of power, it is very 
convenient for allowing change 
of direction in applying the 
power; so that by pulling 
dowuward, or hor, &c, we can 
cause the wt to rise vert. It is 
plain that the rope in this 
pulley is equally strained at all 
points. Theoretically, this is 
the case with any one single 
rope, as rcdfge, Fig 52%, 
passing around any number of 
pulleys, whether fixed, as A or 
1), or movable, as B; and all 
the theoretical calculations of 
the power may be based upon 
this principle alone. They will, however, be incorrect in practice, on account of the friction just 
alluded to. In Fig 52%, where only one rope is used, the lower pulley-block B s, to which the wt W 
is attached, is directly upheld by the two parts df and eg' of the rope. Consequently each of these 
parts is equally strained. Since both parts are vertical the strain on each is that due to a force 
equal to one half the wt*; and since the whole single rope is theoretically strained to the same ex¬ 
tent, that part of it to which the power is applied must be strained equal to half the wt W ; or, in 
other words, the power itself must be equal to half the wt, and will move twice as fast, and, of course, 
twice as far in the same time. 

In Fig 52 the lower pulley-block ty is sustained directly by the 4 parts cccc of the single rope ; 
therefore, each part of the rope, and, consequently the wholeof it, is equally strained by a force equal 
to % of the wt W; and the power P must be = the same %, and will move 4 times as fast as the wt, 
or 4 times as far in the same time. It is immaterial whether the two pulleys in the lower block, Fig 
53, be placed one above the other, as shown, 9c (as usual, and more convenient), side by side ; so also 
with those in the upper block. If there were 3, 4, or 5, &c, pulleys in each block, then there would 
be 6, 8, or 10 sustaining parts c c, &c, of rope, each stretched equal to 1, i, or ~ of the wt W; 

and the power would also be in the same proportions to the wt; in other words, to find the theoret¬ 
ical proportion of the power to the wt, when there is but one rope throughout, as in Figs 52% and 
53. divide 1 by the number of parts ccc c of rope which directly sustain the lower block. In fig 53 
the number of parts is 4. When more than one rope is used in a system of pulleys, the principle is 

the same as above, but the rule must be differently worded. See 
Fig 53%. 

Here the lower block y b, with its attached wt of sav 4 tons, is directly 
sustained by the two parts a and c of one rope. Consequently, each 
part has a strain of % the wt, or 2 tons; which is uniform throughout 
that rope. But all the 2 tons strain on the part c is sustained by the 
hook s; while that on the part a is sustained by the two parts n and m 
of the other rope; each of which plainly sustains one half of it. or 1 
ton, which is uniform throughout this second rope to its very end. 
Therefore the power also Is 1 ton, Or % of the wt W. The mode of 
proceeding Is the same, whatever may be the number of movable pul¬ 
leys. To find the theoretical proportion of the power to the wt. mult 
together continuously as many 2s as there are movable pulleys, and 
div 1 by the prod. Thus, here we have two movable pulleys, and 2 X 
2 = 4; and % = the answer. If there were 4 movable pulleys, we 
should have 2X2X2X2=16; and -jL = answer. 

In all our figs, that end of the rope to which the power P1 is applied 
is represented as hanging vertically, and parallel to the other parts of 
the rope ; but this was done merely because the power is supposed to 
be a weight , and of course acting vert. But if the power is muscular 
force, or any other kind that may act in any direction whatever, then 
the power end of the rope, as mn. Fig 53, may have that direction in 
which it is most convenient. The amount of power required will not 
be thereby changed; for it is plain that leverage from the center of the 
pulley os to m, when the power is at n, and acting in the inclined di¬ 
rection of the rope, is equal to that from the same center to o, when the 
power is at P, and acting vert. 


tlf l/S 



Fig ,531 


* If the two parts, d f and eg, of the rope were inclined, the strain on each would be greater than 
half W, as at ac, ao, Big 5, p 345. 



































344 


THE CORD OR FUNICULAR MACHINE. 


THE CORD OR FUNICULAR MACHINE. 


. .A**** ®9 me allusion to this subject has already been made on page 325 

fl 1 , 1 M 0ry * hat the cor(i - r °P° or String, &C, shall be ab- 

flexible, inextensible, fnctionless, without weight, and infinitely thin; 

«mi iiii UC1 pi , llley . s > P 0 ^’ Pj n . s ? r P e !? s > lo °P s or rings as may be used with the 
cord shall also be absolutely fnctionless; and at times devoid of weight unless 
said wt is included in one of the acting forces. These assumptions cannot of 
course be realized in practice, which however will agree with theory in propor- 
we approximate to them ; and this we can frequently do so far as to render 
the theory ot great use. We know that all cords have wt and thickness and can 

!i?n M Ch r d; a " d tha i s ? far frora bein g flexible, they may require very con- 
fdction 6 Ateo thaf'Tn th n m aro . uud PuHeys, posts, pins, Ac. They also possess 

friction • wl tl, tin h" eyS ’ pins ’ S , ng u nngs or loops Ac, have more or less 
« c when there are many of them, may entirely vitiate all calcu¬ 
lations. A pulley has, in itself, no advantage over a smooth 

a'VomV* ,Cal I> - ,, 1 * °r, P ° S J ^ the case niay be)except that its friction being 
cord ”d a nb? Ac ^ friction which ta kes place between a 

concerned) StU^teonefor^he oTh«’"° WS We “ SUal ‘ y (S ° far as tbeoryls 

strains^on^he^roj^s oTthe^ifferent system^of puHeys? 1 "^ Se ™ S f ° r ,be 

r Ar*;. Assuming then such a theoretical cord, pulleys pins Ac all devoid 

bent out of its course as /e by any num 



. , . -- v j u um- 

her of fnctionless pulleys or pins as n r 
m s, Ac, however placed) is all trans¬ 
mitted along the entire cord to its other 
end,straining it uniformlyin every part. 
In / g or f c this may be regarded as self- 
evident; in / e it is proved by experi¬ 
ment. It is perhaps needless to add that 
a straining torce cannot be imparted to 
one end if there is not an equal one at 


the other end to react or strain against it. 

4 vJL a L an ^£ nethe Pubeys. pins, Ac, as m, we make mo.ma , each equal to the 
force at either / or e, and complete the parallelogram mo da oi forces then the 

fi'o g c n ^ ° r r ® sultantrf m w iH give both the direction and amount of strain which 
the cord produces on that pin. This strain will differ at the seveS pins ac- 

tlfe^brce'at/or^ tW °component,; and may thus be greater, 

a will ahrays^brat T 1 

upou this fact the principle of the USl depeidl * d; aUd 

„ AA* i C * U 1 ? P rinci P le °f the cord applies also when as in Figs 2 3 4 a load 
or third force/ is imparted bet ween the ends of the cord as It a, by means of 



Fig-. 2. 


a frictionless pulley, ring or loop which can move along the cord with nortw 
ease. If such a pulley, ring or loop be first placed nnnn (L ka Wlth P erfect 
will with its load or other force move down along tlm ^LS'l“r?of?h° r ‘f 
until it comes to rest at that point a at which « in .and / the c ‘ ,rd 

b a c, b a o with the direction a b of the force f If both ends a,, ^® es 

at the same height. and the force f a load «nd S . m n thc cord are 
then a will be at the middle of the cord/but if the emSsa^at^ Sent 
























THE CORD OR FUNICULAR MACHINE. 


345 


lieig^lits as in Fig 2, from either of them as n draw a vertical, as n x ; and from 
the other one, as to lay off the whole length m x of the cord : bisect n x at e, and 
draw e a horizontal; a will be the required point. 

It we draw a b to show both the direction and the amount of the force/, and 
complete the parallelogram ^ c a o of forces, then either a c or its equal a o will 
give the uniform strain which f produces from end to end (to to n ) of the cord, 
h rom which it is plain that a force equal to a c or a o may be considered to be im- 
pai ted at each end to, n, of the cord, and there to react against a c and a o, there¬ 
by straining the cord uniformly throughout. 

Rem. 1. Precisely the same result will follow if one or both of the ends to, 
* if instead of being fixed or fastened at those points as is supposed in Figs 2 and 3, 
are continued as at to, big 4, over a frictionless pulley or pin, and prolonged either 
vertically as to l, in which ease it will sustain a load or vertical force l equal 
• *to a o or a c; or in any other direction as to s or to e, in which case it will sustain 
at 5 or e a pull equal to a o or ac acting in said directions. And the same will 
take place if one or both ends as n. Fig 4, be extended over or under staiy mini* 
l>«er of such pulleys or pins, (no matter what angles they form) say to v. The 
strain at v will still be equal to a c or a o, and will be uniform from v to l. The 
strain on any pulley or pin may plainly be found as in Art 2. 


Rem. 2. If we know the angles b a 6 , b a c, and the force or resultant a b, 
we can calculate the strains or components a o, a c; or knowing the 
angles and components we can calculate the force a b, all by the formulas in Art 
I ffb P vvhich are based upon plane trigonometry, which enables us to find un¬ 
known parts of a triangle when certain other parts are given. 

Rem. 3. If the angle m, a, n, is 120°, each component ao,ac, will be 
equal to the resultant a b ; if it is less than 120 °, each component will be less than 
{the resultant; and vice versa. 


Art. 4. A little reflection will enable the student to see that this Art is 
merely a farther illustration of the preceding ones. In Fig 5 a load s , or any 
equal vertical pull will strain the 
strings s a and z v each to an amount 
equal the Ioad.hecausethey both 
act in its own direction. If 
the parts m a, g a of the cord a mn g 
a which passes oyer the friction¬ 
less pulley or pin z were also vert¬ 
ical, each of them would be strained 
equal to one-half the load s; but 
they are not vertical, and if we make 
a b to represent s, and complete the 
parallelogram b o a c, we shall find that 
the resulting strains a c, a o are each 
rather more than half of s; and 
j they will increase if the angles at a 
increase; and maybe calculated 
by the formulas in Art 45, p 334. 



In Fig 5 we had the force s or a b given as a resultant, and from it we found its 
two components or strains a o, n c. In Fig 6 we have the two components hm,h 
g given, (each evidently equal to the load s) and from them we deduce the result¬ 
ant h b, which represents the strain on the pulley, and on the string z w. It is 
plain that this string cannot be vertical as z v is, but must (as z v does) adjust it¬ 
self to the direction of the force or resultant which pulls it. If the load s is the 
same in both Figs, we see that the pull on z w will be about twice as great as on 
zv; and the one along the cord s mgp about twice as great as that along amng a. 
This difference will of course vary with the angles. 

Rem. 1. Let s, Fig 5, instead of the load there represented, be a suspended 
man weighing 200 tbs, and holding in a, g a together at a with both hands; or let 
to a and g a both hang vertically from the pulley, and let him hold their lower 
ends apart by one hand at each end. Now if each part of the rope will bear but 
a little more than 100 fbs, and z v a little more than 200 lbs, he will be safe from 
falling. But if he lets go one end of the rope, patting it into the hands of an- 












346 


THE CORD OR FUNICULAR MACHINE, 


other man standing by to hold it; or if he hooks one end to a projecting spike in 
a wall close at hand, the rope will break, and he will fall, because he at once 
doubles the strain along both the ropes a mn g a, and z v. I 

Rem. 2. Fig 7 shows a device by which boatmen sometimes haul a boat out 
of the water, and up on to the beach, at a landing, when it is too heavy for their ' 
unaided efforts. The rope e m x n b is to be considered as horizontal in this case. |] 
One end b being fastened to the bow of the boat, the rope is carried past one ! 
smooth post n, to another at m, around which it makes a whole turn; ami a man 
stands at that end e to take in the slack while the others, taking hold of the rope 
midway between m and n, pull it into a position manin which, if the angle 
m n n exceeds 120°, (see Rem 3, Art 3) each component an, am of their 
force exceeds said force itself; and a strain equal to one of these components (except 
so far as it may be reduced by the rigidity of the rope and by its friction against tiie 
post n) is transmitted uniformly to the boat b, drawing it a short distance up the». 
beach. The rope is then straightened again from m to n by taking in the slack 
at e , and the operation is repeated as often its necessary. 

To find the strain am ox a n, divide the force by twice the nat cosine 
of the angle x an, or x am. Or to find how many times the strain exceeds the 


force divide the distance a n by twice the distance 


n « x. Here a x represents only half the force, or half 


T, 


m 



the diagonal or resultant of a m and a n. 


The force of a man pulling by jerks at a; or a 
will average between 30 and 80 lbs. 


Art. 5. We will now dispense with thefric-t! 
tionless pulley required by the principle of the* 
cord, and which would of itself roll along the cord t 
until it came to rest at a point which would cause in 


tionless pulley required by the principle of the 
cord, and which would of itself roll along the cord 


equal angles and consequently uniform strain throughout. We will is 
substitute for it a tight kii "' 1 —-" > • -*■ ~ 



tween the direction a b of the force /, and the tworf 
parts a m, a n, of the cord. Here, completing the i ■ 
parallelogram b cao,we find that the part a n of the 
cord is strained to an amount denoted by ac; and in 
that same strain would affect any extension of thew; 
cord on that side, like that to v. Fig 4. The t 
part a m would be strained equal to a o; and that 

strain wo"' J - XI -- ' | 

of any e f 

strain 

the cord, as it would have been if the knot had beentrn 
frictionless; in which case it would have sli J 


n 



part a m would be strained equal to a o\ and that 




along the cord until the angles would be equal. 


Rem. 1. But even in this case of a tight knot at a, there is always one! 
direction, as t a s, in which the force can be imparted so as to cause equal an-b 
gles sam, san, with the direction as of the force t, and then the strain will be 
uniform from end to end, as if a frictionless pulley had been used. 

Rem. 2. With a tight knot at a it is plain that a force may be imparted 
there from any direction as/a, ta <fcc. If the direction coincides witli 
either part a m, or a n of the rope, that part will bear all the strain I 
the other part remaining entirely free. 

Rem. :J. From Rule l,p 150,for drawing an Ellipse, it will be seen that -J< 
whatever point as a. Fig 8, we apply force to an inextensible cord nam with fixed 
ends, that point will be in the circumference of au ellipse, the foci of which are 
at the ends m, n. 





FORCE IN RIGID BODIES. 


347 


PARALLEL FORCES. # 

■ Art. 55 (a). Parallel forces, as a. b , c, etc., Fig. 1, are those whose direc- 
I tions (whether opposite to one another or not) are parallel. 

Any two parallel forces must evidently lie in tile same plane; 

that is, the same flat surface could coincide with both of them \ but three or more 
1 parallel forces may or may not be in the same plane. 



Thus, a and b are in one plane, a x; and, in whatever positions a and b maybe 
1 placed, or at whatever distance from one another, so long as they remain parallel 
i we can always find some one plane which coincides with both of them. So also the 
■J two forces, b and c, or b and d, or c and d, are in one plane d x, while a and c are 
i in the one plane ay. And the three forces 6, c and d, are in one plane dx. But 
the three forces, a, b and c, are not in any one plane, and no single plane can be 
i found that will coincide with all three of them. v 

ej The force b is at once in the two planes, a x and c x. Indeed, we may conceive 
of any number of planes intersecting each other in the line b x; and the force 
t b will be in each of these planes. 

i Forces in the same plane often act in opposite directions. Thus, e is not only 
j in one plane with /, which acts in the same direction, but is also in one plane 

I with any one of the forces, a, b, etc., which act in the opposite direction. With 
c, it is not only in the same plane, but also in the same line. 

We must not confound acting in one plane with acting upon or against one 
plane. Thus, all the forces in Fig. 1 act against or upon the plane m n, but none 
of them act in it, nor do they all act in any one plane. 

I (b). When forces a, b, etc., Fig. 2, are applied to a body, their tendencies to 
\move it are in the directions of their imparted components, a', b', etc., or compo- 
(j nents at right angles to the surface m n, n o, etc., of application. See Art. 25, p. 

1318 e. In order that these components maybe parallel, it is not necessary 
i that the forces, a , b, etc., should be parallel, but only that they be applied either 
,to the same or to parallel surfaces.* Thus, the forces, o, b, c, d, e, and / Fig 



2 are parallel: but the surfaces mn and no, to which b and c are applied, are 
not parallel. Hence the tendencies of b and c to move the body, as indicated by 
their imparted components, b' and c' are not parallel. But a and b are applied 
to the same surface m n. Hence their imparted components «' and b' are parallel; 
and the same is true of e and f and of d and h. Also, the imparted components, 
o' and d', of a and d are parallel, because a and d are applied to parallel planes 
m n and os; and the same is true of c and e and of a and h. 

* In practice this is often modified by the friction at the surface. See Art. 25, 
p. 318 e and Art. 63, pp. 354 and 355. 

















347 a 


FORCE IN RIGID BODIES. 


RESULTANT OP PARALLEL FORCES. 

Art. 56 (a). Tlie resultant of any number of parallel forces. 

whether they are in the same plane or not, and whether in the same directioi 
or not, is parallel to them. If they all act in the same direction, whether in tin 
same plane or not, their resultant is equal in amount to their sum; or, in other 
words, an antiresultant force sufficient to balance them, must be equal to all thci 
forces added together. But if they are in opposite directions, their resultan) 
will be equal to the dilference between the sum of those which act in one direc¬ 
tion, and the sum of those which act in the opposite one; and its direction will 
be that of the greater sum. Thus, in Fig. 1, if the forces pointing to the left 
amount to 10 tons, and those to the right 4 tons; then the resultant will be 10 — 
4 = 6 tons;, and it will point to the left. 

(1>). The resultant of any number of parallel forces In one 
plane is in the same plane with them, whether the forces act in the same 
or in opposite directions; and the leverages, or arms, of such forces, and ol 
their resultant, about any given point in the same plane, are in one straight 
line. Thus, in Fig. 18, p. 347 i, where the five forces, a, b, c, d and e are in one 
plane, their resultant, R, is in that same plane; and the leverages of the forces, 
and of R,about any point,as b or v, in the same plane, are in the straight line It v. 

(c). The resultant, R, or anti-resultant Q, Fig. 3, of two parallel 
forces, o and b, in the same direction, lies between them and in the same 
plane with them. Therefore its direction must intersect any straight line, u v, 
joining the directions of the two forces. It follows, also, that if three parallel 
forces are in equilibrium, they are in the same plane. 



To find the position of the resultant, draw and measure any straight line, uv, 
joining the directions of the forces. It is immaterial whether u v is perpen¬ 
dicular to said directions, or not. The line representing the resultant cuts uv, 
and its position is found thus: 

b a 

“ I==MT; ; and »’ < = «' ) X^ 

In other words, divide uv into two parts, ui and v i, proportioned like the 
forces, but placed inversely to them; that is, place the longer part, v i, on the side 
of the smaller force, b, and vice versa. 

Remark. This may be conveniently done by making u v equal, by any 
convenient scale, to the sum of the forces, as in Fig. 4, where nv = 42. Then 
make u i equal, by the same scale, to the force at v, or r i equal to the force at u 



1 hen a line, R. Fig. 3, drawn through i parallel to a and b, gives the position 
and direction of their resultant; and its amount is equal to the sum of a and b • 
or R = a + b. 

In other words, if a force, Q, parallel and in the opposite direction to a 
and b, and equal to their sum, be applied to the body anywhere in a line 
passing through t, it will Balance a and b, or will be their anti-resultant. 




FORCE IN RIGID BODIES. 


3476 


This rule for the position of the resultant follows from the principle of mo- 
aents explained m Arts. 4(5, etc., pp. 335, etc. For, in order that Q may balance 
i: and b, we must not only have Q equal to the sum of a and b, but the opposite 
1 'laments of a and ot b about any point in the direction of Q must be equal; 
s r u i and v i must be inversely as their respective forces, a and b; or 
( t: b :: v i : u i; so that ay^ui = by^vi. 

: If the two forces are equal, their resultant R is evidently midway between them. 
1 (d). The foregoing, as well as some other facts connected with parallel forces, 
[ pill be more readily recalled to mind by associating them with the idea of the 
[ ornrnon steel-yard, Fig. 5. Here the two forces, a and b, of Fig. 3, are 



-epresented by the two weights, a = 3 pounds at u, and 5 = 1 pound at v. Since 
>ne of these weights is three times as great as the other, we know that when 
hey are suspended at distances, u i and v i (which are as 1 to 3) from the fulcrum 
they are balanced by their anti-resultant, Q, which is equal to their sum 
= 3 _|_ i = 4. Q has precisely the same tendency to pull the steel-yard upward 
hat, the two weights have to pull it downward. Also, R (equal to Q, but acting 
n the opposite direction) is the resultant of the two weights. Its tendency to pull 
he steel-yard bodily downward when applied to it at i, is precisely equal to that 
>f the two weights applied at u and v. Of course their tendency, acting at u and 
; to bend or break the steel-yard, while it is suspended at is very different from 
;hat of their resultant, R, acting at i. Indeed, the latter has no bending ten- 
lency at all, since it acts in the same line with the upward pull of the string at». 

(e). If there are more than two parallel forces, as a, b and c, Fig. 
5, whether in the same plane or not, the process for finding the resultant consists 
in a mere repetition of that in Art. 56 (c). 



tts. e 

♦ 

Thus: between the directions of any two 
straight line, u v , and make 



6 


of the forces, as a and 6, draw any 













347 c 


FORCE IN RIGID BODIES 


ui = uv X 


Then is R the 


Through i draw R', parallel to a and b, and equal to their sum. 
resultant of a and b. 

We now regard the two forces, a and b , as done away with, and replaced by 
their single resultant force, R'; and proceed in the same way to find the 
resultant of R' and any third force at pleasure, as c. That is; from any point 
i in the direction of R', draw iz to any point, z, in the direction of c, and make 
c R' 

ik = iz X , 7 ; ; or z k = i z X „ u , . Through k draw R parallel to a , l 

R the resultant of the three 


c + R' ’ ** * c + K' 

and c, and equal by scale to their sum. Then is 
forces, a, b and c. If there are other forces, proceed in the same way with them 


Remark 1. It is quite immaterial in what order the forces are taken. Indeed, 
we may, if preferred, find first the resultant of any two or more of the original 
forces; then that of some other two or more of them; and then that of these 
two resultants, etc., etc.; in short, we may proceed in any desired order, pro¬ 
vided each of the original forces is taken once, and only once, in finding 
the resultant. 

Remark 2. In Fig. 6 we have shown the forces, a and c, acting upon surfaces 
raised above the general plane, merely in order to illustrate the fact that it is 
not at all necessary that the forces be supposed to act upon or against a plane I 
surface. See also Fig. 2. This is a consequence of the principle stated in Art. 
25(h), p. 318<7; namely, that the effect produced upon a rigid body by an imparted 
force remains the same, no matter at what part of the body it be imparte 1, so 
long as that point is in the line of direction of the force. But unless the surfaces 
to which parallel forces are applied are parallel, the imparted components of 
the forces ( i. e. their tendencies to move the body) will not be parallel. See 
Art. 25, p. 318 e, also Art. 55 (6.) p. 317. 

Remark 3. Although Fig. 6 serves to illustrate the method of finding the 
resultant of parallel forces, yet it plainly does not give the actual relative posi¬ 
tions of the forces and their resultant; because it is necessarily drawn in a kind 
of perspective, and therefore all the parts cannot be measured by a scale. The 
true relative positions may of course be represented in plan, as by the five stars, a, 
b, c, i and k, Fig. 7, corresponding to the points where the forces and resultants 
intersept some one chosen plane. But it is now impossible to represent the 
forces themselves by lines. They must therefore be stated in figures, as is here 
done. It is then easy to find the positions of the resultants, as before. 

If there are also force* acting In the opposite direction, as d and e, 

Fig. 6, find their resultant separately, in precisely the same way. We thus obtain, 
finally, two resultants in opposite directions. These resultants may be equal or 
unequal, and in the same line with each other or not. They are to be treated 
in accordance with the following principles. 

(f). Parallel forces in opposite directions. If two forces are in the 

same straight line, then their resultant is in that same straight line with them. 
If they act in the same direction, their resultant is equal to their sum. If they 
act in opposite directions, it is equal to their difference. Hence, if the two forces 
are equal and opposite, they have no resultant. When they are unequal, we 
may regard the smaller force as merely diminishing the amount of the larger 
one; and the resultant as being simply what is left of the larger force after 
such diminution. 











FORCE IN RIGID BODIES. 


347 d 


COUPLES. 

Art. 56 (g). Couples. Two equal parallel forces, a and b, Figs 8, 9 and 10, 
:mparted to a body in opposite directions, but not in the same straight line , are 
called a couple. A couple has no tendency to move the body as a whole in any 
straight line; in other words, the two forces forming a couple have no resultant. 
iTheir only tendency is to make the body revolve about its center of gravity, G, 
i ind in the plane in which the two forces lie. 



If the plane of the forces is oblique to the surface against which they act, the 
actual plane of rotation of the body (which is always the plane of the imparted 
forces of the couple) will be oblique to that of the forces as applied. See Art. 
25, p. 318 e and Art. 55 ( 6 ), p. 347. 

In order to simplify the treatment of the subject, we shall confine ourselves to 
cases where the center of gravity, G, and any forces besides those ol the couple, 
lie in the same plane with the forces of the couple. If the center of gravity is 
not in the plane of the couple, the tendency to rotation will still be about an axis 
passing through the center of gravity. 

(Ill Tlie moment of a couple. To find the amount of the revolving 
tendency or “moment” of the couple, we may multiply each force, separately, 
by its leverage about the center of gravity, G; i. e., by the shortest distance from 
G to the line of direction of the force. The product thus obtained is the 
moment of that force about G; and the moment of the couple is the sum 
of these two moments of the separate forces. In finding said sum, it is 
imDortant to bear in mind that if the positions of the two forces are (as in 
Fie 9) such that they tend to make the body revolve in opposite directions about 
thf center of gravity, G, then one of them must be regarded as negative or minus; 
so that, in such cases, the sum of the two moments Becomes their arithmetical 

difference. . ...... , , „ 

Thus, in Fig. 8 both forces tend to turn the >'ody about Gin a right-handed 
direction (as tlie Fig. is viewed) or like the hands of a . , 

moment of couple = a X Gu + b X Gv ” ° X uv. 

x) * in TTiCT 9 while the moment of a about G is right-handed, as before, that 
otTulu op^te direction, or le/Mianded. Hence, in F.g. 9, 
moment of couple - » X G u + j^X G „ X « 

In Fig 10, one of the forces, b, acts through G, and hence has no moment about 

it In this case, therefore, 

moment of couple = aXGu —a X uv. 

.. X w in oil three cases (Figs. 8 , 9 and 10) the moment of the 

It will be ? otl , c ®5 s t „ h ^ t p ? = a v m u) no matter how its forces are situated with 

couple remains the sam ( a X j. ovided their am 0 unts, and theiT distance 
reference to the ce “ t ®' 1 The moment of any couple is usually stated, for con- 
apart, * h £ st named lay ; namely, as being equal to the product of either 

»«of n the twoequal forces multiplied by the perpendicular distance between the 

forces. Or (in our figures) 

moment of couple = a X uv ^ b X uv . 

m no resultant (Art. 56 q) it can have no an ((-resultant, 
(i). Since » couple hM lance a CO uplc. To do this requires a second 

i. «., no single *®*®?,®* j to that of the first and in the contrary direction: 
tat P it e is W no°t S ne™eSary thaShe forces of the second couple should be either equal 














347 e 


FORCE IN RIGID BODIES. 


or parallel to those of the first. Thus, in Fig. 11 , let a and b be each = 3 pounds, 
and u v — 6 feet. Then the /e/t-handed moment of a and b is = a X u v = ii X. 



6 — 18 foot-pounds. Now, if c and d are each = 9 pounds, and if to x = 2 feet 
then the r^At-handed moment of c and d is — c X w x — 9 X 2 = 18 foot-pounds! 
1 nerefore it just balances the moment of the first couple, a and b. If we were 
a foo the resultant of either non-parallel pair of the forces, as a and d, by 
Art. 28, p. 319, and then that of the other pair, b and c, these two resultants would 
be in one straight line, and equal and opposite to each other: in ether words 
the four forces, rt, b, c and d, have no resultant. 
t^‘ A- coll pl e and a third force in the same plane, Figs. 12 , 13 and 14 
It will be noticed that the case of two unequal parallel forces In 
opposite directions, as a and (6 + c), Fig. 13, is merely a special case of this. 



mS % e r2, b0V ‘° 4 “ a the eBreet of the 

graTityUtsrnjomen^ab'out Sat‘S/wm $£ ° f 

i 1 q t « er i t r e th ' rd force P arallel t0 the forces of the couple as in Fie-s 19 
3 > or obhque to them, as in Fig. 14, if its moment about the center of era vit v 
of the body is equal and opposite to the moment of the oomile it « ;n gravity 
the tendency to rotation. The resultant of the three forcin’ 
the center of gravity of the body, and move it hi Us own rtSirfit lhS 
r 0 Thns n /f 1 i p S - aid J eSUltant Wil1 be e( l ual and P ara Hel to the thM force 
saS*^ ** app^ed in the 



Fig. 1-4= a 

upward under the action of b (which thus becomes the resultant of tb« *». 
forces) acting through the center of gravity, G of the bJv ThpS^ thre ,e 
similarly be prevented if an upward force 


















FORCE IN RIGID BODIES. 


w 

i than a is, and beyond a; or if an upward force greater than a be applied betwe< n 
G and a, provided the moment of said third force, about G, be equal and contrary 
1 to that ol' a.* 

In other words, if a force c, Fig. 14, act upon a body in any direction not 
passing through its center of gravity, and if we apply to the body any couple, 
as a and 6, having a moment equal and contrary to the moment of said force 
about the center of gravity of the body, the etlect of the couple will merely be 

I to transfer said force (as it were) to the center of gravity of the body. 

And, in general, if any single force, c, Fig. 14, act upon a body, and if any 
couple, as a and b, be then applied to the body, the effect of the couple will 
merely be to shift the line of action of the force, c, to another position parallel 
to its actual one. The distance through which c will be thus shifted is 

, moment of couple 

distance = --„-- * 

force, c 

If the moment of the couple is ^eft-handed, as in Fig. 14, the force c will bo 
shifted towards the right (looking in its owu direction) and vice versa. 

’ (1). To find tile resultant R Fig. 15, of two unequal parallel 

I forces, a and b , in opposite directions, but not in the same straight line. 



cfortimr from anv noint, as it, in the line of the smaller force, ot, draw any 
l ifni”TmfSStW the direction of the greater force, b, as at . * The 
nmS of the resultant R, is = b — a. Its direction is the same as that of the 
greater force,\ and passes through a point i, in urn, the position of which is 

found thus: , _ 

5 a 

V - ; or vi = uv X 


« { = « v 


R 


R 


Through i draw io parallel to the two forces. Then o is the point of applica¬ 
tion of the resultant R The tendency of said resultant, either to move the 
bodv as rwbote in the direction i Q, or to make it revolve about its center of 
r ;' tht. sime as that of a and b combined. In other words, if we sup- 
nose It removed then an anti-resultant, Q, equal to R and acting toward the lett 
Fn the line Q i would eounteraet both the tendency of a and b to move the body 
as a whole in a straight line, and tlieir tendency to make it revolve. _ 

* T>,it in tht^ fi rst7ca.se the upward resultant of the three forces (being always 
equS LThe tWrd foS will bS less, and in the second case greater, than 6. 
















FORCE IN RIGID BODIES. 


347*7 

Art. 57 (a). Graphic Method of finding the point of application of the 
resultant of any number of parallel forces, acting in the same or in opposite 
directions. Let a, b and c, Fig. 16, represent the forces in their relative positions. 



(The lengths of these lines need not represent the amounts of the forces by 
scale.) In any convenient part of the same sheet of paper, draw a straight line, 
X Y Fig. 17, parallel to the lines representing the forces. On X Y lay off dis¬ 
tances representing the forces by any convenient scale. Thus: beginning at a', 
let a' a represent force a; b’ b, force b; and c' c, force c ; and so on, if there are 
other forces. 

The line a' c now evidently represents, by the chosen scale, the amount of the 
required resultant. Or, if we suppose a line, c a', drawn upward from c, said line 
will represent the anti-resultant of the forces. But this would bring us again to 
the starting point a'. We should thus obtain a closed polygon (a' ab c a') of forces 
in equilibrium. (See Art. 38, p. 329.) The polygon here forms a straight line 
because the forces are all parallel. 

From any convenient point, O, on either side of X Y, draw lines O a', O a, etc., 
radiating to the ends a', a, etc., of the several vertical lines a' a, b' b, etc. The 
point O is called the pole. 

Through any convenient point, as m, in the direction of the first force, a, Fig. 
16, draw s h parallel to the first radial line O a', Fig. 17, and of indefinite length. 
From m draw m n parallel to the second radial line O a and ending at n in the 
line of direction of the second force, b. From n draw n p parallel to O 6. Fig. 17, 
and ending at p in the line of direction of the third and last force, c. Through 
p draw w k parallel to O c and intersecting s h in i, which is a point in the required 
resultant. Therefore, a line R drawn through i parallel to the given force*, 
pointing downward, and equal by scale to the sum of the forces, represents their 
r snltaut. 

The result will be the same, in whatever order the forces be drawn in Fig. 17; 
but it is generally most convenient to let them follow each other (beginning at 
the top) in the same order in which their positions follow each other in Fig. 16, 
beginning either at the left, as we have done, or at the right. If such order is 
not followed, see ( d ), p. 347 i. 

(1>). The principle of the foregoing process may be explained as 

follows: If a weight, equal to a, represented to any scale by me, Fig. 16, were 
suspended at m by two ropes having the directions m s and m n, the pulls on those 
ropes would (by the parallelogram of forces, Art. 32, p. 323) be represented by 
the lengths of the lines m t and m u ; or, which is the same thing, by u e and m u. 
But u e and m u, Fig. 16, are parallel respectively to O a' and O a. Fig. 17; and 
a' a, Fig. 17 represents the force a by the scale of that figure. Therefore O a' 
and On represent the amounts of the components of the force a in the two 
directions ms and mn. In other words, O a' a, Fig. 17 is a triangle of forces in 
equilibrium, a ' a representing the force a, and O a', O a the pulls in m s and m n 
respectively. (We of course assume the ropes to be without weight.) 

In the above paragraph we have supposed the rope, m n to be made fast at n. If 
now, instead of this, we attach to it. at r?, a third rope, np, and hang, at n , a sec¬ 
ond weight, equal to b ; then, in order that the ropes m s and m n may remain in 











FORCE IN RIGID BODIES. 


347 A 


their former positions, the rope np must be parallel to the line O b, Fig. 17; for 
there are at n three forces in equilibrium; namely, (1st) the weight b, represented 
by b' b , Fig. 17; (2nd) the pull along n m represented by O b' t Fig. 17 (for, in order 
that the rope « to shall remain in equilibrium, the pull upon it at n must be equal 
to that at w), and (3rd) the pull along np. The direction and amount of this 
last is (according to Rem. 2, p. 330) given by the third side, O b, of the triangle, 
O 6' b, Fig. 17. 

Similarly, the line O c in the next triangle, O c' c, Fig. 17, gives the direction 
of the rope p w, and the amount O c of the pull upon it, when a third weight, 
equal to c, is supported at p. 

We thus see that the radial lines in Fig. 17 represent the inclinations of a 
series of ropes, or of links loosely jointed at their ends, s to, to n, np, pw, Fig. 
16, which would support weights equal to a, b and c, suspended at to, n and p 
respectively; and also the amounts of the pulls in those ropes. 

Tbe lines mn, np, Fig. 16, form what is called a funicular polygon, or 
polygonal frame. If O, Fig. 17, were placed upon the other side of X Y, the 
position of the broken line, to np, Fig. 16, would be reversed, n being then upper¬ 
most. Said line would then represent a sort of arch of rigid bars, which, by 
sustaining longitudinal pressure, would support the weights in unstable equi¬ 
librium (see Art. 59 c, p. 348). The figure so formed is called a linear arch. 
The position of the resultant is of course the same in both cases. 

In practice the ropes to s and p w would transmit their pulls to some firm sup¬ 
ports, as walls, at their ends, s and w. But it would be possible (theoretically) 
to support their pulls by means of two rigid struts, si and iw, placed 
respectively in the same lines with the ropes and flexibly jointed at i. If then 
at i we were to apply an upward force equal to the sum of the three downward 
forces, a, b and c, it would evidently maintain the whole system in equilibrium. 
For, if the force c a', Fig. 17 (equal by scale to the anti-resultant of a, b and c, Fig. 16) 
be resolved in the directions O a', O c, of the two struts, s i and i w, the pushes in 
those struts will be represented by the lines Oa' and cO. They will thus 
exactly balance the pulls of the ropes, to s and p iv. 

We thus see that i is a point in the line of the anti-resultant of the three for¬ 
ces, a, b and c. Hence it is also a point in the line of their resultant, R. 

By choosing other positions for the pole, O Fig. 17, we may obtain an indefi¬ 
nite number of different arrangements of the ropes and struts in Fig. 16. For 
instance, if O were opposite to a', the line s h, drawn through to, Fig. 16, would be 
horizontal, and all the other lines in Fig. 16 would, like the corresponding lines 
in Fig. 17, incline upward from left to right; and the point p would be much 
higher than to. 

If we suppose the body to be upheld by two supports, placed one under each 
of the end forces a and c, then a line drawn from O, Fig. 16, parallel to nip, will 
divide the line a’ c into two segments proportioned like the upholding forces. 
See Arts. 57 (e) and (/.) 

Thus, if O be placed at such a height that a horizontal line drawn from it to 
a ' c divides the latter into two segments proportioned like the pressures on the 
two supports, then to and p. Fig. 16, will be in the same horizontal line. 

If O be placed at a great distance from X Y (compared with the length of a' c) 
then the lines of the funicular polygon in Fig. 16 will be nearly horizontal, and 
will represent a system of ropes in which the stresses are very great, as indicated 
by the great length which the radial lines O a', etc., Fig. 17, would then have. 

But the position of the resultant is not affected by the position chosen for 
O provided the drawing is correctly done. But if any of the angles in Fig. 16 
become very acute, it is difficult to find their exact intersections, and we are thus 
more liable to error. 

(c). Figs. 18 and 19 represent a case where there are forces acting in 
opposite directions. In order to avoid crowding the figure of the polygon 
of forces, Fig. 19, we omit the letters (a', b', c\ etc.) at the beginnings of the verti¬ 
cal lines, a' a, b' b, etc., and show only those (a, h, etc.) at their ends. 

The principle and process are here the same as in Figs. 16 and 17; but in draw¬ 
ing the polygon of forces, Fig. 19, care must be taken to lay off each force in its 
proper direction. The arrows shown on the left of the polygon will enable the 
reader to follow more readily the order in which the forces are drawn in this case. 


347 i 


FORCE IN RIGID BODIES. 


, Thus lines a and 6, Fig. 19, (downward) represent the downward forces, a and 
o, -rig. 18; then c (upward), the upward force c; then again d (downward), the 
downward torce d; and, finally, e (upward), the upward force e. Then the line 
thTfiS 7 0uW be TT.\ lred J 0 * 1 **® the polygon, represents the anti-resultant 
the same direction 15 ' dld ^ wAfite 1UG ca ’ Fig ‘ 17 « where a11 tbe forces were in 


H 


s' \ 


a 


d 


r 


! __ 1 

_,_|--- c 


-' r 




Fig.-18 

It will be noticed that in Fig. 18 the resultant, R, owing to the positions and 

theoi» ato ta Fig'S faUS ° UtsWe ° f thC syste,n Eiven yortes . “ wa8 

Aio^iugTXSSt to e oS"S" ,ar mnp - FIes - 1C <». 

The two lines of the funicular polygon, Figs. 16 and 18, which end at the lire 
of dsection of any given force,, must be drawn parallel to those radial lines 
(figs. 17 and 19) winch extend from O to the ends of that segment which repre¬ 
sents said force. Thus, in Fig. 18, the two lines ending in tie line of the force 
a, are tm and nm, parallel respectively to OX and Oaf Fig. 19 which met the 
*!? ds h ? a e X a representing the force a; and the two’1 infs ending in tlfe linl 
c » Fl S- !'\ are np and a p, parallel respectively to 0 6 and O c Fig. 19 which 
meet the ends of vertical line 6 c representing the force, c. > £• i 


(e). Graphic resolution of parallel forces. In the foregoing 
we have shown how to find, graphically, the resultant of a number of given 
parallel forces. We will now give directions for resolving a given force (such as 
the resultant of a number of forces) into two parallel components The process 
is of course merely a reversal of that already given. 

For instance, suppose Fig. 20 to represent a beam hearing a single con¬ 
centrated load, a, elsewhere than at its center; and let it be required to find 
the pressure on each of the two supports, tv and x. 


X 

























FORCE IN RIGID BODIES. 


347 k 


Draw X a, Fig. 21, to represent the load a by scale, and lines X O, a O, to any 
point O not in the line X a. in Fig. 20 draw is and ir, parallel respectively to 
OX and O a. Join r s, and in Fig. 21 draw Of parallel to r s. Then the two 
segments, w a and X w, of X a, give by scale the pressures upon the two abut¬ 
ments, to and x respectively. The greater pressure will of course be upon the 
abutment nearest to the load ; but we may be guided also by remembering that 
the segment X w adjoining the radial line O X in Fig. 21 represents the pressure 
on that abutment (x Fig. 20) which pertains to the line is parallel to OX; and 
vice versa. 

(f). Wlieii there are two or more loads upon the beam, as in 

Fig. 22; proceed as in Art. 57 (a) with the loads, a, b and c, obtaining the funicular 


X 




polygon, m np, Fig. 22. This would give us the position of the resultant of the 
three loads, passing through i; but we are not now concerned to know this. 
Prolong im and ip upward to intersect the lines of the reactions of the abut¬ 
ments, x and w, in s and r respectively. In Fig. 23 draw O w parallel to r s, Fig. 
22. Then X w, Fig. 23, gives the pressure upon x, and w c that upon w. 

(g). If the parallel forces, as a, b and c, Fig. 24, are in different planes, 

first find their projections, a ', b' and c' upon any plane, as x ?/, parallel to them, and 
then their projections, a", b" and c", upon a second plane, x v, parallel to them and 


x 



at right angles to the first. Next apply the process of Arts. 57 (o) and (r) to the 
projections, a' b ' and c', upon one of these planes, x y, and then to those, a , b and 
c" on the other; thus obtaining two resultants, R' and It , one upon each of the 
two planes. Now, as the lines < b\ c>, and e" are the projections of 

the forces, a, b and c, so R', R", are the projections of the actual resultant, 
R of the forces. The position of R is therefore at the intersection of two planes, 
R’lt' and R R",*perpendicular to the planes, xy and xv, and standing upon the 
projections R' and R", of the resultant, R. 
























347 / 


FORCE IN RIGID BODIES. 


CENTER OF PRESSURE. 

Art. 58 (a). Center of pressure. That point through which the direc¬ 
tion of a single anti-resultant force must pass, in order to balance several parallel 
forces acting at different points; or, in other words, that point through which 
the direction of ihe resultant of those forces must pass, is called their center of 
pressure, or of force. 



For instance, let S, Fig. 25, be a common wooden box; but having one side, as 
o o, loosely fitted, so as barely to allow of pushing it backward and forward. Fill 
the box with dry sand, (clean small gravel will be better) and it will be found 
that there is but one single point, ?, at which we can, by holding to it a thin 
rod ri, balance the pressure of the gravel against the opposite side of oo. If 
we apply the rod at any other point,oo will give way before the sand; thus, if 
the rod is held above i, the bottom of oo will be pushed outward; if held 
below i the top of oo will move outward. This point i is distant above the bottom 
of the sand one-third of the depth of the sand; in other words, the center 
of pressure of sand of any depth is, like that of water, at one-third of that depth 
from the bottom. In the case before us, the depth is supposed to be uniform, so 
that the center of pressure is at the same height above the bottom, clear 
across the box. 

Now the balancing force applied through the rod at i, is the anti-resultant of 
all the pressures resulting from the several particles of gravel against the oppo¬ 
site side of oo; and its effect upon the rigid body oo, (omitting of course any 
tendency to bend or break it, which comes under the head of Strength of 
Materials,) is precisely the same as that of all those forces combined; except 
that it is in the opposite direction. Its tendency to push oo bodily, or as an 
entire mass, toward the right hand, is precisely the same as that of the gravel 
to push it to the left hand; or it is the same a* would result were we to heap up 
sand in front of o o, so as to balance the sand behind it. 

Remark. It is this important principle of the center of pressure, that enables 
us to adopt the convenient practice of representing, bv a single line the effect 
of force actually distributed over a considerable surface. Thus in Fig. 52 p 
342 the horizontal force ha, by which each half of the arch mutually prevents 
the other half from falling, is actually, distributed over an area whose depth is 
the depth cj of the keystone; and its breadth, that of the whole bridge as 
measured across the roadway. Yet the arrow h «, when drawn to a scale per¬ 
fectly represents the effect of this distributed force in upholding the half arch 
considered as an entire rigid mass. So far as regards splitting or cracking the 
stone immediately at a, the effects would of course be different; but as the whole 
force is only supposed, for convenience, to be applied at a, this difference is 
merely ideal in this instance. See Arts. 5 and 8, pp. 226, 227. 

















FORCE IN RIGID BODIES. 




(to). Transverse stresses. Caution. It may be well here to direct par¬ 
ticular attention to the fact that the tendency of a number of forces to bend. 
a body, or break it transversely, like a beam, may be very different from that of 
their several resultants, applied at their centers of force. Thus, in Fig. 3, p. 347 a, 
so far as regards either moving the body to the left against the force Q, or causing 
compressive stresses in their own direction , in the body as a whole, it is imma¬ 
terial whether we employ the two parallel forces, a and b, or their single resultant 
R, equal to their sum and acting at their center of force, opposite Q. But it is 
plain that a, b and Q would tend to bend or break the body transversely , while 
JEt and Q would not. 



(c). An absolutely rigid horizontal beam s s, would sustain any amount of 
load l, without bending; and consequently would always press vertically upon 
its upright supports v, u, without any tendency to press them sideways. But 
! an actual beam nr?, it overloaded, will bend; thereby generating, at its ends, 
forces which are not vertical, but which tend to overthrow the supports 11. 












7 


FORCE IN RIGID BODIES. 


CENTER OF GRAVITY, 

Art. 59 (a). In Fig. 1 the upward pull in the string n (as shown by the 
spring balance) represents the anti-resultant of the parallel downward forces 
of gravity acting upon the innumerable separate particles of the body W. 



Fig. 1 

When a body thus acted on by gravity is kept at rest, or balanced, as in the 
figure, then the direction of the resultant or anti-resultant, or of the string in 
the figure, passes through a certain point, called the center of gravity of 
the body. Upon this point, the body, when acted upon by gravity alone, will 
balance itself, in whatever position it may be placed; and if the entire weight 
or gravity of the body could be concentrated into that single point, its tendency 
to move the entire rigid body would remain precisely the same as it actually 
is with the gravity diffused throughout the entire mass. 

(1>). Tn some bodies, such as the cube, or other par¬ 
allelepiped, the sphere, etc., the center of gravity is 
also the center of the weight of the body; but very fre¬ 
quently this is not the case. Thus, in a body abc, 

Fig. 2 , with its center of gravity at c, there is more 
weight on the side a c, than on the side c b. 

(c). If a body W, Fig. 1, suspended freely from any 
point n, is at rest, its center of gravity is directly 3Tig. 2 

under said point. If the body W be pushed a little 

to one side, and then left to itself, it will plainly tend, under the action of 
gravity, to swing back to its first position; and when this is the case, it is said 
to be in stable equilibrium. But if the body, instead of being suspended, be 
balanced on top of a slim rod, and if we then push it a little to one side, it will 
not tend to return, but will fall over; and therefore the equilibrium of a body 
so balanced is said to be unstable. Again, in such cases as that of a grindstone 
supported by its horizontal axis passing through its center of gravity, or of a 
sphere resting upon a horizontal table, if we cause it to revolve a short distance, 
stop it, and then leave it to itself, it will have no tendency either to return, or 
to resume its revolution; and its equilibrium is called indifferent. 


(d). GENERAL. RULES. 

( 

The following general rules ( 1 ) to ( 0 ), form the basis of the special rules, i 
(7) to (39). i 

In speaking of the center of gravity of one or more bodies, we shall assume, 
for simplicity, that they are homogeneous ( i . e., of uniform density throughout) 
and of the same density with each other. The center of gravity is then the 
same as the center of volume , and we may use the volumes of the bodies (as in 
cubic feet, etc.) in the rules, instead of their weights, (as in pounds, etc.) 

In applying these general rules to surfaces, use the areas of the surfaces, 
and in applying them to lines, use the lengths of the lines, in place of the 
weights or volumes of the bodies. 

In all of the rules and figures, pp. 349 to 351 h, G represents the center of 
gravity, except where otherwise stated. 












FORCE IN RIGID BODIES. 


34U 


(1). Any two bodies. Fig. 3. Having found the center of gravity, gr, <7', 
1 each body, by means of the rules given below: then Gt is in the line joining 
and g'; and 

~ weight of q' 

g G — 9 9 X 6 J 


g'G — gg' X 


sum of weights of g and g' 
weight of g 


sum of weights of g and g' 

This rule is based upon the principle explained in Art. 56 (c), p. 347 a. 



(2). Any number of bodies, as a, b and c, Fig. 4, whether their centers 
f gravity are in the same plane or not. , 

First, by means of rule (1) find the center of gravity, gr, of any two of the 
■odies, as a and b. Then the center of gravity, GJ> ot the three bodies, a, b 
nd c, is in the line g g' joining g with the center of gravity, g ot c; anu 


gG = gg'X 
g'G = gg' X 


weight of c 


sum of weights of a , b and c 

sum of weights of a and b 
sum of weights of a, b and e 


,nd so on, if there are other bodies. 

tax Tn manvca^es a single complex body maybe supposed to be divided 
nto'Varte v.Se several cfnters ofVavity can be readily found Then he 
*enter of gravity of the whole may be found by the foregoing and following 
; ules. Thus, in Fig. 5, we may find separately the centers ot gravity ot the 



vo parallelepipeds an 4^K0^h^e”o?idy e and e in ^i^G^iecente^^of gravity 
Imon center of gravity as directed in rules (1) and (2). 




















JQU 


FORCE IN RIGTD BODIES. 


(4). Any hollow body, or body containing one or more openings, Fig. 7. 
Find the common center of gravity, g', of the openings by rule (1) or (2), and 



the center of gravity, g, of the entire figure, as though it had no openings. 
Then G is in the line g g\ extended, and 


sum of volumes of openings 


0 G = (](/ X 


volume of entire body — volumes of openings 

volume of entire body 


<]'&• = gg f X 


volume of entire body — volumes of openings 


Remark. For convenience, we have shown the several centers of gravity, 
U, g,[ upon the surface of the figure. In the real solid (supposed to be of 
uniform thickness) they would of course bo in the middle of its thickness, 
and immediately under the positions shown in the figure. 

(5) . In any line, figure or body, or in any system of 1 ines, figures or bodies, any 
plane passing through the center of gravity is called a “plane of gravity ? » 
for said line, etc., or system of lines, etc. The intersection of two such planes 
of gravity is called a “ line of gravity ” The center of gravity is (1st) the 
intersection of two lines of gravity; (2nd) the intersection of three planes 
of gravity, or (3rd) the intersection of a plane of gravity with a line of gravity 
not lying in said plane. 

® figure or body has an axis or plane of symmetry (i. a line or plane 
dividing it into two equal and similar portions) said axis or plane is a line or 
plane of gravity. II a figure or body has a central point, said point is the 
center of gravitj’. >. 

- 1 ! 1 Fi .?- P-.348, the string represents a line of gravity; and any plane with 
w’hich the string coincides is a plane of gravity. Thus G may often be con¬ 
veniently found, especially in the case of a flat body, by allowing it to hang 
iree.ly a string attached alternately at different corners of it, or by bal- 
ancmg it in two or more positions over a knife-edge, etc., and finding'G in 
either case by the intersection of the lines or planes of gravity thus found. 

(6) . The fjfrapliic method of finding the resultant of parallel forces (Art. 
o7 (a), p. 347 g) may often be advantaeeouslv used for finding tho nmio. 



and in the following instructions WSuppose this to be done. 








FORCE IN RIGID BODIES. 351 

radiaf'lme«^3 


X 



fTS2fi ra i W 1 ing ° f th - e ®® Cond f,3 ? i ? ular polygon is often less simple than that 
i the first, because m the.second the parallel lines through the several centers 
t gravity do not necessarily follow each other in the same order as in the first, 
here be guided by the principle embodied in the rule in Art. 57 (d) 
3 ^ 7? A nam . e - y ’. *' he tw ? hnes (as n'p', »' m') meeting in the parallel line 
is on) pertaining to any given pait, 6, of the figure, must be perpendicular 
espectively t° those radial lines (Oa,G6) which meet the ends of the line 
5, that represents that same part. ’ 

digs. 10 and 11 show the application of the same process to an irregular fig- 
re composed of three rectangles, a, b and c. The lettering is the same as in 
lgs. 8 and 9; but in Fig. 10 it happens that t'and p f of the second funicular 
olygon fall upon the same point. 



f the centers of gravity of the several bodies, or of the several parts of the 
1}% etc., are in more than one plane, we must find their projections upon 
•tain planes, and apply the process to those projections, upon the principle 
.plained in Art. 57 {g ), p. 347 k. 


25 


















351 a 


FORCE IN RIGID BODIES. 


(e). SPECIAL. RULES. 

LINKS. 

17). Straight line. G* is in the line, and at the middle of its length. 

(8). Circular arc/ nob, Figs. 12 and 13 (center of circle at c). Gis in th< 
lino co joining the center of the circle with the middle of the arc, ana 

c g = radius a c X chord a b _ # (For length of arc, see p. 141). 

length of arc nob 



Fig. 13 

(8 a). If the arc is a semi-circle,* 

2 

— radius n c X - 



cG 


7T 


radius a c X 0.6366. 


(8 h). Approximate rules for distance iG, Fig. 12, from chord to center < 
gravity. 


If rise s o 

— 

.01 chord a b: sG — .666 8 o 

Tf rise so — .30 chord 

44 4. 

44 

—- 

.10 

44 

; “ = .665 s o 

44 

44 

“ = .35 

it 

44 44 

44 

—— 

.15 

44 

“ ; “ = .663 S o 

44 

44 

“ = .40 

44 

a u 

44 


.20 

44 

M ; “ = .660 S O 

44 

44 

“ = .45 

44 

44 44 

U 

= 

.25 

44 

“ ; “ = .657 » o 

44 

44 

“ = .50 

44 


44 . 
9 

44 . 
9 

a . 


= .649 s 
“ = .646 s 
1 ‘ = .641 .s 
“ = .637 a 


(9). Triangle, a be, Fig. 14. The center of 
gravity, Gt, of its three sides* is the center of the 
circle inscribed by a triangle, d ef, whose corners 
are in the centers of the sides of the given triangle. 

(1(>). Parallelogram (square, rectangle, 
rhombus or rhomboid;. The center of gravity 
of the four sides* is at the intersection of the 
diagonals. 

(11) . Circle, ellipse, or regular polygon. 

The center of gravity of the outline or circumfer¬ 
ence* is the center of the figure. 

(12) . Regular prism, right or oblique, and riglit regnlar pyramt 
or frustum. The center of gravity of the edges* is the center of the ax 
In the prism, the position of G is not affected by either including or excludi! 
the sides of both of the polygons forming the ends. 



Fi 


SURFACES. 

A. Plane surfaces. 


We now treat of the centers of gravity of plane surfaces, which may 
regarded as infinitely thin flat bodies. The rules for surfaces may be us 
also lor actual Hat bodies, in which, however, the center of gravity is in t 
middle of the thickness, immediately under the points found by the rules. 

(13) Parallelogram (square, rectangle, rhombus or rhomboid), circ 
ellipse or regnlar polygon. CJ is the center of the figure; or the int 
section of any two diameters, or the middle of any diameter. In a Parallel 
grant, G* is the intersection of the two diagonals. 

(14) . Triangle, Fig. 15. G is at the intersection of lines (as a e and i 
drawn from any two angles, a and e, to the centers, eand d , of the sides, 


* We are now treating of lines only; not of the surfaces bounded by th( 
For surfaces, see rules (13), etc., etc. 













3516 


FORCE IN RIGID BODIES. 


and a ^respectively opposite to said angles. Such lines are called “ medial lines.” 


{G== K«e; 



/« 


A b f if being the middle of ac). 

% - w 



Fig. 16 


(14a). Fig 15. Or,on either one of the sides (as a b), meeting at anvane-le a 
?««» = % «&. . Draw op parallel to the other side, a c. ThenoG = S 
yp, and G is at the intersection ot op with any medial line, as a e, etc. /2 

If « 0 ', b b', cc' and GG'are the distances of the three cor- 
lers and of G from any straight lino or plane a' c'; then 

G G' = A (« «' + b b' + c c'). 

This gives us the position of the line of gravity G G". In the same wav we 
ind the distance G G" of G from any second line or plane, b" c" This gives 

r G S G' a e nd°GG ? ' n ° f & SeC ° nd line ° f gravityGG ' G i8 at the intersect?^ of 

(14c). Fig. 17. The distance Gn of G iu any direction from any side as a r 
intended it necessary) is = % the distance n' b measured in a parallel’direc- 
9r lon the same side to the opposite angle, b. 



It follows from this that the shortest distance, Go, of G from any side (as 
«)« = A the shortest distance, o' 6 , from the same side to its opposite angle b 
It follows also that p G — %p b, as in Rule (14). 

,0(15). Trapezium or trapezoid, Fig. 18. For trapezoids, see also Rule 
xi ,J )- Draw the two diagonals, ac and bd. Divide either of them, as a c into 
ro equal parts, a m and c m. From b, on b d, lay off b n = d s (or from d lay off 
n =>= o s). Join m n. G is in m n, and 
N wG = % m n. 

* is the center of gravity of the triangle a c n). 



f (15a). Or, Fig. 19, find first the centers of gravity, m and n, of the two tri- 
, gles, c bd and abd, into w r hich the trapezium is divided by one of its diago- 
' .Is, bd. Join mn. Then find the centers of gravity, o and p , of the two 
angles, dac and bac, into which the trapezium is divided by its other 
agonal, a c. Join op. Then G is the intersection of m n and op. 







351c 


FORCE IN RIGID BODIES. 


(16). Trapezoid only, Fig. 20. G is in the line ef joining the centers, 
c and/, of the two parallel sides, a 6 and cd. To find its position in said line, 
prolong either parallel side, as a 6, in either direction, say toward i; and make 
6 i equal to the opposite side, cd. Then prolong said opposite side, cd, in the 
opposite direction, making dh = a b. Join hi. Then G is the intersection of 
hi and ef. Or 


fG = 


ef 


w 2 ab + cd 
X ab + cd 5 


or 


o G = 


en 


X 


2 ab -f cd 
ab + cd 



C 11 o f 
Fig. SO 




(17). Regular polygon. G is the center of the figure. 

(17 a). Irregular polygon. If the polygon be divided into any two 
portions, as by any diagonal, G must be in the line (of gravity) joining the 
centers of gravity of those two portions. If we again divide the whole polygon 
into two other parts by another diagonal, and join the centers of gravity ol 
those two parts, G is the intersection of the two lines of gravity. 

(176). Or we may divide the polygon into triangles, find the center ol 
gravity of each triangle, by Rules (14), etc., and then find G bv general Rule 
(1), (2) or (6). 




(18). Circular sector, aobc, Fig. 21. (Center of circle at c). 

^ _ 2 chord ab radius 2 X chord 

c ' radius ci c X 


3 arc a o b 

For length of arc, see p. 141. 

(18a). If the sector is a sextant, 

2 

= radius X 0.6366. 


3 X area 


cG 


radius X 


7r 


(186). If the sector is a quadrant, Fig. 22, 


cG 


cx 


4 vx v/ 2 

— radius X - 

3 7T 


radius X 0.6002. 


4 1 

x G = — radius X - 

3 7T 


(18 c). If the sector is a semi-circle, 
c G 


4 1 

— radius X - 

3 7 r 


radius X 0.4244 
14 


14 

(approximately) radius X • 

00 






















FORCE IN RIGID BODIES. 351 d 

(19). Circular segment, nobs, Fig. 23. (Center of circle at c). 

„ n __ cu be of chord a b ,^ 

~ • (For areaoi se g me nt,see p. 146). 

(19 a). If the segment is a semi-circle, 


cG 


4 l 

~ radius X - 

a 7T 


radius X 0.4244 


= (approximately) radius X 

33 



Fig. S3 



(30). Cycloid, Fig. 24. (Vertex at v). 

5 

»G «= — v d. 

O 

(21). Parabola, a b c, Fig. 25. ac 
is the base; ax and cx, ordinates; 
and the height or axis, bx , an ab¬ 
scissa. Center of gravity at G, in the 
axis x b, and 

2 

x G -= — x b. 
o 


(31 a). Semi-parabola, a b x or 

cbx. Center of gravity at G', and 


I 


x G 


= T xb: 



(23). Ellipse, mnop, Fig 2G. The center of gravity, r, of the whole 
dlipse is at the center of the figure. & ’ 


n 



m 


G is the center of gravity of the quarter ellipse, one. 

g' “ “ “ “ » Koif *r , nop$ 

“ mno. 


G 

G" “ 


half 

U U 


e g / 

C G" 


4 1 

T oe *T 


o c. 


G'G 


~ C n X 


0.4244 o c = (approximately) 

33 

14 

= 0.4244 cn» (approximately) —— 

33 


en 























351 e 


FORCE IN RIGID BODIES. 


(23). Any plane figure. Draw the figure to scale on stout card-board. 

Cut it out and balance it in two or more positions over the edge of a table or 
on a knife-edge; and mark on it the several positions of the supporting edge. 
Where these intersect is the center of gravity. Considerable care is of course 
necessary to obtain very close results by this method. Before balancing the 
card, its upper edges should be marked off into small equal spaces. Otherwise 
it will be difficult to locate the positions of the supporting edge. The paper 
on which the figure is prepared must of course be so stiff that the figure will 
not bend when balanced on the knife-edge. See Rule (5), p. 350. 


B. SURFACES OF SOLIDS.* 

(24) . Curved surface* of sphere or spheroid (ellipsoid). G is the center 
of the figure. 

(25) . Curved surface* of any spherical zone, as a spherical segment, 
hemisphere . etc., Figs. 27. G is the center of the axis or height, a o.f 

In the hemisphere, o G =34 radius.f 


a a a 



(26) . Right or oblique prism, whose ends are either regular figures (p. 110) 
or parallelograms (this includes the cube and other parallelopipeds); and 
right or oblique cylinder (circular or elliptic). Surface* (either including 
both or excluding both of the two parallel ends). G is the center of the axis, 
or line joining the centers of the two parallel ends. 

(27) . Curved surface *f of rightcone, Fig. 28 (circular or elliptic), or slanting i 
surfaces*! of right regular pyramid, Fig. 29. G is in the axis o a (the line; 
joining the apex and the center of the base); and 

o G — % o a. 

In an oblique cone or pyramid, the perpendicular distance of G* from the 
base is one-third of the perpendicular height, as in the right cone and pyramid; 
but does not lie in the axis. 



o G 


— OCL X — 


circumfe rence of o -f 2 circum ference of a. 
circumference of o -f circumference of a. 


* We treat now of the surfaces of solids, not of their contents or volumes o 
weights. For these, see Rules (29), etc., pp. 351/, etc. 

t If the top or base is to be included, see Rules (1) and (2), p. 349 . 





































FORCE IN RIGID BODIES. 


351/ 

In the conic frustum, Fig. 30, we may use the radii of the two ends; and 
in the frustum of a regular pyramid, Fig. 3L, any fide of each end (as 
b c and d e) instead of the circumferences. * 




Fig- 30 


In the following rules for center of gravity of solids, the solid is supposed 
to be homogeneous; i. e., of uniform density throughout; so that the center of 
gravity is the center of magnitude or of volume. 

(39). Sphere and spheroid (ellipsoid). G is the center of the body. 

(30). Hemisphere, Fig. 32. (Center of sphere at c). Height eT = radius 
G is in the axis, c T, and 


c b. 


cG 


= I tT 


(31). Spherical sector, Fig. 33. 


cG 


«= — radius cb. 

8 

(Center of sphere at c). 

V 


— (radius cb — 
4 


2 



(33). Spherical segment, a mb T, Fig. 34. Center of sphere at c. Center 
of base at m. Rise or height of segment = m T — h. G is in the axis m T; and 

3 (2 radius c b of sphere — height h)- 

4 ^ 3 radius cb of sphere — heights 



m G 


height, h 2 (radius mb of base) 2 + (height, h ) 2 

^ X 


height, h 


X 


3 (radius m b of base) 2 + (height, h ) 2 

4 X radius c b of sphere — height, h 
3 X radius cb of sphere — height, h 



Fig. 35 

(33). Spherical zone, Fig. 35. 

o t 2 (radius o b of base) 2 + 4 (radius t c of top) 2 + (height o t) 2 


0 G 2 ' X 3 (radius o b of base) 2 + 3 (radius t c of top) 2 + (height o t) 2 

(34). Prism, regular or irregular, right or oblique (including-the cube 
and other parallelopipeds), and cylinder, circular or elliptic, etc , regular 
for irregular, right or oblique. G is the center of the axis joining the centers 
of gravity of the two ends. 



































FORCE IN RIGID BODIES. 


351 g 

(3 4a). A very short cylinder or prism; as a flat body, such as an iror 
plate, etc. Find the center of gravity of its surface. (Rules 13 to 23). Tht 
center of gravity of the body is in the middle of its thickness, immediately 
under the point so found. 

(35). Ungula of a cylinder, circular, or elliptic (provided one of the axes 
of the ellipse'coincides with the oblique cutting plane); right or oblique 
Figs. 3G and 37. 



Let O T be the axis (joining the centers of gravity of the ends), and X N t 
line drawn parallel to the axis, in the plane, A B C D, passing through th< 
axis and through the uppermost and lowermost points C and D of the obliqu< 
cutting plane. Then the position of G in the plane A B C D, is found thus: 


OX 

XG 


O B 


X 


XN 


2h + a ’ 

~l(2h+» + i - V ). 

4 ' 2h+»/ 


(35 a). Figs 38 and 39. If the oblique plane C D meets the base, A B, at A, s< 
that h = 0, while C D remains a complete ellipse or circle, this becomes 




OX 


O B 


X G = 


XN 




(36). Cone, Figs. 40 and 41, circular, 
elliptic, etc., right or oblique; or pyra¬ 
mid, regular or irregular, right or 
oblique. The center of gravity G is in 
the axis O T, drawn from the apex, or 
top, T, to the center of gravity O of the 
base; and 

O T 

OG . 




Fig. 4A. 


(37). Frustum of a cone, Figs. 42 and 43, circular or elliptic, right o 
twoS a:B and P 2 D “fpara^r " ,rregU ‘ ar ' r ‘ Eh ' ° r ° bIi< ’ Ue • «« 

let h b, 






















































FORCE IN RIGID BODIES. 


351 A 




G is in the axis O Z, which joins the centers of gravity O and Z of the two 
ends; and its distance from the base, A B, measured along the axis, is 


O G = 


h_ 

4 


X 


A -f- 2V^Aa 4- 3a 


A + 


/A 


+ 


(37 a). In a frustum of a circular cone, right or oblique, with parallel 
ends, this becomes 

R2 + 2Rr + 3 r 2 


O G 


= ~ X 

4 


r 2 


R2 + R r + 

where R and r are the radii of the large and small ends of the frustum 
respectively. 

(38). Figs. 44 and 45. Frustum, A B C D, of a cone, circular, elliptic, 
etc., right or oblique; or of a pyramid, regular or irregular, right or oblique; 
whether the ends are parallel or not. By rule (3<>) find the center of gravity 
N of the entire pyramid (or cone, as the case mav be) A B T, of which the 
frustum forms the lower part; and the center of gravity S of the smaller 
pyramid or cone DCT(= entire pyramid or cone, minus‘the frustum). Also 
Und the volume of each: thus, 




Volume of pyramid or cone = area of base X perpendicular height # 

>nd 

Volume of volume of volume of 

the frustum = entire pyramid — smaller . 

ABCD or cone, A B T one, D C T 


Then the center of gravity G of the frustum ABCD is in the extension of 
he line S N; and 

volume of smaller pyramid or cone, D C T 
NG = SN X — 


volume of frustum, ABCD 


(39). Paraboloid. G is in the axis, and at one-third of its length from 
' he base. 
























352 


FORCE IN RIGID BODIES. 


Art. 60. Tlie Inclined Plane is a rigid straight plane surface, as ab. 
Fig. 63, not horizontal. If a vertical line b c be drawn trom the top b ol the 
plane, to meet a horizontal line a c, drawn from its bottom a, then b c is galled 
the height of the plane; a c its base; and a b its length The angle ba c, which 
the plane forms with the horizontal linear, is called its inclination, slope, or 
steepness; which, however, is frequently expressed also by the proportion 
which the base bears to the height; thus, if the length a c of the base be 1 , 
\y 0 , 2 , &c., times that of the height b c, the inclination or slope is said to be 
1 to 1 l l A to 1, 2 to 1 , &c. The angle bac is the angle of inclination of the plane. 



When one rigid body as N or M, Fig. 63, is placed loosely upon another, 
as upon the rigid plane a b, the effect produced bv its weight is the same as it 
all that weight were concentrated at its center of gravity g, and acted in the 
direction of a vertical line g v drawn through said center. When we assume 
the weight to be thus concentrated at the point g, we must remember that all 
other parts of the body must be considered to be without weight; although still 
retaining their inherent cohesive force, or strength. 

If as in N this vertical line g v, which now represents the direction of the 
entire weight of the body, passes beyond, or outside of the base, the body must 
fall. 

But if, as in the body M, the line g v falls within the base r s, the body will 
not upset; but we shall have a force g v equal to the weight of the body, and 
applied obliquely to a rigid surface a 6 , at the point v; and consequently 
(Art. 25 b, p. 318 f), resolvable into two components; namely, iv, perpendicular to j 
the surface a b, and therefore imparted to it as a pressure; and x v, parallel to* 
the surface, and consequently not imparted to it. All these lines may be drawn 
by scale, to represent their respective forces. When we consider a single force 
as 7 v to be thus resolved into two components, with a view to ascertaining their 
effects, it is plain that said single force must then be considered as no longer 
existing; but as being replaced by its components. Now the component/ore' 
x v being parallel to the plane, it follows (Art. 25, p. 318 e), that the pressure o. 
strain i v, cannot oppose the cross action of the sliding force x v, and x v must 
therefore produce motion in the body, causing it to slide down the plane, 
were it not for the friction, the amount of which depends upon that of the; 
pressure; as also upon the degree of smoothness of the surfaces in 
contact, and upon whether they are lubricated or not. If the friction 
is equal to the force x v in the opposite direction, the body of course can 
not move; but if less, it will move, under the action of a force equal tc 
the excess of x v over the friction. It must be remembered that the, 
pressure component i v, which produces friction on an inclined plane« 
is not equal to the weight of the body, but is less than it. It is equa^ 
only when the surface is horizontal, so that the vertical force g v, represent., 
ing the entire weight of the body, is at right angles to the joint, and when > 
consequently, it all acts as pressure. Therefore, the steeper the plant 
becomes, the less is the friction ; because then less of the weight of the bodj 
acts as pressure, and more of it as moving force. Hence, a locomotive has, 
less adhesion on an inclined grade, than on a level; for the so-called adhesior 
is in reality nothing but friction. But although both the perpendicular pres 
sure and the friction become less in amount as the plane becomes steeper 
yet they retain nearly the same proportion to each other. 










FORCE IN RIGID BODIES. 


353 



Friction : Perpendicular pressure :: Height: Base; 
Base : Height :: Perpendicular Pressure : Friction. 


herefore, when a body barely begins to slide, measure a c horizontally, and 
e vertically; divide the last by the first, and the quotient will be the pro- 
jrtion which the friction of the bodies experimented upon, bears to the 



Pig 6 3 £ 


essure which causes it. Or, measure the angle b a c in degrees, Ac.; the 
ngent of this angle will be that same proportion. This proportion is called 
e coefficient of friction for those bodies; a table of which will be found on 
.ge 373. A horizontal line d g, drawn from g, and terminating in v i extended, 
II, when measured by the same scale as g v, i r, x v , give a horizontal force 
lich, without the aid of friction, would prevent the body from moving down 
j e plane. Or, if the length a b of the plane be taken by scale to represent the 
J ught of a body, then b l , perpendicular to a b, to meet a c produced at 1, will 
ve that same horizontal force. 

Art. 6/4. If the length m n , Fig. 03%, of an inclined plane, be taken by a 
ale, to represent the weight in lbs., tons, Ac., of any body placed upon it; 
en the base o n will, by the same scale, give the perpendicular pressure in 
?., tons, Ac., which the body imparts to the surface of the plane; and the 
ight mo will give the amount of force parallel to m v, and which tends to 
love the body down the plane, either by sliding or rolling. If the pressure 
in be multiplied by the proper coefficient of friction, the product will 
linly be the actual amount of frictional resistance in lbs., Ac., which the 
1 rface m n is capable of offering. If the friction thus obtained proves to be 
i eater than or equal to the sliding force m o, then the body will remain at 
I st on the plane; but if less, then sliding or rolling down the plane will be 
I e result; and the amount of force which starts or begins the motion, will be 
ual to the excess of m o over the friction. As the motion continues, this 
; cess, if it continues to act, will accelerate the velocity. When a body is 
j iced upon an inclined plane, rn n Fig. G3%, whether it slides or not, the 
Sliding force (parallel to. surface of plane) == weight X sine of m n o. 
Pressure (perpendicular “ “ “ ) = weight X cosine of m n o. 

Maximum friction (parallel “ “ “ )== weight X “ “ “ X co- 

icient of friction. For the friction does not vary as the angle of slope of 
s plane, but as the cosine of that angle; or, in the same manner as the per- 
ndicular pressure varies. 

lx. 1. Suppose we wish to slide a wooden box M, Pig. 63, filled with stone, and weighing in all 








354 


FORCE IN RIGID BODIES. 


1200 lbs, up the iron rails of an inclined plane, sloping 5°; what force must we use, parallel to tl 
plane; assuming the coeff of wood on iron to be .4, or A- of the perp pres ? Here we have to ove 
come the parallel force x v, and the fric. Now, as just stated, this parallel force at v is equ 
to, wt X nat sine of slope, = 1200 X .087 = 104.1 lbs. The fric is equal to, wt X nat cos of slope 
coeff of fric; = 1200 X .006 X .4 = 478. Consequently, 104.4 478 = 582.4 fts, is the force reqd. I 

fact, however, this force merely balances the downward tendency of the box, together with its frii 
thus rendering them incapable of resisting any additional upward force; but it is plain that we mu 
apply some additional force, in order to impart motion to the now uuresistiug box. 

Now, suppose we wish to slide the box down the plane, what force must we use? Here nothing r t 
sists us but the fric, just found to be 478 fts. The parallel force helps us to the amount of 104.4 ft, ‘ 
therefore we need only to add 478 — 104.4 = 873.6 fts. 

The box will then be upon the point of moving. Any additional force will move it. 

For acceleration on inclined planes see p 363. 

The following table will facilitate calculations respecting the draft required on grades, incllD 

planes, &c. In practice, allowance for friction must be mad 
in the last 2 cols. Origins 


Inclination or Slope of the Plane. 

..vertical height 

The sloping length is =-r-——* 

sine, col 6. 

Pres, on 
Plane, in 
parts of the 
wt. Or, nat. 
cos. of angle 
of Plane. 

Pres, on 
Plane, in fts 
per ton. 

Tendency 
down the 
Plane, in 
parts of the 
wt. Or, nat. 
sine of angle 
of Plane. 

Tendency 
down the 
Plane, in fti 
per ton. 

Vert. 

Hor. 

Ft. per mile. 

Deg. 

Min. 





I 

in 

3. 

1760.00 

18 

26 

.9487 

2125 

.3162 

708. 

1 

in 

4. 

1320.00 

14 

2 

.9702 

2173 

.2425 

543. 

1 

in 

5. 

1056.00 

11 

19 

.9806 

2196 

.1962 

439. 

1 

in 

6. 

880.00 

9 

28 

.9864 

2210 

.1645 

368. 

1 

in 

8. 

660.00 

7 

8 

.9923 

2223 

.1242 

278. 

1 

in 

9. 

588.66 

6 

20 

.9939 

2226 

.1103 

247. 

1 

in 

10. 

528.00 

5 

43 

.9950 

2229 

.0996 

223. 

1 

in 

11.4 

461.94 

5 

00 

.9962 

2231 

.0872 

195. 

1 

in 

12. 

440.00 

4 

46 

.9965 

2232 

.0831 

186. 

1 

in 

14.3 

369.23 

4 

00 

.9976 

2232 

.0698 

156. 

1 

in 

15. 

352.00 

3 

49 

.9978 

2233 

.0666 

149. 

1 

in 

19.1 

276.73 

3 

00 

.9986 

2237 

.0523 

117. 

1 

in 

20. 

264.00 

2 

52 

.9987 

<4 

.0500 

112. 

1 

in 

23.1 

229.04 

2 

30 

.9990 

44 

.0436 

97.7 

1 

in 

25. 

211.20 

2 

17 

.9992 

2238 

.0398 

89.2 

1 

in 

28.6 

184.36 

2 

00 

.9994 

4* 

.0349 

78.2 

1 

in 

30. 

176.00 

1 

03 

** 

44 

.0334 

74.8 

1 

in 

32.7 

161.47 

1 

45 

.9995 

2239 

.0305 

68.4 

1 

in 

35. 

150.86 

1 

38 

.9996 

44 

.0285 

63.8 

1 

in 

38.2 

138.22 

1 

30 

.9997 

2240 

.0262 

58.6 

1 

in 

40. 

132.00 

1 

26 

14 

“ 

.0250 

56.0 

1 

in 

45.8 

115.29 

1 

15 

44 

44 

.0218 

48 8 

1 

in 

50. 

105.60 

1 

9 

.9998 

44 

.0201 

45.0 

1 

in 

57.3 

92.16 

1 

0 

4 4 

44 

.0175 

39.1 

1 

in 

60. 

88.00 

0 

57 hi 

.9999 

44 

.0167 

37.4 

1 

in 

70. 

75.43 

0 

49 

4 4 

44 

.0143 

32.0 

1 

in 

76.4 

69.12 

0 

45 

44 

44 

.0131 

29.3 

1 

in 

80. 

66.00 

0 

43 

44 

44 

.0125 

28.0 

1 

in 

90. 

58.67 

0 

38 

44 

44 

.0111 

24.9 

l 

iu 

100. 

52.80 

0 

34 

1.0000* 

44 

.0100 

22.4 

1 

in 

114.6 

46.07 

0 

30 

44 

44 

.0087 

19.6 

1 

in 

125. 

42.24 

0 


<4 

44 

.0080 

17.9 

1 

in 

150. 

35.20 

0 

23 

44 

44 

.0067 

15.0 

1 

in 

175. 

30.17 

0 

19^ 

44 

44 

.0057 

12.8 

1 

in 

200. 

26.40 

0 

17 

44 

It 

.0050 

11.2 

1 

in 

229.2 

23.01 

0 

15 

<4 

4* 

.0044 

9.77 

1 

iu 

250. 

21.12 

0 

14 

44 

14 

.0041 

9.18 

1 

in 

300. 

17.60 

0 

11^ 

» 

44 

.0033 

7.H9 

1 

in 

343.9 

15.35 

0 

10 

44 

11 

.0029 

6.52 

1 

in 

400. 

13.20 

0 

8% 

•• 

4 4 

.0025 

5.60 

1 

in 

500. 

10.56 

0 

7 

44 

14 

.0020 

4 48 

1 

in 

600. 

8 80 

0 

6 

44 

1 4 

0017 

3.81 

1 

in 

800. 

6.60 

0 

*% 

44 

44 

.0013 

2.91 

1 

in 

1000. 

5.28 

0 

3?i 

44 

l< 

.0010 

2.24 1 

1 

in 

3437. 

1.54 

0 

1 

4 1 

44 

.0003 

0.65 


Level. 

0.00 

0 

0 

»4 

« 1 

.0000 

0.00 


For other tables of grades, see pp 176, 723, 724, 725. 

Art. 63. The following principle is one of great practical importance. Wh 


* Near enough for practice; actually .99995, or less by one pa.t in 20000, or about 1 ft in 9 tons. 





























FORCE IN RIGID BODIES. 


355 


lM i* P /* ne T ?’ Fig 1 64, h f 3that inclination, yxw, at which the fric of any given 
w body is balanced; and sliding is about to commence, from ® 

“ tne action of any force h o, applied to the plane, through 
i , ) the body, in any direction ho, not perp to it; if from the 
q point o of application, we draw a line op, at right angles to 
the surf of the plane, then the angle hop will always be 
equal to the angle of fric yxw of the body. If the plane 
«is so steep that sliding must take place, then the angle 
formed between the force h o, and a perp op to the plane, be¬ 
comes greater than the angle of fric; but if the steepness is 
so slight that the body rests firmly on the plane, then said 
angle is less than the angle of fric. 


j The practical applications of this principle are very numerous; they 
d extend to pressures in any directions whatever; and apply to plane 
a surfs in any position whatever, whether inclined, vert, or hor; for any 
given pres produces precisely the same amount of fric, whether we im¬ 
part it to the ceiling, the floor, or the wall of a room ; provided they all 
be of the same material. The angle of fric of cut stone 
■ upon cut stone is about 32°; that is, one block of cut stone 
will not slide upon another at a less slope than about 32°; 

11 the fric then being full ^ of the pres. Therefore, if the 
floor /, Fig 65, ceiling c, and walls w, w, of a room be of 
3ut stone; and p,p,p, p, lines at right angles to them ; we 
- may press a piece s of cut stone against them with any 
[force whatever, applied in the direction of the stone itself, 
without danger of its sliding; provided only that the di¬ 
rection of the pres along s does not form with the perp p 
in angle exceeding 32°. But sliding will take place, 
whether the pres be great or small, if, as at o, o, o, o, said 
ingle exceeds 32°. The angle of fric is, by some writers, 
called, in such cases, tEie limiting’ angle of 
resistance. 

Rem. 



Fitf 64- 



*^\\\\\\^\^ 


Fitj 65 


The friction at the feet of 
rafters when highly inclined diminishes very 
nuch their horizontal pressure and tendency to 
split off the ends of the tie-beams. 

The angle of fric of oak endwise against hard limestone, is, according to 
forin, 20%°; therefore, if the walls, &c, of a room consisted of such lime- 
tone, we could not press a piece of oak endwise against it without sliding, 
f the angle withp exceeded 20%°; and the legs of a wooden trestle, Fig 66,’ 
tould not spread, on the level surf of such limestone, under any wt w, if 
he angle a b c be less than 20%°; but certainly would if it be greater, unless 
tner preventives besides fric at the feet be depended on. In this case the 
ric amounts to very nearly of the pres; that being the proportion cor- 
(espondiug to 20%°. These two illustrations show how wide is the applica- 
jion of this principle: for the announcement of which we are (the writer 
elieves) indebted to Moseley. 



Eio66 


Art. 64. To find the effect of an extraneous force (fg, Fig- 67.1 
in parted in any direction, to a rigid body (R) on an inclined 
»lane. ip% when we know the angle of fric, and the wt of the body. The prin- 
iple laid down in the preceding Art 
nab]e8 us to do this. : 

Through the cen of grav c, of the 
ody, draw avert line a tv ; and extend 
he direction f g of the force, to meet 
his line, as at o. Make o a by scale, to 
epresent the wt of the body; and o z 
o represent the amount of the force 
g. Then is o a point at which we 
day assume both these forces to be im- 
arted to the body. (Art 29.) Complete 
he parallelogram of forces a x z o, by 
rawing a x, and z x, parallel and 
qual to o z, and o a. Draw the diag 
o, and extend it to meet the plane, as 
1 1. Make the line t v perp to the surf of the plane. This done, we have a single 
jrce x o, equal in effect upon the rigid body, to its wt, and fg combined. 

This single force may be considered as imparted to the body at any point that lies in its 

ne of direction x t ; therefore, we will assume it to be Imported at t, where it encounters the force 
f fric acting in the direction e e , of the joint formed between the body, and the plane. Now, if . 
trikes within the base s e, t v being at right angles to this joint, it follows from the last Art, that 
’ the angle xtv is less than the angle of fric corresponding to the nature of the materials which 



Fid 67 








































356 


FORCE RIGID BODIES, 




•ompo.se the body and plane, then the body will remain at rest on the plane. But if the angle * t 

e greater than said angle of fric, the body will slide up or down the plane, (according to circum 
stances, stated in the next paragraph;) if the angles be equal, the body will be just on the point < 
beginning to slide either up or down. 

11 ben x t v is on the down hill side of v t, as in the fig, the tendency of the body will ev 

demly be to move up the plane; but if, in consequence of a diff direction of the force/ a, (and cons< 
queutly of the resultant x o ,) the angle x t v is on the up hill side of v t, then the tendency will 1 
down the plane. 

Rem. 1 If the direction of the resultaut x o, or the point t, falls outside of its base * e, the bodt 
instead of sliding, will upset. It will fall up hill, if t strikes p i up hill from the base; and down hi! 
if t sinkes down hill from the base. See Art 65. 

Rem. 2. In order to draw the parallelogram of forces a x z o, and its resultant diag x o, the line a < 
which represents the wt, may sometimes have to be regarded as pulling instead of pushing dowi 
ward at the point o, where the other force meets it. See Fig 9%, and Fig 69, below. 



Rem. 3. It follows from the foregoing, that when at the joints p q, ri 
Fig 68. of a mass of masonry; or at the joints of timbers in carpentry 
iron work. &c. the fric alone is depended on to prevent sliding, the rt 
sultant asac, co, on, <fec, of all the forces acting at anv joint, must m 
form an angle m c i. c o a, o n e, with a perp c i, o a, n e, to the joint 
greater than the angle of Trie corresponding to the nature of the ms 
terials whose surfaces constitute the joint. 

Rem. 4. The extraneous force reqd to move a body up a plane, wit 
be the least when its direction, i ti. Fig 67. makes with surf ip, of th 
plane, an angle, nip, equal to the angle of fric. 


Art. (»•>. To find the force required to prevent i 

body S, Fig 69, from moving; when the direction, ow, of its wt 
strikes outside of its base tt'. Thus, suppose we wish to impar 
a pulling force at e, and in the direction en, to prevent the bod' 
from sliding either up or down the frictionless plane ip, am 
from upsetting down the plane. Through the ceil of grav c, drav 
a vert line xw ; and continue the line of direction of a e to mee 
it at o. From o draw oy at right angles to the plane ip. Bv seal 
make ow equal to the wt or the body : and from w draw wy paral 
lei to o a. Make e a equal to wy; then is e a the reqd force, whicl 
will resist all tendency of the body to move. For in the par 
allelogram of forces o wy z, we have the force o w tending to mak 
the body fall; and the force o z (equal to e a) tending to preven 
it from falling; and the resultaut o y, of these two forces, equal t 
their joint effect, is at right angles to the surf of the plane; an, 
is consequently all imparted to it as pres; no part bein 

left unresisted, to produce motion in any direction. For as befor 
said, when two forces, as o w, o z, are compounded into one result 
ant force o y, those two forces must be considered as no longer ex 


. ttUI ' w y* vuus »° lorces must oe cons me red as no longer ex 

lsting ; thug, in this case, so long as we regard the joint effect of o w and o z as being concentrate 
in their resultant o y, we cannot of course, consider them as acting in other directions at the sam 
time ; so that there is, as it were, no longer any wt. o w, tending to make the bodv fall; nor anv fore 
e a, tending to uphold it; but only the single force o y , which presses the inert body against and t 
right angles to, the surf ip; imparting to it a tendency to move only in the direction o y ; which tei 

deucy is reacted against by the inherent 
cohesive force, or strength, of the plane. 

If the body is prevented, by friction or 
by a stop at its lower toe t, from sliding 
down the plane, and we wish to know 
the least force in the direction oz which 
will just prevent the body from over¬ 
turning about t ; the line oy, instead or 
being drawn perpendicular to the plane 
ip as in Fig 69, must be drawn from o 
through the lower toe t as in Fig 69 a. 

The lines ow and zy of the parallel¬ 
ogram ozyw, representing the weight 
of the body, will of course remain the 
same as for the same body in Fig 69, but 
the lines o z and w y, representing the 
extraneous force, although the same in 




^ 

p 

\ ° 


X 


\ c 

\ 

X 




Xt 


/ 

3 

y 


JX 

IV 




Fig. 


j.69 a 

* VAM«*MWMP IVIVt| 01 

direction as before, will evidently be much shorter. 



Erf.70 

o 


Art. 60. Stability. The stability of a structure, or of anv body, is, strictl * 
speaking, that resistance which its wt alone enables it to oppose against forces tenc ' 
ing to change its position. Such resistance may be assisted by extraneous wts, c 11 
by other forces properly applied; but such must be distinguished from the stabiiit 
inherent in the structure, or body itself. To insure the stability of a structure, th 1 
disposition of its parts, as well as that of the entire mass, must be such that neithf * 
of them shall move, either by sliding , or by overturning , under the action of the in 
parted forces. Stability is therefore a branch of Statics; or of forces at rest or i 
equilibrium with each other; Art 16. 






















FORCE IN RIGID BODIES. 


357 


Stability must not be confotimleri with strength. A structure 

stron S- a ?. d very unstable. A block of stone is quite as strong while sliding down 

onlv in ?he i» n < e ,’r 0r a 8 d °" U a , 8t , eep ban *’ as when restin « on a firm hor bas e : but it has stability 
last case - f Pyramid of weak chalk may have great stability : while a globe of granite 
Of haS Ve P htt e ‘ W . e generally have to examine into the strength, as well as the stability 

oL -h str ' lcturcs: , bu * u “ ust be done by diff processes. The stability has reference to the structure 
U A 'P cons l SU! ‘g of one or more rigid bodies, which may be moved as entire masses, but not 

broken, or changed inform , by the applied forces. See Remark 2, Art 29. 

Those forces which tend to impair the stability of a structure, are called acting ones; and those 
which tend to maintain it resisting ones. This distinction is merely a matter of convenience; for 
aji tne torces act, and resist. 

The forces which affect the stability of a rigid structure considered as one mass, are its wt; extra¬ 
neous wts, or strains, and the foundation, or support; which last reacts as an antiresultant 
against the others. W hen these three balance each other, the structure is stable. When the struc¬ 
ture is to be considered as composed of several rigid bodies, then the joints or surfaces of contact be¬ 
tween these bodies must also be regarded as so many secondary foundations, and these also must re¬ 
spectively balance the forces acting upon them; otherwise these parts may slide, or overturn, while 
Other parts may remain firm. 

Art. 6 7. In order to guard against accidents, a structure must generallv be so designed as to 
be capable of resisting much greater forces than those which it sustains under ordinary circum- 
stances. The proportion which, with this object, we give to the resisting forces, in excess of the act- 
mg ones, is called the coefficient of stability ; or simply the stability, or the safety, of the structure, 
rnus, if we make it capable of resisting 2, 3, or 6 times the amount of the ordinary* acting forces, we 
say it has a stability, or a coeff of stability, or a safety, of 2, 3. or 6. 


. Art. 68. Since the stability of a structure, considered apart from its founda¬ 
tion, consists entirely in the resistance which its several parts, as well as the entire 
mass, can present against both sliding and overturning, it follows that two precau- 
" tions, already adverted to in previous articles, must be resorted to. Namely, 1st, 
it against, sliding, take care that the resultant of all the forces acting upon 
! any joint, (including that between the base and the foundation,) shall act either at 
’ right angles to said joint; so as to be entirely imparted to it as strain, (press or pull,) 
leaving no part unresisted to tend to produce motion ; or else that it shall not de¬ 
viate from a right angle, to a greater extent than the angle of fric corresponding to 
the materials which compose the joint; so that the portion of it which is not im- 
J parted at right angles, shall be resisted by friction; and thus be prevented from 
\ producing a* motion of sliding. 


it Otherwise,, instead of relyiDg upon the position of the joints, resort must be had to the cohesive 
III strength of joint-fastenings, such as bolts, spikes, cramps, joggles, mortises and tenons, mortar, 
t cement, &c, to prevent sliding. As to mortar and cement, however, it is important to remember that 
a frequently, and especially in very massive work, they have not time to harden, or acquire their full 
tl strength, before the acting forces are brought to bear upon them ; therefore, great care is necessary, 
[■ when we use them as substitutes for position. On this account we frequently cannot consider a mass 
[• of masonry to be a single rigid body, but must regard it as composed of several detached rigid 
til bodies; the stability of each of which must be separately provided for, before we can secure that of 
e the whole. Therefore, in large massive structures of importance, we should, as far as possible, omit 
f all consideration of the strength of the mortar, and rely for stability chiefly upon placing the joints 
i. at or nearly at right angles to the forces acting upon them. 


Art. 69. Moment of stability. We have already stated (see Arts 46 
and 49) that the resistance which any rigid body as B, Fig 71, opposes against being 
overturned about any given point a, is equal to that 
which would be produced if the entire wt of the 
body were concentrated at its cen of grav g ; and 
acted at the end i of a straight lever a i, of which a 
is the fulcrum; or at the end o, s, or », of any straigh t 
lever (Art 49) a o, as, an; or of any bent lever a i n, 
a i s, a to, a s o, provided that in every case there is 
| the same leverage at, measured from the fulcrum a, 
and at right angles to the direction mn of the force 
I of grav of the body. So far as regards tendency to 
I resist overturning, it is immaterial at what 

point of the body, in this line of direction mn, we 
. conceive the grav to act: or whether as a push at o, 
or a pull at t, as denoted by the arrows. We have 
also said that the tendency, or moment, of this force 
of grav, or wt, to produce or to resist motion about the fulcrum a, through the me¬ 
dium of any of these levers, is found bymult the force or wt in ibs, by the leverage 
u i in foot. The prod in ft-ths is generally called the moment of the force about the 
point a; but in cases like that before us, in which this moment becomes the measure 
of the stability of the body, it is called the moment of stability (or simply the star 














358 


FORCE IN RIGID BODIES. 




bility) of the body, about that point. Therefore, if bodies of the same size and shape 
and with their centers of gravity in the same position, have diff wts, or sp gravities, 
their respective stabilities will be in proportion to their wts, or sp grs. 

A body may have diff moments of stability, about diff points. Thus, it would be 
far more difficult to overturn B about the point b, than about a; because the lever¬ 
age bi is 2% times as great as at; and since the wt and the point of the cen of grav 
remain unchanged, the moment about b is 2^ times as great as about a. 

Rkm. 1. Let a b c o, Fig 72, be a squared block of stone 6 feet long; on a hor base ; and weighing 12 

tons ; and h, a force applied to overturn it about the toe o. Since its 
length o c, is 6 feet, its cen of grav t, will be dist o g, or 3 ft back from 
o. Consequently, the moment with which the block resists being 
overturned, is 12 (tons) X 3 (ft leverage) = 36 ft tons. Now. suppose 
the upper half a b o, to be removed; the remainder ob c will weigh 
but 6 tons. But its cen of grav n, is farther from o, than that of the 
whole block was. Being now triangular in shape, the dist o y will be 
% of o c; or will be 4 ft. Consequently, the resisting moment will be 
6 (tons) X 4 (ft leverage) =24 foot-tons. So that although the block 
has but half the wt of the first one, it has % as great resisting 
LV/C ^7 O power. It is on this principle, that in order to save masonry, the 

# w faces of retainiug-walls. Ac, are sloped, or battered back. 

Rem 2. In Fig 72^ let the upper part, 

f l ,_ § cm, of A, and the lower part, oy, of B, be 

- } C "7 "> 111111 1|| n C ... made of lead, and the remaining part of 

\ each, of cork. Then the center of gravity 
7 of the entire body A will be near the dot t ; 
/ and that of B, near the dot s. Let the 
/ weights of A and B be equal, and their cen- 
' ters of gravity, t anti s, at the same hori- 
zontal distance, ao , from the fulcrums, o 
and o, around which the bodies are to be 
overturned. Their moments of stability 
must then be equal. Consequently they will 
require, in order to begin to overturn them, 
equal forces, applied in any same given 
direction, and at any same given point; 
as, for example, the equal forces, / and g, 
applied horizontally at i and t. ' But, in 
order to entirely overthrow B, the overturn¬ 
ing force must act through a greater sjmce, 
i e, must do more work, than is required to overthrow A; for A will be overthrown 
when the force,/, has acted through only the small space necessary to move it into 
the position of the dotted lines J. Its center of gravity being then at e, which is 
beyond the base, A must necessarily fall. But the force, g, acting upon B, must move 
through the longer space necessary to move B into the position N ; for not until then 
will its center of gravity, s, be in a position, l, beyond the base. As either A or B 
turns, about its fulcrum o, from its original position (A or B) to that (J or N) in which 
it is about to fall, its leverage, ao, of stability plainly diminishes, until the center 
of gravity is over o. The leverage is then = 0. Since the weight remains unchanged, 
the moment of stability decreases (and ceases) with the leverage of stability, as does 
also the overturning moment required to keep the body moving. The overturning 
leverage varies with the rising, and subsequent falling, of the corner, i, at which 
the force is applied; but in bodies shaped like A and B this variation is but slight, 
and hence the horizontal force, f or g, required, decreases nearly as the leverage of 
stability decreases, ceasing when it ceases. The object of the civil engineer is gen¬ 
erally to secure his structures against beginning to overturn. Hence he is chiefly 
anxious about the amount of the force, f or g, (in pounds, tons, etc) required to start 
motion; and seldom needs to concern himself about the amount of icork (in foot¬ 
pounds, foot-tons, etc) required to complete the overthrow. 

Rem. 3. It is not alone the wt of the body itself, which contributes to its stability in all cases; for 
this may be assisted by extraneous wts or loads. Thus, the wt of a pier P, Fig 73, gives it in itself a 
certain degree of stability; but wheu we add the wt of the two equal arches, 
its stability is thereby increased, supposing t he foundation to be secure. And 
a passing load, when it is directly over the pier, increases it still more. It is 
true that the wt of the arches might crush the pier to fragments, if the stone 
be soft; but this is a matter of Strength of Materials ; not of stability; and 
must be examined into by itself. If the two arches be of unequal sizes, or if 
there be but one arch, the stability of the pier may become either increased 
or diminished, according to ciroumstances; as will appear farther on. 

Whether the force acting for or against the stability of a structure, be 
gravity, or pushes, or pulls, produced from other sources, is, as in other cases, 
a matter of no importance: for force is simply force, no matter whence de¬ 
rived. We have, therefore, only to look at the diff forces acting upon our 
structures, as so many tendencies to produce motions in certain directions. If these tendencies are 
reacted against, or destroyed by others, they will not produce it; but if they meet no resistance, mo¬ 
tion must take place. The onlv peculiarity we need assign to grav. is that the direction of its action 
v always vert downward ; while other forces may be imparted in either that,, or any other direction, 
hie resultant of grav combined with other force or forces, may be in any direction". 


Fig 7Z{ 




Fio73 






























































FORCE IN RIGID BODIES. 359 




W 
'er -} 
rar ’ 

?ir 

: ::i 

MU 

H 

u 

M-i 

1 bf i 
11* 

i 

! 

'si 


I.H 

A 

it f 

> 


A 

l)9j 

!tf 

1 


•» 8taU,ity tliat present th ™ 8eiv< » 
forces to a:, /,/,/, Ac, may all be considered to be 
imparted and acting in the same plane; which is a 
vertical one, eloc, passing through the center of 
gravity, v, of the structure, mnpqrstu: and of 
course, coinciding with the line of direction w x, 
of its weight, or force of gravity. The plane eloc, 

. .j. the forces > therefore, may be considered as 
coinciding with a leaf of paper standing vertically 
on one edge. This renders the calculations much 
oiore simple than if the forces were in different 
plaues; in which case diagrams alone would not 
mthce for determining the resultants, leverages 
moments, Ac. See Art 44. Whereas, when they 
ire in the same plane, such diagrams, neatly 
irawn to a convenient scale, will usually possess 
ill the accuracy required in practice. 



Thus, In examining the strains on the diff pieces composing tbe 
russ of a roof, or of a bridge, &c, not only their wts, but all the 
training forces, may be assumed to act in a vert plane passing 
engthwise of the truss from end to end : splitting it into two equal 
•arts. In the case of a structure of masonry, such as a retaining- 
vall, or a stone bridge B, Fig 74, we base our calculations upon a 
hin vert slice of it, having a length a o, i t, or y t, of only one ft; 
10 matter what may be the height y s; or the thickness y c, of the 
tructure. Through the center of this one ft of length, we suppose 
• ver t plane to pass ; splitting, as it were, the body B into two pre- 
iselv similar parts; and all the forces actually diffused equally 
hroughout the whole one ft of length, are supposed to be coneen- 
rated. and to act upon one another, in this plane. See Remark 
Irts 49 and 57, 



Pig 7 4- 


-.4 


11 

'"’l 

1 

ltd 

li 

V8 

n 

8 

■ li 

: it 

■I, 

? 

t 

i 

i • 

5? 

i 

it* 

fi 

% 

id 

ii 


Art. 71. In order to ascertain the effect of diff forces to 
produce either sliding-, or overturning, of either an entire rigid 
)°d.v, or of one composed of several rigid bodies placed together without joiDt- 
astenings; we first find the direction of the resultant of those forces. If the body 
•r structure is not composed of diff parts ; or if, being so composed, these parts are 
o firmly united together by joint fastenings, as to constitute virtually hut a single 
•igid mass, then we need do nothing more than (as in Art 35) find the*resultant ad 
* ig 19 , of ail the forces. If the direction of this resultant strikes inside of the base’ 
he body will not overturn (Remark 2, Art 72, ); and if, besides striking inside 

ff the base, it forms with a line b x, at right angles to the base, and angle abx, less 
han the angle of fric between the body and its support n m, then it cannot slide. 

At Fi s 67, the direction of the resultant x o strikes at t. inside or the base s e ; therefore the 

ody B cannot upset; whether it will slide or not, depends upon whether the angle o t v is greater or 
sss than tbe angle of fric corresponding to the materials composing the body, and the plane. If both 
re of ordinary dressed granite, this angle must not exceed about 32°. See other angles, under head 
friction. 

Questions on overturning, may also be solved on the principle of leverage. See Art 49. 

Art. 72. We will now consider a case in which the structure is assumed to be 
:omposed of several rigid bodies, merely placed together without joint-fastenings of 
ny kind, such as cramps, bolts, mortar, &c; but depending entirely upon their wt 
i.nd positions, for securing their stability. The process in this case is the same as 
n Fig 19, except tliat instead of assuming the body to have but the one joint 

' m; and finding the effect of the resultant ad with reference to this joint alone; 

. e consider it to have several joints, as P Z, F L, W X, Ac, Fig 75; and then examine 
he effect of the diff resultants which they must respectively sustain in consequence 
•f the diff wts of the several parts NMPZ, NMFL, NM W X, resting on them. 

I.«‘t ft AT J T be one half of a stone arch ; or rather a vert slice of 
t, 1 ft thick ; and let N M W X be a similar slice of a dressed stone abut which has 
•een designed to sustain the thrust of the arch; and the fitness of which for the 
mrpose, we wish to ascertain. 


o e 

if 

jd 


be 

A 

e* 

Bf 

re 

(V 

•o 


Suppose the thrust of the slice of the arch, (that is. the resultant of its wt, and of its hor pres,) to 
ave been previously ascertained by Ex 2, p 330, to be 30 tons; that this is concentrated at o. (the 
enter of the skewback :) and that its direction is o h. Also, suppose the wt of the part N M P Z be 
>und to be 10 tons, and to be concentrated at its cen of grav G ; and, consequently, to act in the 
ert direction G a. Now (Arts we may suppose the 30 tons resultant of the arch, and the 

0 tons grav of the part NMPZ, to act at the same point c, at which their directions Go and o l meet, 
lake c d by scale equal 30 tons, and ca 10 tons; complete the parallelogram of forces cdya,-, and 
raw its diag cy, which by the same scale will give the resultant or all the forces acting upon the 
art NMPZ. 

Now we see that the direction cy of this resultant strikes at i, or ivithin the base P Z; consequently 
Art 60, &c) NMPZ cannot upset, no matter how great may be the pres cy ; see Remark 2. From i 









































M to bring the pres of the arch vpo« the abut; or by iron cramps, stone joggles, &c; but these are 
expensive. 1 be most obvious remedy, as well as the least expensive, is simply to incline the joint 
FZ into a direction somewhat like from K to Z: so as to receive the pres of the resultant cy more 
uearly at right angles ; at least so nearly as to be fairly within the limits or the angle or trie. If this 
is done, stability is secured ; for the part N M PZ. being now safe against both sliding and overturn- 
lng, can move m no other way ; unless the strength of the stone composing the masonry is insufficient 
to bear the pres, and may therefore -crush to pieces under it. Hut this is a question of Streugth of 
Materials.' The point i, of Fig 75, comes much nearer to Z than would bo 

desirable in practice ; for it might cause crushing at Z. See Item 2, following. 

Having thus provided for both the sliding and the overturning stability of the abut as far down as 

f b * P VI « n .° W eXi \ miuo do " n M the joint F I,. Taking the entire part NMFL of 

the abut, we first find its weight, say 2 o tons; and this we assume to act at its cen of grav K, and in 
the vert directions K l. The amount and direction of the thrust of the arch at o, of course, remain 
as before. Therefore, from the point v. where the two directions meet, lav off ve to represent as 
nr f fnr-^ e 3 ° t /° nS t , h 'i Ust °! the .a'’C*i 1 and y l, the 25 tons wt of the abut. Complete the parallelogram 
of forces ves t, and draw us diag tin; which, measd by the same scale, will give the resultant r.r all 
Ihe forces acting upon the part N M F L. Now we see that the direction of this resultant does not 
fall within the base 1-1,; but, on the contrary, passes out of the body at j; outside of which it me-ts 
no force to resist ,t. Consequently, since this resultant must be considered as an 

ar M, d r fh act !"f >'P" n inert body or abut. NM F L, without wt. (Art 35.) that body must upset 
around the point L; or around the nearest joint in the masonrv between T, and j ; and cannot ran- 

* X r sta " d of ‘V l ^r les3 it3 . base be above >- U i- s true that by p-acing earih behind"t espe- 

uoon the haseTT- a e ^ rn r ^ nm ’ ng ’ ° f 9 8,naU arch ,ui K ht be "'■■ule to stand safely even 

.if a 6 )r X V a f rt tbe , case of arches of moderate spans, this aid may he resorted to for 

rains 8 a h nd rhn« th » e ah,ltS , - , wben there ls “° danger that the earth may be washed away hy floods or 
raius, and thus expose them to ruin ; and this is generally and properly done. J 

w'v' ho V . Uth f S . ame ™ anner that tbe P°' nt * found in the joint P Z. others between P Z and 
be determined also: then a curve, commencing at the skewhack o, and drawn through them 
will represent the line of pressures or of resistance, or of thrust. , r ,' 

abut. At any point whatever in this line, say at i, the entire pres above said point may be si nnUed 
to be concentrated ; while the entire length, as cy, of that resultant which cuts said point , 

amount of said pres at that point; and the direction, as «, or the same resultantis also the direr 
tion in which said pres acts upon said point. See Art 13 of Hydrostatics, p 231. tA d ec ’ 

2 ' , The . line of pr ® s ? nables us t0 determine another very important point connected with the 
stability of a structure. It is not sufficient, in practice that this line should strike merely within the 
base; it must strike at a considerable, dist within. If the structure aud its found^tfon were ahsof 
lately rigid, so that no conceivable force could bend or break them, this would not he necessary • but 
all materials are more or less weak, so that if great, pressures come too near to their edges there is 

1r a X7-il Pl ' Ui ^ ° r °T. ,in f at th Vu poi " ts: or if ncar tbe ed ee of a base, an unequaf wttlernem 
•:{ tbe .l°j', ,e " e ? kb mRT f take fl P ,ane - Therefore, even in structures of but small size, the dist i Z Fie 
7o of the line oi pres, from the outer point Z. should never be less at any joint than V of the width 
sf that joint; except, perhaps, in a case like that of a small arch in which the earth filling is depos- 


360 


FORCE IN RIGID BODIES. 


draw if, at right angles to the line P Z ; and measure the angle cif, which the resultant cy forms 
with it. This, we find, is greater than 32°; that is, it exceeds the angle of fric between surfaces of 
dressed stone. Therefore, the part NMPZ must slide, along the joint P Z. This might possibly be 
prevented by good mortar, if time be allowed it to solidify properly, before the centers are eased so 



















FORCE IN RIGID BODIES, 


361 




ed 1 


[ Bd behind the abuts before the centers are removed. In Important works, it should not be less than 
e )out % of the width of the joint; and it is still better, when possible, at % ; or, in other words, at 
te center of each joint. When, as at Q, a footing U is added at a base, W X should be taken as the 
int; not W Q. 


Rem. 3. The line of pressure in an arch itself, as H J N T, Fig 75, 

tay also be found much in the same way, thus: First divide the half arch FI J N T, 
ud the filling above it, by vert lines ru, w x, &c, which need not he at equal dists 
(part. Four such lines will suffice for a flat arch, and about six for a semicircular 
• lie. We then consider in turn, and separately, each part, as r u H J, wx H J, 
D T 11 J, &c, which extends from these lines to the center II J of the half arch; 
le last of these being the entire half arch. The cen of grav, and the wt, of each 
!f these parts, must be found; also, (Fix 6, p 341,) the hor pres at the keystone. 


Now each of these parts, like the beam in Ex 1, or the half arch in Ex 2, p 330, is acted upon, and 
j»pt in equilibrium, by three forces; namely, the hor pres at the keystone, (see Ex 6. p 341;) its own 
t. acting vert; and the reacting force of the part next behind it. We proceed with each part sepa- 
itelv, as v/e did with the entire beam alluded to, thus: Beginning with the part rw H J, from its 
:u of grav, to, draw a vert line to/. From the center E of the keystone, draw a hor line F. n. to meet 
] ,f. From n lay off n f by scale, to represent the wt of the part r u H J; and from / lay off f g. hor 
y scale, to represent the hor pres at the key. Draw the diag n q; which will give, by scale, the re- 
iltant of these two forces. The point b, at which the diag n g intersects the vert ru. is a point in 
le line of pres reqd. Next go to the part w x H J ; and in the same manner find another point in 
lie line wx, using the cen of grav and the wt of that part. Finally, treat the entire half arch ,N D 1' 

I t J in the same way. The resultant diag of this last will pass through o, the center of the skewback, 
’ the archstones have the same depth throughout,. A curve drawn by hand through the points thus 
vund, will be the reqd line of pres. These points will pot all fall equally well within the thickness 
f the archstones. 

Art. 73. The stability of bodies on inclined planes, as regards 
verturning, is measd in the same way as when the base is hor; namely, by mult 
iieir wt, by the perp dist (an, or c t, at A, B, and D, Fig 76,) from the fulcrum, or 
irning-point a or c, to the vert line of direction (g o) drawn from the cen of grav 



f the body Hence, it is evident that the body B has less overturning stability 
bout its toe a than the similar body A has, when the force, n tends to upset it 
own hill. But it lias more than A, when the force tends to upset it up hill, or about 
te toe c; for the leverage t c of B is greater than that, o c, oi 

The bodv C, which would overturn upon a level base, because the line ffo strike.' th « 

D the line 

f direction n o cuts the cmfer of the base a c. The two leverages, a o and t c, are therefore equal as 
re, consequently, the moments of stability of 1) about a and c respectively Sum arly a given up- 
•■mi vertical force would have the same upsetting effect whether applied at a (to upset up hil ) oral 
Tto up et doTOa hiU) because its leverage, a o + f c, is the some in both cases. But a horizontal 
rree applied^any given height, as at g, has a greater leverage ( = go) when pushing downhill 
han when pushing up hill. For in order to upset down hill the body must rotate on the corner 
,-hereas. in orde/to^ipset up hill, it must rotate on the corner c, which gives to a horizontal force 
pplied at g, only the shorter leverage = g t. 

Structures built upon slopes are, however, liable to slide; 

•it is they are deficient in frictional stability. In practice this is remedied by cutting the slope 
to hor steps as at E. But works so constructed are not as strong as it the base were a continuous 
,r Mne because the vert faces of the steps break the bond of the masonry; and because the mortar 
the higher portions s d. being in greater quantity than that in the lower portions e y. necessarily 
lows more settlement of the masonry in the former; and thus renders the wot k liable to crack, or 
>Iit open vertically. The case is analogous to that of a foundation, firm in someparts, a»»<l com- 
■essible in others" Therefore, when circumstances permit, the foundation should be levelled off as 
Tv or if the masonry has to sustain down-hillward pressures, v should be lower than d; and the 
lurses of masonry be laid with a corresponding inclination. 


























362 


GRAVITY—FALLING BODIES. 


GRAVITY, FALLING RORIES. 


Art. 1. Bodies falling' vertically. A body, falling freely in vacuo 

from a state of rest, acquires, by the end of the first second, a velocity of about 
32.2 feet per second; and, in each succeeding second, an addition of velocity, or 
acceleration, of about 32.2 feet per second. In other words, the velocity receives ic 
each second an acceleration of about 32.2 feet per second, or is accelerated at the 
rate of about 32.2 feet per second, per second. This rate is generally called (for 
brevity, see foot-note,f p. 310), simply the acceleration of gravity (but see * 
be.owj, and is denoted by g. It increases from about 32.1 feet per second, per 
second, at the equator, to about 32.5 at the poles. In the latitude of London it is 
32.1S). These are its values at sea-level; but at a height of 5 miles above that level 
it is diminished by only about 1 part in 400. For most practical purposes it may be 
taken at 32.2. 

Caution. Owing to the resistance of the air none of the follow¬ 
ing rules give perfectly accurate results in practice, especially at great vels. 
Th« greater the specific gravity of the body the better will be the'result. The air 
resists both rising and falling bodh s. 

If a body be thrown vertically upwards with a given vel, it will 
rise to the same height from which it must have fallen in order to acquire said 
vel; and its vel will be retarded in each second 32.2 ft per sec. Its average ascend¬ 
ing velocity will be half of that with which it started ; as in all other cases of 
uniformly retarded vel. In falling it will acquire the same vel that it started 
up with, and in the same time. See above Caution. 

In the following, the falls are in feet, the times in seconds, 
and the velocities and accelerations in feet per second.* 

Acceleration acquired* 
in a given time — ff X time 


in a given fall from rest 
in a given fall from rest ) 
and given time j 

Time 

for a given acceleration 


= V2 g X falT. 
twice the fall 
time 
required 
acceleration 


9 


See table, p 258 


for a given fall from rest 

for a given fall from rest) 
or otherwise j 


V fall 

Hff 


fall 


fall 


^ final velocity 
_ fan _ 

]/ 2 (initial vel + final vel) 


mean vel 

Fall 

in a given time (starting from rest) = time X A final vel = time 2 x Yg 

in a given lime (starting X = , |m>> v ..initial vel + final vel 

from rest or otherwise) J * ean ' el — tlme X -- 

reqd for a given acceleration ) _ acceleration 2 
(starting from rest) 2g See table, p 258. 


during any one given second (counting from rest) 

= 9 X (number of the second (1st, 2d, &c) — ^ 
during any equal consecutive times (starting from rest) a 1, 3, 5, 7, 9, &c. 


Calling g = 32.2 
we have 


l 


At the end of the 

1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 10th. 

seconds 


Velocity; ft per sec. 
Dist fallen since end 

32.2 

64.4 

96.6 

128.8 

161.0 

193.2 

225.4 

257.6 

289.8 

322.0 

of preceding sec; ft. 

16.1 

48.3 

80.5 

112.7 

144.9 

177.1 

209.3 

241.5 

273.7 

305.9 

Total dist fallen; ft. 

16.1 

64.4 

144.9 

257.6 

402.5 

579.6 

788.9 

1030.4 

1304.1 

1610.0 


* By acceleration,” in this article, we mean the total acceleration • i e the whnl« 
change of velocity occurring in the given time or fall. For the rate of a’ccelerari ... 
wo use simply the letter g. m»»uou 








































DESCENT ON INCLINED PLANES. 


363 



Art. 2. Descent on inclined planes. When a body, U, is placed 
upon an inclined plane, AC, its whole weight W is not employed in giving it 
velocity (as in the ease of bodies falling vertically) 
but a portion, P, of it (= W X cosine of o = W X 
cosine of a*) is expended in perpendicular pressure 
against the plane; while only S, (= W X sine of o 
= W X sine of a,*) acts upon U in a direct ion parallel 
to the surface A C of the plane, and tends to slide it 
down that surf. See Art. 60, p 352. 

The acceleration, generated in a given body in a 
given time, is proportional to the force'acting upon 
the body in the direction of the acceleration (Art. 
db, p. 310). Hence if we make W to represent by scale 
the acceleration g (say 32.2 ft per sec) which grav 
would give to U in a sec if falling freely, then S will 

give, by the same scale, the acceleration in ft per -*-lE 

sec which the actual sliding forces would give to U in one sec if there were 
no friction between U and the plane. We have therefore 

theoretical acceleration down the plane = g x sine of a. 

Therefore we have only to substitute “ g. sin a” in place of “ q and the 
sloping distance or “slide” A C in place of the Corresponding vertical distance 
or tall A E in the equations, p. 362, in order to obtain the accelerations etc as 
follows: 

on an inclined plane without friction. 

In the following*, the slides A C are in feet, the times in 
seconds, and the velocities and accelerations in feet per 
second.f 

Accelerationfof sliding velocity 

in a given time = vert ac <j el squired in falling 1 . 

6 vert during the same time / A &1U u 

= g. sin a X time 

in a given slide, as AC, 
from rest 


'}= 


slide 
A time 


(vert aecel acquired in falling) 

= < freely thro the corresponding > = 1/ 2 g. A E 
( verthtAE j 

= Y 2 g. sin a X slide 


Time required 

- . .... ... sliding acceleration 

for a given sliding acceleration = — 


for a given slide, as A C, from 
rest 


for a given slide, from) 
rest or otherwise 1 


g. sin a 
slide 


A final sliding velocity 


4 


slide 
A 9- sin a 


time reqd to fall freely thro the correspond¬ 
ing vert ht A E 


sin a 


slide 


slide 


Cosine a 


base E C 


mean sliding vel A (initial + final sliding vels) 

horizontal stretch, as E C, 

of any length, as AC I/AC 2 — AE 2 


length A C 


that length 


AC 


„. _ height AE fall, A E, in any given length, AC 1 / A C 2 — EC 2 

,ie a length A C that length — A^C 


* Because o and a are equal. 

f By acceleration, in this article, we mean the total acceleration, i. e., the whole 
change in velocity occurring in the given time or slide. For the rate of acceleration 
we use simply the letter g. 































364 


GRAVITY—PENDULUMS 


in a given time, starting from 

in a given time, starting from 
or otherwise 


Slide, as A C 

rest = time X final sliding vel 
= time 2 X g. sin a. 
rest • 

= time X mean sliding vel 


= time X X A (initial + final, sliding vels) 


required for a given sliding aecel- _ sliding acceleration 2 
eration (starting from rest) 2 g. sin a 


But in practice the sliding on the plane is always op¬ 
posed l»y friction. To include the effect of friction, we have 
only to substitute 

“ 9 X j^sin a — (cos a. coeff fric)J ” in place of “ g. sin a ” in the above equations. 
Because 

Friction — Perpendicular pressure P X coefficient of friction 
= weight W X cosine a X coefficient of friction 

and 

retardation of friction = g X cosine a X coefficient of friction. 
(For table of coefficients for various substances, see p 373.) 
Resultant sliding acceleration 

= theoretical sliding accel (due to the sliding force, S) — retardation of frit 
= ( g. sin a) — {g. cosine a. coeff fric) 

= g X £sin a — (cosine a. coeff fric)^j 

If the retardation of friction (= g. cos a X coeff fric) is not less than the tota 
or theoretical accel (“ g. sin a ”) the body cannot slide down the plane. 


-■> i tm - 

PENDULUMS. 


The numbers of vibrations which diff pendulums will make in any given place ii 
a given time, are inversely as the square roots of their lengths; thus, if one of then- 
is 4, 9, or 10 times as long as the other, its sq rt will be 2, 3, or 4 times as great; bu 
its number of vibrations will be but %, or % as great. The times in which difl | 
pendulums will make a vibration, are directly as the sq rts of their lengths. Thus 
if one be 4, 9, or 16 times as long as the other, its sq rt will l>e 2, 3, or 4 times a 
great; and so also will be the time occupied in one of its vibrations. 

The length of a pendulum vibrating seconds at the level of the sea. in a vacuum 
in the lat of London (51%° North) is 39.1393 ins; and in the lat of N. York (40%' 
North) 39.1013 ins. At the equator about y 1 ^ inch shorter; and at the poles, about yl 
inch longer. Approximately enough for experiments which occupy but a few sec j 
we may at any place call the length of a seconds pendulum in the open air, 39 ins 
half sec, 9% ins ; and may assume that long and short vibrations of the same pen 
dulum are made in the same time; which they actually are, very nearly. For meas 
tiring depths, or dists by sound, a sufficiently good sec pendulum may be made of ■, 
pebble (a small piece of metal is better) and a piece of thread, suspended from i 
common pin. The length of 39 ins should be measured from the centre of the pebble 















365 


PENDULUMS, ETC. 


Tn starting the vibrations, the pebble, or bob, must not be thrown into motion, but 
merely let drop , after extending the string at the proper height. 

To find the length of a pendulum reqd to make a given number of 
vibrations in a min, divide 375 by said reqd number. The square of the quot will be 
the length in ins, near enough for such temporary purposes as the foregoing. Thus, 
for a pendulum to make 100 vibrations per min, we have = 3.75; and the square 
of 3.75 — 14.06 ins, the reqd length. 

To find the number of vibrations per min for a pendulum of 

given length, in ins, take the sq rtof said length, and div 375 by said sq rt. Thus, 

375 

for a pendulum 14.06 ins long, the sq rt is 3.75; and — — 100, the reqd number. 

Hem. 1. By practising before the sec pendulum of a clock, or one prepared as just 
stated, a person will soon learn to count 5 in a sec, for a few sec in succession ; and will 
thus be able to divide a sec into 5 equal parts; and this may at times be useful for 
very rough estimating when he has no pendulum. 

Centre of Oscillation and Percussion. 

Rem. 2. When a pendulum, or any other suspended body, is vibrating or oscillating 
backward and forward, it is plain that those particles of it which are far from the 
point of suspension move faster than those which are near it. But there is always 
a certain point in the body, such that if aU the particles were concentrated at it, so 
that all should move with the same actual vel, neither the number of oscillations, 
nor their angular vel, would be changed. This point is called the center of oscilla¬ 
tion. It is not the same as the cen of grav, and is always farther than it from the 
point of suspension. It is also the centre of percussion of the suspended vibrating 
body. The dist of this point from the point of susp is found thus : Suppose the body 
to be divided into many (the more the better) small parts; the smaller the better. 
Find the weight of each part. Also find the cen of grav of each part; also the dist 
from each such cen of grav to the point of susp. Square each of these dists, and 
mult each square by the wt of the corresponding small part of the body. Add the 
products together, and call their sum p. Next mult the weight of the entire body 
by the dist of its cen of grav from the point of susp. Call the prod g. Divide p by g. 
This p is the moment otTnertia of the body, and if divided by the wtof the 
body the sq rt of the quotient will be the Ratlins of Gyration. 

Angular Velocity. 

i When a body revolves around any axis, the parts which are farther from that 
axis move faster than those nearer'to it. Therefore we cannot assign a stared 
I linear velocity in feet per second, or miles per hour etc, that shall apply to every 
part of it. But everv part of the body revolves around an entire circle, or 
through an angle of 360°, in the same time. Hence, all the parts have the same 
i velocity in degrees per second, or in revolutions per second. This is called the 
angular velocity. Scientific writers measure it by the length of the arc de¬ 
scribed by any point in the body in a given time, as a second, the length of the 
arc being measured by the number of times the length of its own radius is.con- 
tained in it. When so measured, 

Angular velocity linear velocity (in feet etc) per sec 
in radii per second = length of radius (in feet etc) 

I Here, as before, the angular velocity is the same for all the points in the body; 
because the velocities of the several points are directly as their radii or dis¬ 
tances from the axis of revolution. . . . 

In each revolution, each point describes the circumference of the circle in 
which it revolves = 2 nr {n = 3.1416 etc; r = radius of said circle). Conse¬ 
quently, if the body makes n revolutions per second, the length of the arc de¬ 
scribed by each point in one second is 2 tt rn ; and the angular velocity of the 
body, or linear velocity of any point measured in its own j adii, is 


a = 2ir . ™ = 2 n n = sav 6.2832 X revs per second = say .1047 X revs per minute, 
r 

Moment of Inertia. 

Suppose a body revolving around an axis, as a grindstone; qr oscillating, like 
a pendulum. Suppose that the distance from the axis ot revolution (which, m 
the pendulum, is the point of suspension) to each individual particle of the 
body, lias been measured: and that the square of each such distance has been 
multiplied by the weight of that particle to which said distance was measured. 





366 


MOMENT OF INERTIA. 


The sum of all these products is the moment of inertia of the body. Or 

Moment _ J the sum, ) , [weight square of dist 

of inertia 1 for all the particles f of ) ot . , X of particle from 
^ f particle axis of revolution 

or, I = 2 <&■ w. 


Scientific writers frequently use the mass of each particle ; 
ip __its weight 

acceleration (g) of gravity, or about 32.2 * ns ^ ea< ^ 01 weight, in calculating 
the moment of inertia. 

in practice we may suppose the body to be divided into portions measuring 
a cutnc inch (or some other small size) each ; and use these instead of the thei> 
letical infinitely small particles. The smaller these portions are taken the 
more nearly correct will he the result. * 

When the moment ot inertia of a mere surface is wanted (instead of that of a 
body) we suppose the surface to be divided into a number of small areas and 

am])le, e “e p S 48G d ° f U>e Weights of the s,ua11 portions of the body. For an ex- 

__ „ weight of bodv, 

Moment of inertia = or x square of 

area of surface radius of gyration * 

For definition of Radius of gyration, see p 440. A body may have anv number 
ol radii ot gyration, depending upon the position of the axis of revolution. 


Table of Radii of Gyration. 


Body 

Revolving' 

around 

Radius of Gyration 

Any body or 
figure 

any given axis 

/moment of inertia around the eiven axis 



\ weight ot body, or area of surface 

Solid eyliif 

its longitudinal 

radius of cylinder X 
~ radius of cylinder X about .7071 

der 

axis 

ditto 

ditto, infinitely 

a diam, mid wav 
betweeu its ends 

/lengths ( radius 2 of cylinder 

Y 12 + 4 -- 

short (circular 
surface) 

a diameter 

radius of cylinder 

2 

Hollow cyl- 

its longitudinal 

- - - - ... 

inder 

axis 

^ /inner rad 2 + outer rad 2 

ditto, infinitely 
thin 

ditto 

\ 2 

radius of cylinder 

ditto, of any 
thickness 

a diam midway 
between its ends 

/inner rad 2 -f outer rad 2 lengths 

\ 4 + 12 ’ 

ditto, infinitely 
thin 

ditto 

/radius 2 of cylinder lenirth 2 
' 2 + 12 

ditto, infinitely 


radius of cylinder X 

♦ bin and infinitely 
short (circumfer- 

a diameter 

enee of a circle) 


— radius of cylinder X about .7071 

Solid sphere 

a diameter 

/radius 2 of sphere 



A 2.5 


1 

— radius of sphere X "jXTi - 
~ ra,Jius of sphere X about .6324G 





























































RADII OF GYRATION 


367 


Table of Radii of Gyration.— Continued. 


Body 

Revolving 

around 

Radius of Gyration 

Hollow 

sphere of any 
thickness 

a diameter 

/ 2 (outer rad 5 — inner rad 5 ) 

\ 5 (outer rad a — inner rad 3 ) 

ditto, thin 

ditto 

approx (outer rad + inner rad) X .4085 

ditto, infinitely 
thin (spherical 
surface) 

Straight line, 

ab 

ditto 

any point, r , in its 
length 

radius of sphere X 
= radius of sphere X about .8165 

lax 3 -fa-63 

A 3 ab 

a x c b 

either end, a or b 

length ab X 



= length ab X about .5775 


its center, c 

flC x Vir 



= length ab X about .2887 

Solid cone 

Circular 

plate, of rect¬ 
angular cross sec¬ 
tion 

Circular 
ring, of rectan¬ 
gular cross section 

its axis 

See Solid cylin¬ 
der 

See Hollow cylin¬ 
der 

radius of base of cone X 1/^3“ 

= radius of base of cone X .5477 

For the thickness of plate or ring, 

. measured perpendicularly to the plane 
of rhe circumference, take the length of 
the cylinder. 

Square, rect¬ 
angle and 
other sur¬ 
faces 

For least radius of gyration, or that around the longest axis, 
see p 440 and 441. 

























368 


CENTRIFUGAL FORCE. 


CENTRIFUGAL FORCE. 

Vk hen a body a, Fig 3 moves in a circular path abd, it tends, at each point, a? 
a or b, to move in a tangent at or bt' to the circle at that point. But at each 



point, as a etc, in the path, it is deflected from the tangent by a force acting 
toward the center, c, ot the circle. This Lrce may be the tension of a string or uj 
the arm of a fly-wheel, ca, or the attraction between a planet, c, and its moon, a or 
the inward pressure of the rails, a 6, on a curve, etc. Like all force, it is an act’ion 
between two bodies, tending either to separate them or to draw them closer to-eiher 
and acting equally upon both. (See Art. 5 6, p. 308). In the case of the string, it 
vtii s the body, a, toward the center, c, and the nail or hand, etc., at c, toward the 
body at a or 6 etc.; i. e.,from the center. In the case of a car on a curve it pushes 
the ear toward the center, and the rails from the center. The pull or push on the 
revolving body toward the center is called the centripetal force: while the 
pull or push tending to move the deflecting body from the center is called the cen- 
tiifugal force. These two “forces,” being merely the two “sides” (as it were) 
of the same stress, are necessarily equal and opposite, and can only exist together, 
ihe moment the stress or tension exceeds the strength (or inherent cohesive foroel 
of the string, etc., the latter breaks. The centripetal and centri^RtoSTSS 
fore instantly cease; and the body, no longer disturbed by a deflecting force, moves 
o n at a uniform velocity f, m a tangent, a t or b t', etc., to its circular path, or at 
right angles to the direction which the centrifugal force had at the moment it ceased. 

Centrifugal force = weight of body X veIocit y 2 in feet per second 

' radius, ca Fig 3, in feet X g 
= we . lg " t of x ^°- of revs P er min X n-2 x rad ca Fig 3, in ft 
° y flOO^r ■ 

7T = about 3.1416. See page 123. 7r2 = about 9.8696. See caution, p 369 

g ----- 32.2. See page 362. v 

(lst) Let „ - a b ' Fig 3 ’ be the minute space described bv the re- 
vdvmg body® in an exceedingly short time t* so that we may consider S 
straight line , or chord, ab, as equal to the arc, ab 7 consider the 

Now (Euclid, vi. 8) ’ 

diameter ad : ab :: ab : an, or an — ab \ 

But an is = lb ad 

= !, h / e t^5 a Vn through w . hich the body a deviates from the tange. 

— L/^r rd th< T center c 1,1 the time t, or while moving from a to b 
in the time e t ^ ia(i0n toward c I )roduced by the centripetal fort 


tom^r^^ 

SSliaiiSftsB 

t neglecting mctiou, gravity, the resistance of the air, etc. 


























CENTRIFUGAL FORCE. 


369 


whence we have 


% centripetal acceleration in time t — 


a b 2 
ad' 


The same proportion holds good if we let 

ab = the velocity of revolution in feet per second 
an= half the centripetal acceleration in one second. 

mi • aft 2 a ft 2 

Therefore, since an =—5- or 2 an = - 

ad %ad 

centripetal acceleration velocity 2 of revolution in feet per second 
given in one second - radius ac in fat * 

But (see Art. 9, page 1510) forces are proportional to the accelerations which 
they can produce in a given body in a given time: and gravity (or the weight 
of the body) would, in one second, give to it an acceleration, g, of about 32.2 feet 
per second (see page 362). Therefore, 

n . centrip accel . . n . vel 2 of rev in ft per sec .. weight . centrip or 
** ' given in 1 sec '' 3 ’ radius iA feet ' body ' centrif force 
or, 


centrifugal force — weight X 
as in the first formula. 


velocity 2 of revolution in ft per sec 
radius X <7 


(2d) Velocity in feet per second 

revolutions per minute X circumference in feet 

60 (seconds per minute) 
revolutions per minute X radius in feet X 2 tt 
= 60 ’ 
Hence, velocity 2 in feet per second 

revolutions 2 per minute X radius 2 in feet X 47r 2 

3600 

. revolutions 2 per minute X radius 2 in feet X ^r 2 
— 900 

But, by the first formula, page 368, 


velocity 2 in feet per second 
radius X g 

revolutions 2 per minute X radius 2 X n 2 
radius X g X 900 

revolutions 2 per minute X *r 2 X radius 
900 g • 


centrifugal force = weight X 
Hence, 

centrifugal force = weight X 
= weight X 

as in the second formula. 

The centrif force thus found will be in lbs, tons etc, according as the wt of the 
body is taken in lbs, tons etc. 

Caution. If the dimension at of the body, Figs 1 and 2, measd in the direc¬ 
tion of the rad ca, does not exceed about one-third of the entire rad ca, we may, 
near enough for ordinary practice, measure the rad from the cen c of the circle 
to the cen of grav n of the body Fig 2, or to the cen of grav n of cross section 
of a revolving ring Figs 1 and 4. 

When at is equal to one-third of ac, the rad thus found will be but about one- 
fiftieth part too short; when one-half of ac , about one-nineteenth part too short; 
and when two-thirds of ac, about one-ninth part too short. But when, as is 
often the case, as in fly-wheels, etc, at is much less than one-third of ac, the 
error is not worth notice in practice. 

But if greater accuracy is required, or if, as in Fig 4, at is greater than about 

one-third of ca, use ilie radius of g-yration, see p 366, and the 

velocity of the center of gyration. 

Velocity of centerof gyration Velocity ofouter particles w radius of gyration 
in feet per second = in feet per second X outer radius 

ca Figs 1, 2 and 4 


Revolutions per minute 
60 


X radius of gyration X 2 n 


The ring Fig 1, or the body, Fig 2, must be united to the center c, either by arms, or by a string 
or wire, etc, and the weight of these arms, etc, will slightly shorten the radius of gyration. But in 
practice this effect is usually too small to be regarded ; or a trilling allowance is made for it by guess. 




























FRICTION. 


370 


FRICTION. 

Art. 1. When one rough body rests upon another, the projections and de¬ 
pressions forming tlie rough nesses of their surfaces of contact interlook 
to a greater or less extent; and, in order to slide one over the other, we must 
expend a portion of the sliding force, either in separating the bodies (as by lift¬ 
ing the upper one) sufficiently to clear the projections, or in breaking off some 
of the projections and clearing the others. 

Thus, lei li fig. lap. 318 e, represent one of the minute projections (highly magni¬ 
fied) on the surf of the lower body: a one of those of the upper body; and Fg 
a force tending to slide the upper body hor over the lower one. To do so it must 
separate the two bodies (by pushing the upper one up the inclined plane st) un¬ 
til a clears t; but it may reduce the height of this lift by grinding or breaking 
off a part off or of a. It is immaterial whether the surf of contact is hor, in¬ 
clined or vert. The separation of the two bodies must take place in opposition 
to whatever force, acting perp to the surf of contact, tends to hold them to¬ 
gether. 

The surfs of all bodies are more or less rough. Hence this interlocking takes 
place to some extent between even the smoothest surfs. Without it, the most 
powerful vise could not prevent the lightest wt front falling out of its jaws; and 
the smallest conceivable force would slide the greatest conceivable wt. 

The resistance which the sliding force thus encounters is called friction.* 

Art. 2. Friction always tends to pi'event relative motion of the two bodies 
between which it acts; i e, motion of one of the bodies relatively to the 
other. In doing so, however, it tends equally to cav-se relative motion be¬ 
tween each of those two and a third, or outside body. Thus, the fric between 
a belt and the pulley driven by it, tends to prevent slipping between thorn; 
but thus tends to make the belt si ip on the driving pulley, and sets the 
driven pulley and its shaft in motion relatively to the bearing in which the shaft 
revolves. This motion is resisted by the fric between journal and bearing ; and this 
fric, in turn, tends equally to make the bearing revolve with the journal, and to 
make the belt slip on the driven pulley. 

Art. 3. _ The fric between two bodies at rest relatively to each other is called 
static friction, or fric of rest. That between two bodies in relative motion 
is called kinetic friction or fric of motion. 

Art. 4. (a). The ultimate or maximum static fric between two 
bodies, as U and L Fig 1, (or the greatest fric resistance which they are capable 

of opposing to any sliding force when at rest) is 
equal to a force (as that of the wt F) which is just 
upon the point of making U begin to slide upon 
L.f Thus fric, like other forces, may be expressed 
in weights, as in lbs. 

(1») A resistce cannot exceed the force which it 
resists.;}; Therefore if F is less than the ult static 
fric bet ween U and L, the frictional resistce actually 
exerted by them is also less. When F is = the ul't 
fric (and U is therefore on the point of sliding) the actual resistce is = the ult 
stat fric. If F exceeds the ult stat fric, the excess gives motion to U. 

Art. 5. If, when a body is in motion, all extraneous forces and resistces are 
removed or kept in equiiib, it moves at a uniform vel. Hence, if the force, F 
Fig 1, is just = the ult kinetic fric between U and L, their vel is uniform. If F 
exceeds this, the excess accelerates the vel. If the ult kinetic fric exceeds F, the 
excess retards the vel. Thus tlie actual fric resistce exerted by two 
bodies in relative motion is = their ult kinetic fric = that force (as F) which 
can just maintain their relative vel uniform. 

Hence, if the hor surfS upon which L rests, could be made perfectly friction¬ 
less, the pres of L against the lug m (which would then always be = the actual fric 
resistce between U and L) would also be = their ult fric so long as U continued in 
motion over L, and might therefore be greater or less than or = F; but when 


miii/iu 

er 


IF, 


IS 



Fig.l 


* Friction" (meaning rubbing) is a misnomer in so far as it implies that rubbing must take 
place in order to produce the resistance. For we meet this resistance, not only during rubbing, tpit 
also before motion (or rubbing) takes plaoe. “ Resistance of roughness” would better express its 
nature. 

t We here neglect the fric of the string and pulley, and assume that all the force of the wt F is 
transmitted by the string to U. 

{ If a resisting force exceeds the force resisted, the excess is not resistce, but motive force. 









FRICTION. 


oTT 


U was at rest the pres against m would be 
the ult trie. 


F, and less than (or at most just =) 


Art. 6. Since no surface can be made absolutely smooth, some separation of 
the two bodies must in all cases take place in order to clear such projections as 
exist. Hence the trie is always more or less affected by the amount of the perp 
pres which tends to keep them together. 

The proportion which the ult frtc, in a given case, bears to the perp pres, is 
called the coefficient of friction for that case. Or, 


Coefficient of friction = 


ultimate friction 


perpendicular pressure 


and 


Ultimate friction = perp pres X coeff of fric 


Thus, if a force F Fig 1, of 10 lbs, just balances the ult fric between U and L, 
and if the wt of U (the perp pres in this case since the surf bet ween U and L is 

hor) is 50 lbs, then the coeff of fric between U and L is = „ 8 = .2. 

' 50 lbs 

The coefT is usually expressed decimally, or by a common fraction; 
but sometimes, as in the case of railroad cars and engines, in lbs (of fric) per ton 
(of perp pres). Or by the “angle of fric” in degs and mins. (Art. 61, p 353.) 

The coeff is diff for diff materials; and, in a given material, varies with the 
smoothness, cleanness, dryness etc of the surf. 

Art. 7. The coefT of static fric may he found experimentally, 
either as in the above example, or by inclining the surf of contact, as in Art. 61, 
p 353. Expts on fric with unguents cannot well be made on a small scale in the 
latter way; on account of the stickiness or cohesion of the unguent. 

Art. 8. (a) To find the coeff of hinetic fric, allow one of the bodies, 

U Fig2, to slide down an inclined plane AC 
formed of the other one and having any con¬ 
venient known steepness ACE greater than 
the angle of fric (Art. 61, p 353). Note the vert 
dist AE through which U descends in sliding 
any dist as AC, (A E = AC X sine of A C E. 
table of sines etc, pp 60 etc); also its actual 
sliding ve\ in ft per ^ec on reaching C. Calcu¬ 
late the vert dist AD through which it would 
have to descend along the plane (from A to B) 
velocity 2 in ft per sec 



D 

E 


to acquire that vel if there were no fric. ^A D — — 




wice the accel g of grav*, 

Find DE( = AE —AD), and the hor dist EC corresponding to AC (EC = AC X 
cosine ACE = l/AC 2 — ATE 2 ). Then 

DE 

Coeff of the average fric in sliding from A to C = 

because if we let A E represent the total sliding force expended (in moving U 
from A to C and in overcoming the fric); then A D represents the portion of 
A E expended on vel; and DE that expended on fric , and therefore - the fric 

A E sliding force 
itself. And (Art. 62, p 353) ^ pre s ' 

„ DE friction , r 

Hence =-—— = coeff. 

EC perp pres 

(to) Or, find the sine and the tangent (table pp 60 etc) of ACE; and the dist 
AC = time 2 in secs X h V* X sine of A CE) through which U would slide m a 
trjvpn time if there were no fric. Measure the dist A15 through >vliieli it uctimly 
slides in that time; and find BC = AC AB. Then 


coeff of the average 
fric in sliding from A to B 


| = tan DCE = tan ACEX 


BC 

AC 


because 


* g = about 32.2 ; 2 g = about 64.4. 


















372 


FRICTION. 


(1st) AC : A B : BC 


A E : A D : D E 


: ' due : the actual velocit ? : the f riclional retardation 


:: the total sliding force 


sliding force employed 
in giving the actual 
velocity 


the friction, or the sliding 
force required to balance 
the friction. 


And, if A E is = the total sliding force, then E C is = the perpendicular pressure; 
aud (Art. 62, p. 353.) p g 

the coefficient of friction = tangent of D C E. 


E C 


(2nd) Owing to the similarity of the two triaugles, A B D and ACE, we have 
- - - - - - AEDE 


AC: BC:: AE: DE 


: tangent ACE: tangent D C E. 


EC EC 

Art. 9. In 1831 to 1834, Gen*l Arthur Klorin* experimented with 

pressures not exceeding about 30 lbs per sq in; and arrived at the following 
conclusions in regard to sliding fric where the perp pres is considerably less 
than would be necessary to abrade the surfs appreciably. These were for a long 
time generally regarded as constituting the three fundamental laws of 
frie. But see Art. 11, p 374. 

1st. The ult fric between two bodies is proportional to the total perp force 
which presses them together; i 6, the coeff is independent of the perp 
pres and of its intensity (pres per unit of surf). Hence 

2d. For any given total perp pres, the coetr is independent of the 
area of surf in contact. 

If upon a hor support we lay a brick, measuring 8X4X2 ins, first upon its 
long edge (8X2 ins) and then upon its side (8X4 ins), we double the area of 
contact, while the total pres (the wt of the brick) remains the same, and thus re¬ 
duce the pres per sq in by one-half. Consequently (the coeff remaining practically 
the Same) we have only half the fric per sq in. But we have twice as mauv sq ins 
of contact, and therefore the same total fric. 

But if we can increase or diminish the area of contact without affecting the pres 
per sq in, the total pres will of course vary as the area, and the total fric will vary 
in the same proportion, for the coeff remains the same. Thus, if we place two 
similar sheets of paper between the leaves of a book (taking care not to place 
both sheets between the same two leaves) and then squeeze the book in a letter¬ 
copying press, it will require about twice as much force to pull out both sheets 
as to pull out only one of them. 

3d. Although thecoeff o i static fric between two bodies is often much greater 
than their coeff of kinetic fric; yet the coefT of kinetic fric is inde¬ 
pendent of the vel. 

This applies also (approx) to the fric, and hence to the work (in foot -pounds etc) 
of overcoming fric through a given (list; for then the work (== resistce X di«t) is 
independent of the vel But in a given time, the dist (and consequently the 
work also) of course varies as the vel. See Art. 21, p 374/. 4 y 

Art. 10. (a) Some kinds of surfaces appear to interlock their projections 
much more perfectly when at rest relatively to each other, than when in even 
very slow motion ; and in some cases the degree of interlocking seems to in¬ 
crease with time of contact. Hence there is often a great diff in amount between 

Initu nl'f 0 “ 01 °“' Thus, Gen’l Morin found that with oak upon 
rlV a, bl i i> f riie two pieces at right angles, the resistce to sliding while stiff at 
rest, and alter being for some time in contact,” was about one eighth greater 
than when the pieces had a relative vel of from 1 to 5 ft per sec. S S 

(b) But experience shows that even very slight jarring suffices to remove this 
d.ft; and since all structures,even the heaviest, are subject to occasiona? fwrini 
(as a bridge, or a neighboring building, or even a hill,' during the passage of a 
train , or a large factory by the motion of its machinery ; or in numberless cases 
action of the wind) it is expedient, in construction, not to rely on fric for 
stability any further than the coeff for moving fric will justify. When it is to be 
* resistce, which we must provide force for overcoming, it should be 
taken at considerably more than our tabular statement, p. 373. 


New S York, S i860. ndamenUl IdeaS ° f Mecha,lics translated by Jos. Bennett; D. Appleton & Co„ 













FRICTION. 




! Table of moving- friction, of perfectly smooth, clean, 
itlry, plane surfaces, chiefly from Morin. 


ami 


Materials Experimented with. 


Coeff of 
Fric: or 
Propor¬ 
tion of 
Fric to the 
Pres. 


Oak on oak ; all the fibers parallel to the motion. 

“ “ moving fibres at right angles to the others; and to the motion. 

“ “ all the fibres at right angles to the motion. 

“ “ moving fibres on end; resting fibres parallel to the motion.... 

“ cast iron, fibres at right angles to motion. 

Elm on oak, fibres all parallel to motion. 

dak on elm, “ “ “ . 

Elm on oak, moving fibres at right angles to the others, and to motion. 

4sh on oak, fibres all parallel to motion. 

Fir on oak, " “ “ “ . 

Beech on oak “ “ “ “ . 

Wrought iron on oak, fibres parallel to motion. 

Wrought iron on elm, “ “ “ “ . 

Wrought iron on cast iron, fibres parallel to motion. 

“ “ on wrought iron, fibres all parallel to motion. 

Wrought iron ou brass. 

IWrouglitiron on soft limestone, well dressed. 

“ “ “ hard “ “ “ . 

ti ti *i «« <* “ “ wet.... 


“ “ or steel on hard marble, sawed. By the writer.about.. 

“ “ •< “ “ smoothly planed, and rubbed mahogany, fibres par¬ 
allel to motion. 

*< << 11 “ 11 smoothly planed wh pine....... 

ast iron on oak, fibres parallel to motion. 

“ “ “ elm, “ “ “ “ . 

“ “ “ cast iron. 

“ “ “ brass. 

Steel on cast iron.... 

Steel on steel. By the writer.•. 

Steel on brass... ; ... 

Steel on polished glass. By the writer.about.. 

quite smooth, but not polished; ou perfectly dry planed wh pine, fibres 

parallel to motion.about.. 

“ quite smooth,but not polished; on perfectly dry planed and smoothed 

mahogany, fibres parallel to motion..,.about.. 

fellow copper on cast iron. 

“ “ on oak..... 

Brass on cast iron. 

“ on wrought iron, fibres parallel to motion..._. 

“ on brass.,. 

“ on perfectly dry planed wh pine, fibres parallel to motion.about.. 

k i* •> “ “ and smoothed mahogany, fibres parallel to mo¬ 
tion.,..,.about.. 

Polished marble on polished marble. By the writer.average. 

• • “ on common brick.. “ — 

Common brick on common brick. “ . 

Soft limestone well dressed, on the same.. 

Common brick, on well-diessed soft limestone.. 

<• «' “ “ hard “ ..■. 

3ak across the grain, on soft limestone, well dressed. 

• • “ “ “ “ hard “ “ “ . 

Hard limestone on hard limestone, both “ M .‘. 

“ “ “ soft “ “ “ “ . 

Soft “ “ hard “ “ “ “ .. 

Wood on metal, generally, .2 to .62,.,...mean.. 

Wood, very smooth, on the same, generally, .25 to .5 . j( •• 

Wood, “ “ on metal, “ .2 to .62. ( •• 

Metal on metal, very smooth, dry “ .15 to .22... ‘‘ 

Hasonry and brickwork, dry " .6 to .7 . •• 

■ « •• •• with wet mortar.about.. 

it it a “ slightly damp mortar. “ •- 

“ on dry clay. (i •• 

“ “ moist" ...•.*- •• 

Marble, sawed ; on the same ; both dry. By the writer.#.average 

.. .. . both damp. “ . “ “ 

«> “ on perfectly dry planed wh pine. 

<< " on damp planed wh pine. 

<< polished, on perfectly dry planed wh pine “ . " •• 

White nine, perfectly dry : planed; on the same; all the fibres parallel to 

motion ...about.. 

“ “ damp, planed ; on the same.. “ •• 


Angle of 
Fric. 


.*. 


.16 


.24 

.16 

.44 

.64 

.64 

.65 

.60 

.38 

.38 

.38 

.67 

.65 

.41 

.38 

.41 

.18 

.65 

.47 

.74 

.51 

.33 

.4 

.55 

.45 

.6 

.26 

.4 

.6 


9 6 


13 30 

9 6 

23 45 
32 38 

32 38 

33 2 

31 00 
20 48 
20 48 
20 48 
33 50 
33 2 

22 18 

20 48 
22 18 

10 12 

33 2 

25 30 
36 30 

27 00 
18 15 

21 49 

28 49 

24 14 
31 00 

14 35 

21 48 
31 00 


* But after a few trials the surfaces become so much smoother as to reduce the angles as much as 
rom 2° to 5°; the sliding blocks weighing about 30 fi>s each. 
























































































FRICTION. 


3Y4 


Art. 11. Recent experiments, with much greater variations of pres ami of 
vel, and with more delicate apparatus for detecting slight changes in the coeff, 
although giving conflicting results,* show that the three laws in Art. 9 
are far from correct for surfs moving at high vels, and under great pres; and 
that they are only approximately correct for ordinary vels and 
pressures; for the coeff is found to vary both with the intensity of the pres and 
with the vel, as also witli the temperature.* But in the cases with which the 
civil engineer has mostly to deal, slight diffs in the character of the surfs, or 
even in the dampness of the air, will often cause much greater changes of coeff 
than those due to any probable changes of pres, vel and temp: so that, within 
the limits of abrasion, we may generally take Morin’s rules as sufficiently cor- i 
rect for such cases. ' 


Art. 12. Prof. A. S. Kimball, of 

the YVorcestet (Mass) Inst of Industrial 
Science, has made some very delicate experi¬ 
ments upon the fric between surfs of pine 
wood.f The results are given in Fig 3. 
merely to show how the coeff varied 
vel and pres. Our table gives a coeff 
for pine on pine. 



100 00 80 70 CO 50 40 30 20 

^ eloeity of Sliding, In Inches per second. 


Line A shows coeffs at diff vels under a pres of 1.58 lbs per so in. 

“ B “ “ ** ] 59 «» «4 

“ C “ “ “ i.fio “ “ 

“ p " “ “ j pj «• t< 

“ E “ “ “ 4.i7 « »• 

It. will be seen that at low vels the coeff decreased when the pres per sq in was 
almost imperceptibly increased; but this diff disappeared as the vel increased. 
At vels from 4 to 1‘20 ins per sec, the coeff generally decreased as the vel in¬ 
creased ; rapidly at first, but more slowly as the vel became greater. This agrees 
with other recent expts. But at very low vels (.08 to 5 ins per sec) Prof. Kimball 
found the coeff (line E) increasing very rapidly with the vel. 

We have made the scale of coeffs large in order to show their variations, which 
are so slight that they would otherwise be scarcely perceptible. Less delicate 
expts would have failed to show them at all. 

Art. 13. (a) In 1878 Capt. Douglas Gallon and Mr. George West* 
ingbouse, Jr., made careful experiments in England to ascertain the effect of 
friction in connection with railway brakes. J The friction and pressure were 


* This is not surprising in view of the extent to which the coeff is affected by the nature of the 
surf. If the shape of the minute projections is such that they fit into each other'as perfectly under 
small pressures as uuder great ones, and if they are too strong to be broken by the pressures applied 
the coeff, as stated in the 1st law, should he independent of the pres. Hut if high pres wedges the 
projections of one body more closely between those of the other, the coeff should increase under such 
pres. On the other hand, if the higher pres breaks down the projections while the lower ones are 
unable to do so. the coeff should decrease under the higher pres. The particles thus broken off may 
either act as a lubricant and thus still further reduce the fric and its coeff, or (if angular and hard) 
may increase it. Change of area of contact, under a given total pres, may, by affecting the intensity 
of the pres, make changes in the coeff similar to those just mentioned. 

At high vels the roughnesses have not time to interlock as perfectly as at low vels. Hence we 
should expect a less coeff at high vels. But high vel generally increases the number of projections 
broken away ; and these may either increase or diminish the' coeff, as explained above. High vel 
often indirectly affects it by increasing the temperature. 

t Sillimans Journal (American Journal of Science) March 1876 and May 1877. 
t See Prnc, Instn of Mechl Kngrs, London, June and Oct 1878 and April 1879: and “Engineer- 
ing, London, 1878; vol. 2b, pp 432, 469, 490; vol. 26, pp 153, 386, 395. 6 












































































FRICTION. 


374a 


automatically recorded by means of hydraulic granges. With cast iron brake-blocks 
land steel-tired wooden wheels, 43*4 inches in diameter, they found coefficients about 
las shown in Pig. 4. 

The points in lines A, B and C show the average brake coeffs. or coeffs of slid¬ 
ing fric between the tread of a rolling wheel and the brake-block. 


Speed of Car, in miles per hour. 



Speed of Car, in miles per hour. 


Line A shows brake coeffs obtained irnmed’y after application of brake 
“ B “ “5 secs “ “ 

15 “ “ “ 


B 

C 


D shows rail coeffs or coeffs of sliding fric between the tread of a slid¬ 
ing or “ skidding" wheel (held fast by the brake) and the rail. 

'(h) From lines A B and C it appears that the brake coeff obtained at a 
riven length of time after the application of the brake was generally greater 
at low than at high vels. But where the vel was maintained uniform 
he brake coelf diminishert as block and wheel remained 
longer in contact. Thus, lines A and B show that at 37^ miles per hour 



.096 at x). . ,. ,, , 

The diminution of the rail coeff with length of time of application of brake, 

tvas scarcely noticeable. 

(c) When the brake fric (owing to the reduction of vel and consequent m- 
irease of coeff) becomes = the “ adhesion ” or static fric between the rail and 
he tire of the rolling wheel, the vel of rotation rapidly falls below that due to 
the vel of the car; i e, the wheel begins to “skid” or slide along the 
rail: and in from .75 to 3 secs the rotation of the wheel ceases entirely. 

(d) The rail coeff, line D, is generally much less than the 
brake coeff lines A, B and C. The pres on the rail ( = the wt on a wheel) 
was about 5000 lbs per sq in, or greatly in excess of the limit, of abrasion. That 
U the brake was about 200 lbs per sq in. A few expts were made with brake 
Dlocks having but £ of the usual area of contact, and therefore 3 times the pres 
per sq in under a given total pres. They failed to show conclusively that this 
jaused anv marked change in the coeff. 

(e) The rail coeff line D, like the brake coeff, increases as the vel 
diminishes; slowly at first., but much more rapidly as the speed becomes 
less: until, at the moment ofstopping.it is generally even greater than the 
arake coeff just before skidding. With steel tires on iron rails at high vels it was 
somewhat greater than on steel rails, but this dill disappeared as the vel dimin- 

(f ) Locomotives overcome resistces = from l to ^ or more of the wt on 
ill the drivers; i e, they have a coeff of .33 or more, although the experimental 
;oeff for steel on steel in motion at low pres, is only about .15. But, the cases are 
io diff that, a similarity in their coeffs could hardly be expected. The great wt, 
lay from 2 to 6 or even 7 tons, on a driver, is concentrated on a surf (where the 
wheel touches the rail) about 2 ins long X about } inch wide, or = sav 1 sq in. 
rile pres per sq in thus greatly exceeds not only that upon which the tables are 
jased, but also the limit of abrasion. Besides, any point in the tread, during 


27 


































































































































































































































574 b 


FRICTION. 


the instant when it is acting as the fulcrum for the steam pres in the cyl, is 
stationary upon the rail, its trie (miscalled “ adhesion ”) is therefore static. 

Capt. Galton found that the coeff of “ adhesion ” was independent of 
the vel, and depended only on the character of the surfs in contact. With a 
four-wheeled car having about 5000 Ihs load on each wheel, it was generally over 
.20 on dry rails; in some cases .25 or even higher. On wet or greasy rails, with- ji 
out sand, it. fell as low as .15 in one case, but averaged about .18. With sand i 
on wet rails it was over .20. Band applied to dry rails before starting gave .35 « 
and even over .40 at the start, and an average of about .28 during motion; hut i 
sand applied to dry rails while the car was in motion was apt to be blown away 
by the movement of the car and wheels. 

(g) Owing to the constancy of the coeff of “ adhesion ” under given conditions { 
of tire and rail, the brake fric necessary to “skid ” the wheels in any case was ‘ 
also practically constant for all vels. But at high vels, owing to the lower brake 
coeff, a higher brake pres was reqd to produce this iixed amount of brake fric j 
Tlie skidding also reqd a longer time than at low speeds. 

Art. 14. If the pres is sufficient to produce abrasion (indeed, while it is s 
much less) the frie often varies greatly, but no precise law has yet been discov¬ 
ered for estimating it. Rennie gives the following table of coeffs of fric ti 
of dry surfaces, under pressures gradually increased up to i 
the limits of abrasion. It will be noticed that in this table the i 
coelf generally increases with the intensity of the pres: 

Coeffs of friction of dry surfaces, under pressures grad¬ 
ually increased up to the limits of abrasion. (By G. Rennie, C E ; J 


Fres. in Lbs. 

Wrought Iron 

Wrought Iron 

Steel 

Brass 

per 

on 

OD 

on 

on 

Square Inch. 

Wrought Iron. 

Cast Iron. 

Cast Iron. 

Cast Iron. 

32.5 

.140 

.174 

.166 

.157 

186 

.250 

.275 

.300 

.225 

224 

.271 

.292 

.333 

.219 

336 

.312 

.333 

.347 

.215 

448 

.376 

.365 

.354 

.208 

560 

.400 

.367 

.358 

.233 

672 

. 

.376 

.403 

.233 

700 


.434 


2*54 

784 




232 

821 




.273 


Art. noiiing ineiion, or that between the circumf of a roll 

jug body and the surf upon which it rolls, is somewhat similar to that of 
pinion rolling upon a rack. In disengaging the interlocking projections or it 
lilting the wheel over an obstacle o, Figs 5 and 6, the motive force F, instead of 
dragging one over the other, as in big 2. p. 318/, acts at the etid of a’bent level 
FR W Figs 5 and 6, the other end W of which acts in a direction pern to tin 
contact stirf; and in practical cases of rolling fric proper the leverage RW of 
the resisting wt of the wheel and its load is very much less, in proportion t( 
that (FR) of the force F, than in our exaggerated figs. Hence the force F reqc 
to roll a wheel etc is usually very much less than would be necessary to slide it 

. There are usually two ways of applying the force ifa overconv 
tng rolling fric: 1st (Fig 5) at the axis of the rolling body; as the force of i 

horse is applied *at the axle of s 


-.f 


°i 

R 

w / 







Fig. <3 


Fig. 6 


wagon-wheel; or that of a man at tin 
axle of a wheel-barrow : 2d (Fig 6) a: 
the circvmf ; as when workmen pusl 
along a heavy timber laid on top of 
two or more rollers; or as the endsol 
an iron bridge-truss play backwart 
and forward by contraction and ex 
pansion, on top of metallic rollers o 
halls (p614). In Fig 5 we have, in ad 
dition to the rolling fric of the cir 
cumf of the wheel on its support, tin 
sliding fric of the axle in its bearing 
In lig 6 we have only rolling fric 


hut at both top and bottom of the wheel. 

smooth V hIl-V roads St or e nf 0 ^ small >. as in the case of cart-wheels ot 
smooth hard roads, or of car-wheels on iron or steel rails, the leverag 




































FRICTION. 


374 c 


(F R) of F becomes, practically, in Fig 5 the radius, and in Fig 6 the diam, of the 
wheel; while that (R W) of the resistce is very small. Hence, neglecting axle 
frie in Fig 5, the force F reqd to overcome rolling fric in such cases is directly 

8 as w t W of and on the wheel, and inversely as the diam of the wheel. 

r The few expts that have been made upon the coeffs of rolling fric, apart from 
axle fric, are too incomplete to serve as a basis for practical rules. See Art 20. 
i and “Traction,” p 375. 

I (d) The fric (or ‘‘adhesion”) between wheel and rail, which enables a 
j locomotive to .move itself and train, or which tends to make a car-wheel revolve 
notwithstanding the pres of the brake, is a resistce to the sliding of the wheel 
on the rail ; and is therefore not rolling but sliding fric ; static when the wheels 

9 either stand still or roll perfectly on the rails; and kinetic when they slip or 
is skid ”. See Art. 13 (c, d, e and /). 

'! Art 16. The friction of liquids moving in contact with solid bodies 
is independent of the pressure, because the “lifting” of the particles 
of the fluid over the projections on the surf of the solid body, is aided by the 
is pres of the surrounding particles of the liquid, which tend to occupy the places 
!• of those lifted. Hence we have, for liquids, no coefF of fric corresponding with 
{ that (= resistce pres) of solids. The resistce is believed to be directly as the 
o area of surf of contact. Recent researches indicate that Resistce = a coetf'X 
e area of surf X vel», in which both n and the coefF depend upon the vel and 
upon the character of the surf; and that at low vels n = 1, but that, at a certain 
I “critical” vel (which varies with the circumstances) n suddenly becomes = 2, 

- owing to the breaking up of the stream into marked countercurrents or eddies. 
The resistance of fluid fric arises principally from the counter currents thus set 

- in motion, and which must be brought into compliance with the direction of 
the force which is urging the stream forward. 

Art. 17. Table of coefficients of moving friction of smooth 
plane surfaces, when kept perfectly lubricated. (Morin.) 


Substances. 

Dry 

Soap. 

Olive 

Oil. 

Tal¬ 

low. 

Lard. 

Lard & 
Plum¬ 
bago. 

Dak on oak, fibres parallel to motion. 

.164 


.075 

.067 


“ “ “ fibres perpendicular to motion. 

.... 

.... 

.083 

.072 


“ on elm, fibres parallel to motion. 

.136 

• . • • 

.073 

.066 


“ on oast iron, fibres parallel to motion. 

• . • • 

• • • . 

.080 



“ on wrought iron, “ “ “ . 

• • • • 

«... 

.098 



lieech on oak, fibres “ “ “ . 

.... 

• • • • 

.055 



iil m on oak, “ “ “ “ . 

.137 

• • • • 

.070 

.060 


) “ on elm, “ “ “ “ . 

.139 





“ east iron, “ “ “ “ . 

• • • • 

• • . • 

.066 



iVrought iron on oak, fibres parallel, greased and wet, .256. 






“ “ “ “ fibres parallel to motiou. 

.214 

.... 

.085 



“ “ on elm, “ “ *' “ . 

.... 

.055 

.078 

.076 


44 14 on cast iron, lt li il . 

.... 

.066 

.103 

.076 


“ “ on wrought iron, “ “ “ . 

.... 

.070 

.082 

.081 


“ “ on brass, fibres “ “ “ . 

.... 

.078 

.103 

.075 


Hast iron on oak, fibres parallel to motion. 

.189 





“ “ “ “ “ “ “ “ greased aud wet, .218 

.... 

.075 

.078 

.075 


“ “ on elm, “ “ “ “ . 

.... 

.061 

.077 

.... 

.091 

“ “ on cast iron, with water, .314. 

.197 

.064 

.100 

.070 

.056 

“ “ on brass. 

.... 

.078 

.103 

.075 


Hopper on oak. fibres parallel to motion. 

.... 

.... 

.069 



fellow copoer on cast irou. 

.... 

.066 

.072 

.068 


trass on cast irou. 

.... 

.077 

.086 



“ on wrought iron. 

.... 

.072 

.081 

• . . • 

.089 


• • • • 

.058 




Steel on east iron. 

.... 

.079 

.105 

.081 


“ on wrought iron. 

• • • • 

.... 

.093 

.076 


“ on brass . 

• • . • 

.053 

.056 

• • • • 

.067 

’aDned oxhide on cast iron, greased and very wet, .365 . 

• • • • 

.133 

.159 



“ “ on brass. 

.... 

.191 

.241 



“ “ on oak, with water, .29. 







% 

The launching friction of the wooden frigate Princeton was found by a 
sommittee of the Franklin Institute in 1844, to average about .007 or one-fifteenth 
)f the pressure during the first .75 of a second and .022 or one forty-fifth for the 
lext 4 seconds of her motion. The slope of the ways was 1 in 13, or 4 degrees 24 
ninuteg. They were heavily coated with tallow. Pressure on them = 15.84 lbs. 
ler square inch, or 22S0 lbs. per square foot. In the first .75 of a second the vessel 
did 2.5 iuches; in the next 4 seconds 15 feet 6.5 inches; total for 4.75 seconds 15.75 
eet. 




























































374c? 


FRICTION. 


Art. 18. The friction of lubricated snrfaces varies greatly with 

the character of the surls and wilh that of the lubricant and the manner of its 
application. If the lubricant is of poor quality, and scantily and unevenly ap¬ 
plied under great pres, it may wear away in places and leave portionsof the dry 
surfs in contact. The conditions then approximate to those of unlubricated , 
surfaces. But if the best lubricants for the purpose are used, and supplied reg¬ 
ularly and in proper quantity, so as to keep the surfs always perfectly separated, 
the case becomes practically one of liquid fric (Art. 16), and the resistce is very 
small. Between these two extremes there is a wide range of variations (see 
table, Art. 19 (d)), the coeff being affected by the smallest change in the condi¬ 
tions. Where any degree of accuracy is reqd, we would refer the reader to the 
experimental results given in Prof. Thurston’s very exhaustive work,* devoted 
exclusively to this intricate subject. i 

Art. 19. (a) Expts by Mr. Arthur M. Wellington upon the fric of ' 
lubricated journals f gave a gradual and continuous increase of coeff as 
the vel of revolution diminished from 18 ft per sec ( = a ear speed of 12 miles 
per hour) to a stop. This increase was very slight at high vels, but much more 
rapid at low ones ; as in Figs 3 and 4. At vels from 2 to 18 ft per sec the coeff 
was much less under high pressures than under low ones; hut at starting there 
was little diff in this respect. The coeff increased rapidly as the tempera¬ 
ture rose from 100° to 120° and 150° Fahr. 


(b) Prof. Thurston, also experimenting with lubricated journals t 
found that at starting, the coeff increased with increase of pres, as it did also 
when in motion, if the pres greatly exceeded the max (say 5U0 to 600 lbs per sq 
in) allow able in machinery. He also found that at high vels the coeff increased 
very slowly (instead of continuing to decrease) as the vel increased. 

(c) Prof. Thurston gives the following approx formulae for journal 
friction at ordinary temperatures, pressures and speeds, with journal and 
bearing in good condition and well lubricated: 


it 

I* 

li 

j| 

it 

ra 

« 


Coeff for starting; = (.015 to .02) X V P res i* 1 fbs P er sq in. 


Coeff when the shaft 

is revolving; 


.(.02 to .03) X |H ! r min 


j/pres in lbs per sq in. 


At pressures of about 200 lbs per sq in : 


Temperature of minimum 

fric; in Fahr degs 


3_ 

15 X 1/vel in ft per min 


I 


Caution. Tbe leverage, with which journal fric resists motion, in¬ 
creases with the diam of the journal. 


(d) The following figures, selected from a table of experimental results given 
by Prof. Thurston, merely show' the extent to w hich the coeff of 
journal fric is affected by pres, vel and temperature: and 
hence the risk incurred in rigidly applying general rules to such cases. In 
these expts the character of journal and hearing, the lubricant and its method 
of application, remained the same throughout. Where these vary, still further, 
and much greater, variations in the coeff may occur. 


Steel journal in bronze bearing;, lnbricated w ith standard 

sperm oil. 


I 

loa 

¥ 

mu 

He 

'till 

(to 

c* 


«r 

it 

a, 


• 

4) 

U 

S- 

'2 « 

cs _s 

h a 

W 


1300 

90° 


30 feet per minute 

Speed of revolution 

100 feet per minute JoOO ft per min 

1200 ft per min 

Pressures. 


200 

100 

4 

200 

100 

4 

200 

100 

200 

100 

lbs per sq in 

lbs per sq in 

lbs per sq in 

lbs per sq in 

Coeff 

Coetf 

Coeff 

Coeff 

Coeff 

Coeff 

Coeff 

Coeft' 

Coeff 

Coeff 

.0160 

.0044 

.125 

.0087 

.0019 

.0630 

.0053 

.0037 

.0065 

.0075 

.0056 

.0031 

.094 

.0040 

.0019 

.0630 

.0075 

.0061 

.0100 

.0150 


* Friction and r.n=t Work in Machinery and Mill Work. John Wiley & Sons, New York, 1885. 
i Trai.s Amor Soc of Civil liner*, Xew York. Dec. 1884, 
t Journal of the Franklin Institute, Nov, 1878. 



l 

k 


(l 

k 













































FRICTION. 


374 e 


(e) Where the force is applied first on one side of the jour¬ 
nal and then on the opposite side, as iu crank pins, the trie is less 
than where the resultant pres is always upon one side, as in fly-wheel shafts; 
because in the former case the oil has time to spread itself alternately upon both 
sides of the journal. 


(f) Friction rollers. If a journal J, in¬ 
stead of revolving on ordinary bearings, be sup¬ 
ported on friction rollers R, R. the force required 
to make J revolve will be reduced in nearly the 
same proportion that the diam of the axle o or 
o of the rollers, is less titan the diam of the 
rollers themselves. 

Mr. Wellington experimented with a patent 
hearing on this principle, invented by Mr. A. 
Higley. Diam of rollers RR, 8 ins; of their 
axles oo If ins; of the journal c, 3£ ins. Here, 
theoretically, 

• F . . . , .. . . , .. diam of axles oo 1? ins 

fric of patent journal == fric of 3£ in journal X ~-——-- = 4 — 

8 diam of rollers RR Sins 

or as 1 to 4.6. Under a load of 279 lbs per sq in, Mr. Wellington found it about 
as 1 to 4 when starting from rest; and about as 1 to 2 at a car speed of 10 miles 
per hour. 

Art. 20. (a) Resistance of railroad rolling- stock. This con¬ 
sists of rolling fric between the treads of the wheels and the rails (the treads 
also sometimes slide on the rails, as in going around curves); of sliding fric be¬ 
tween the journals and their bearings, and between the wheel flanges and the 
rail heads; of the resistce of the air; and of oscillations and concussions, which 
consume motive power by their lateral and vert motions, and also increase the 
wheel and journal fries. 

Its amount depends greatly upon the condition of the road-bed and rails (as 
to ballast, alignment, surf, spaces at the joints, dryness etc); upon that of the 
rolling stock (as towr carried", kind of springs used, kind and quantity of lubri¬ 
cant, condition and dimensions of wheels and axles etc); upon grades and curv¬ 
ature; upon the direction and force of the wind; and upon many minor con¬ 
siderations. Experiments give very conflicting results. 

(b) During the summer of 1878, Mr. Wellington experimented with 
loaded and empty box and flat freight cars, passenger and sleeping cars, and at 
/speeds varying from 0 to 35 miles per hour. The cars were started rolling (by 
grav) down a nearly uniform grade of .7 foot per 100 feet, or 36.5 ftet per mile 
and 6400 ft long. Their resistces were calculated as in Art, 8 (a). “The rails 
were of iron, 60 lbs per yd, and the track was well ballasted and in good line and 
surf, but not strictly first class.” The following approx figures are deduced 
from Mr. Wellington’s expts upon cars fitted with ordinary journals:* 

Car Resistance in pounds per ton (2240 lbs) of weight of 
train, on straight and level track in good condition. 



Speed of 
train in 
miles per 


Empty cars 



Loaded cars 


Axle, 

Oscilla¬ 

tion 



Axle, 

Oscilla¬ 

tion 



hour 

tire and 
flange 

and 

con- 

cuss’n 

Air 

Total 

tire and 
flange 

and 

con- 

cuss’n 

Air 

Total 

0 

14 

0 

0 

14 

18 

0 

0 

IS 

10 

6 

.6 

.4 

7 

4 

.6 

.4 

5 

20 

6 

2.7 

1.3 

10 

4 

2. 

1 . 

7 

30 

6 

5.3 

2.7 

14 

4 

4.7 

2.3 

11 


(c) With the Higley patent anti-fric roller journal, the resistce to starting was 
jut about 4 lbs per ton. But see Art. 19 (f). 


(d) About midway in the track experimented upon, was a curve of 1° de- 
lection angle (5730 ft rad) 3000 ft long, with its outer rail elevated 3 to 4 ins 


* Trnnsactions, American Society of Civil Engineers, Feb 1879. 



































FRICTION. 


374 / 


above the inner one. The rise of the outer rail was begun on the tangent, about 
500 ft before reaching the curve. In the first 500 ft of the curve the resistce was 
greater than that encountered just before reaching the curve, by from .6 to 2.1 
(average 1.1) lbs per ton. In the last 500 ft of the curve this excess had diminished 
to from .2 to .9 (average .6) lbs per ton. Owing to the continuance of the down 
grade on the curve, the vel increased as the train traversed the curve; but it 
does not clearly appear whether the decrease in curve resistce was due to the 
increase in vel*or to the fact that the oscillations caused by entering the curves 
gradually ceased as the train went on. 

(e) Mr. I*. M. Dudley, experimenting with bis “ dynagrapli ”* ob¬ 
tained results from which the following are deduced: 


91 

di 

ti 

of 

h 

»« 


Train Resistance in pounds per ton (22-10 lbs) of weight of 

train, including- grades. 


Description of train 


Loaded 

cars 

Empty 

cars 

Weight tons 
(2240 lbs) 

29 

2 

526 

37 

0 

633 

25 

2 

458 


Trip 

Average 

speed. 

Miles per 
hou r 

Average 

resist¬ 

ance. 

Toledo to Cleve¬ 
land. 95 miles 

20 

8.34 

Cleveland to Erie 
95.5 miles 

20 

7.67 

Erie to Buffalo. 
88 miles 

20 

8.89 


“With the long and heavy trains of the L. S. & M. S. Ry, of 600 to 650 tons, it 
reqd less fuel with the sameengine to run trains at 18 to 20 miles per hour than 
it did at 10 to 12 miles per hour”, owing to the fact that at the higher speeds 
steam was used expansively to a greater extent, and hence more economically. 

Art. 21. The work, in ft-lbs. reqd to overcome fric through 

any dist, is = the fric in lbs X the dist in ft. In order that a bodv,started slid¬ 
ing or rolling freely on a hor plane and then left to itself, mav do'this work ; ie, 
may slide or roll through the given dist, its kinetic energy (= its wt in lbsX its 
vel 2 in ft per sec -=- 2g\) must = the first-named prod. Conversely, the dist 1 
in ft through which such a body will slide or roll on a hor plane, is 

_ its kinetic energy in ft-lbs, at start 

fric in lbs 


wt, of bod y in l bs X initial ve l 2 in ft per sec _ initial vel 2 in ft per sec 
wt of body in lbs X coeff of fric X 2 p-f coeffof fric X 2 

dist in ft, so found dist in ft 


The time reqd, in secs, is = 


mean vel, in ft per sec £ initial vel in'ft per set 


Suppose two similar locomotives. A and B, each drawing a train on a leve 
straight track ; A at 10 miles, and B at 20 miles, per hour. The total resistce of 
each eng and train (which, for con venience, we suppose to be independent of 
vel) is 1000 lbs. Hence the force, or total steam pres in the two cvls reqd t< 
balance the fric and thus maintain the vel, is the same in eaeh eng.’ In travel 
ing ten miles this force does the same amount of work (1000 lbs X 10 mile: 
= 10000 pound-miles) in each eng, and with the same expenditure of steam it 
each; although B must supply steam to its cyls twice as fast as A, in order t< 
maintain in them the same pres. In one hour the force in A does 10000 lb-mile 
as before, but that in B does (1000 lbs X 20 miles =) 20000 lb-miles, and witl 
twice A’s expenditure of steam. 

But in fact the resistce of a given train is much greater at higher vels. Se< 
table Art. 20 ( b) And even if we still assumed the resistce to be the same a 
both vels, B must exert more force than A in order to acquire a vel of 20 mile 
per hour while A is acquiring 10 miles per hour. 


* An inst for measuring the strain on the draw-bar of a locomotive, or the force which the latte 
exerts upon the train. 

t g = acceleration of gravity = say 32.2 ; 2 g= say 64.4. See p 362. 




If 

bti 

fcj 

kit 

V: 

Wt 

k 

% 

kill 

flv 

Wt, 

ft 

111 

w, 

Ik, I 

til. 


to 
































TRACTION. 


375 


A double purchase crane, with a weight of 7000 lbs suspended from it, showed a fricof }j the weight, 
vr nearly 800 lbs. One ton suspended at each end of a chain passing over 2 cast-iron sheaves of 2 fcet. 

diaui, with wrought-iron journals, working in brass beariugs, well oiled, gave A- of the weight; or 
2 tons 14 

^—— 320 B»s ; or 160 lbs per ton. 

f Y,?ji r L sa - vs tlle fr * c of a s led on dry ground is % of the pres. Babbage states that a bl-ocik of stone 
or lo o lbs was drawu along a rock surface by 758 lbs: or fric 70 per ct: on a woodeu sled on a wooden 
noor, 00 per cent ; with both wooden surfaces greased, only 6 per cent; and with the block on top of 
iwooden rollers 3 ins diam, only 2.6 per cent, Kubbie masonry on wet clay .2 to .35. 


TRACTION. 


Traction on common roads, and canals; or the power rend to draw 

vehicles and boats along them. In connection with this subject read the preceding and tlie following 


The following table shows tolerable approximations to the force in lbs per ton,reqd to dTaw a stage 
toach and passengers, up asceuts on the Holyhead turnpike road in England, (a fine road,) bv horses ; 
is ascertained by means of a dynamometer. The entire weight was tons ; but in the table, the 
results are given per single ton. From the nature of such cases, no great accuracy is attainable. 


Proportional 

Ascent. 

Ascent in Ft. 
per Mile. 

At 4 Miles 
per Hour. 

At 6 Miles 
per Hour. 

At 8Miles 
per Hour. 

At 10 Miles 
per Hour. 



Lbs. 

Lbs. 

Lbs. 

Lbs. 

1 in 15X 

340.7 

210 

216 

225 

240 

1 “ 20 

264. 

196 

202 

212 

229 

1 “ 26 

203.1 

155 

160 

166 

175 

1 “ 30 

176. 

137 

142 

147 

154 

1 “ 40 

132. 

114 

120 

124 

130 

1 “ 64 

82.5 

109 

115 

120 

126 

1 “ 118 

44.7 

102 

107 

113 

129 

1 “ 138 

38.3 

99 

103 

109 

117 

1 “ 156 

33.9 

98 

101 

106 1 

112 

1 “ 245 

21.6 

93 

96 

101 

107 

1 “ 600 

8.8 

81 

85 

91 

96 

Level. 

0 . 

76 

80 

85 

91 


The following results, most of them with the same instrument, are also in Sis per ton ; with a four- 
wheeled wagon, at a slow pace, on a level; and the loads in fair condition. 

' On a cubical block pavement. 32 lbs per ton............to 50. 

*• Me Adam road, of small broken stone. 62 44 44 44 proba&ly to 75. 

44 gravel road. I’M “ u ** 

Telford road, of small stone on a paving of spawls 46 44 44 44 44 4 to. 

44 broken stone, on a bed of cement concrete. 46 44 44 44 44 44 75. 

44 common earth roads .. 200 to 300. On a iplank road JO, to 50 lbs. 


The tractive power of a horse diminishes as his speed in¬ 
creases; and perhaps, within certain limits, say from % to four miles per hour, 
dearly in inverse proportion to it. Thus, the average traction of a horse, on a level, and actually 
pulling for 10 hours in the day, may be assumed approximately as follows: 


Miles per hour. Lbs. Traction, 
% . 333.33 

1 . 250. 

IX. 200. 

IX.166.66 

1%.142.86 

2 .125. 


Mites per hour., Lbs. Traction. 

2 % .lll.U 

21C . 100 . 

2% . 90.01 

3 83.33 

3X. 71.43 

4 62.50 


If he works for a smaller number of hours, his traction may increase as the hours diminish ; down 
to about 5 hours per day and for speeds of about from IX to 3 miles per hour. Thus, for 5 hours per 
day his traction at 2% miles per hour will be 200 lbs, &e. When ascending a hill, his power dimin¬ 
ishes so rapidly, from having partiallv to raise his own weight, (which averages about 1O00 to 1100 
Tbs,) that up a slope of 5 to 1. he can barely struggle along without any load. Ou such an ascent, (see 
table, p 354, of Force in Rigid Bodies,) he must exert a force equal to 439 tbs per ton, or of 196 lbs tor 
the 1000 lbs of his own weight. Assuming that on a level piece of good turnpike, he would when haul¬ 
ing a cart and load, together weighing 1 ton, have to exert a traction of 60 fts; then on ascending a 
hill of 4° inclination, (or 1 in 14.3; or 36!)X ft per mile,) he would hare to exert 156 tbs, against the 
gravity of the 1 ton : and 67 lbs, against that or his own weight; or 223 lbs in all. He may, for a few 
initis, exert without injury, about twice his regular traction. This calculation shows that up a hill 
o f 4°’ an average horse is fully tasked in drawing a total load of one ton ; and should, therefore, he 
allowed in such a case, to choose his own gait; and to rest at short intervals. A fair load for a single 
horse with a cart, at a variable walking pace, working 10 hours per day, on a common undulating 
road in good order, is about half a ton. in addition to the cart, which will be about half a ton more. 
With two horses to this same cart, the load alone may be about IX tons. 

Kem. Since the action of gravity is the same on good roads and bad ones, it follows that 

ascents become more objectionable the better tlie roan is. 

Thus, on an ascent of 2°, or 184.4 ft pe’r mile, gravity alone requires a traction of 78 lbs per ton ; 








































376 


TRACTION, 


which is about 10 times that on a level railroad at fi miles an hour; but only about equal to that on a 
level common turnpike road, at the same speed. Therefore, (to speak somewhat at random,) it would 
require 10 locomotives instead of 1; hut only 2 horses instead of 1. A grade of 1 in 35; or 150 ft to a 
mile; or 1°38', is about the steepest that permits horses to be driven down a hard smooth road, in a 
fast trot, without danger. It should, therefore, not be exceeded except when absolutely necessary, 
especially ou turnpikes. 

On canals ami other waters, the liquid is the resisting medium that 

takes the place of friction on level roads. But uulike friction, its resistance varies as the squares ol' 
the vels; (see Art 26 of page 280.) at least from the vel of 2 ft per sec, or 1.364 miles per hour; tt 
that of 11 tjj ft per sec, or 7.84 in per h. As the speed falls below 1 % m per h, tbe resistance varies less 
and less rapidly; and this is the ease whether the moved body flouts partly above the surface; or is 
entirely immersed. In towiug along stagnant canals, Ac. the vel is usually from 1 to 2^ in per h; 
for freight mo3t frequently from 1to 2. Less force is required to tow a boat at say 2 m per h, where 
there is no current, than at say per h, agaiust a current of % m per h. because in the last case 

the boat has to be lifted up the very gradual iuclined plane or slope which produces the current. 

Tbe rorce required to tow a boat along a canal depends greatly upon the comparative transverse 
sectional areas of the channel, and of the immersed portion of the boat. When the width of a canal 
at water-line is at least 4 times tliatof the boat; and the areaof its transverse section asgreatas ut least 
6 3-S times that of the immersed transverse section of the boat, the towiug at usual canal vels will be 
about as easy as in wider and deeper water. With less dimensions, it becomes more difficult. (D'Au- 
buisson.) Much also depends on the shape of the bow and other parts of the boat; and on the propor¬ 
tion of its length to its breadth and depth. Hence it is seen that the mere weight of the load is by no 
means so controlling an element as it is on land. The whole subject, however, is too intricate to be 
treated of here. Morin states that naval constructors estimate the resistance to sailing and steam 
vessels at sea, at but from about .5 to .7 of a Jb for every sq ft of immersed transverse section, when 
the vel is 3 ft per sec, or 2.046 miles per hour. It is far greater on canals. 


On the Schuylkill Navigation of Pennsylvania, of mixed canal 

and slackwater, for 108 miles, the regular load for 3 horses or mules, is a boater very full build; and no 
keel: 100 rt long, 17ft beam ; and 8 ft depth of hold ; drawing 5M ft when loaded.* Weight of boat 
about 65 tons; load 175 tons of coal, (2210 lbs;) total weight 240 tons, or 80 tons per horse or mule. 
On the down trip with tbe loaded boats, for 4 days, the animals are at work, actually towing, (except 
at the locks,) for 18 hours out of the 24; thus exceeding by far the limits of time usually allowed for 
continuous effort. 


Ou the canal sections, (which have 60 ft water-line; and 6 ft depth,) the speed is 1 y. miles per hour* 
and on the deep wide pools, 2 miles. r ' 

On the up trip with the empty 65-ton boats, the average speed is about 2 14 miles per hour. The 
empty boats draw 16 to 18 ins water; and frequently keep ou without stopping to rest day or nicht 
through the entire distance of 108 miles. The animals generally have 2 or 3 days' rest at each end of 
the trip ; but are materially deteriorated at tbe end of the boatiug season. 

If our preceding assumption of 143 lbs traction of a horse at 1% miles per hour, is correct, the 

143 lbs 

1.83 lbs per ton. 


preceding assumption 
traction of the loaded boats on the canal sections is 


80 tons 

The intelligent engineer and superintendent of the Sch Nav, James F Smith, gives as the results 
or his own extensive observation, that ono or these large boats loaded (240 tons in all) mav without 
distressing the animals, be drawn along the canal sections, for 10 hours per dav, as follows’* Bv one 
average horse or mule, at the rate of 1 mile: by two animals, at 1 J 4 miles; and by three atl 4 f miles 
per hour. M hen four animals are used the gain of time is very trifling. At a time of rivalry among 
the boatmen, one of them used 8 horses ; but with these could not exceed 2 M miles per hour in the 
--.portions. Two or more horses together cannot for hours pull as much as when working sepa- 

If our preceding short table of the traction of a horse at diff vels for 10 hours is correct then th# 
traction of the above loaded coal boats (240 tons) on the canal sections of the navigation, is as follows • 
The last column shows the traction in lbs per sq ft of area of immersed transverse section where largest* 

f flDOUt JO 8Q it. ® I 


rses. 

1 . 

Miles per Hour. 

250 

Lbs. per Ton. 

Lbs. per Sq Ft. 

2 . 


3 3 3 



3. 


428 



3 ou pools 


3 75 



8 . 


8 00 



3 up-trip.. 

. 2% . 

300 




ILachine Canal, Canada, 120 ft wide at water-line; 80 ft at bottom ; dei>th 

mitre sills 9 ft; 6 horses tow loaded schooners with ease- , * 



«S* C R OS , t ? f , V.° at ?’ 1884 > (Schuylkill Canal) about $1800. Annual repairs about 

Boats last 16 to 20 years. Length 102 ft* lie#m 171 / ft- n / * r * cUmmII 

tons ; weight about 58 tons ; speed, with 3 mules,’1^ miles ptr hour. ’ A * ' Capa<nty ’ ls ' ] 

and^cost JSWTOO^ier^ni?^ fo^the^en^ar^ment only ^TheTcos^of the * rt: ,* 2 ft: a " d 7 « °f water; 
has ranged between $23000 and $500W p™r mile. J ' ‘ Caua ‘ 8 iu ^“usylvuuia 


I 

j 

i 


i 




: 


: 1 i 


































ANIMAL POWER. 


377 


m7le f r?"n a sma 1 " er . canal - (the boats nearly touching bottom,) the traction at 
lbs per sq nof Lmersed section 1 tW ‘ Ce great as the above L78 fts ’ U would be 5.7 

For traction on railroads, see Art 20, pp 374 e and 374/. 


ANIMAL POWER. 


„n£, r i* lm ®° f ar , a ® re R ards horses, this subject has been partially considered 

owfnc to'\he^iffTtreoTt < h's Tr T IOn 'H AI1 ® st ‘ ma ; es ou this subject must to a certain extent be vague, 
» d,ff stieugths and speeds of animals of the same kind; as well as to the extent of their 
“‘“S, 0 any particular kind of work. Authorities on the subject differ widely- and sometime* 
♦ hnthofif** el ' es 1 “ aIo ” se ® anner tha t throws doubt ou their meaning. We believe however 
To J lo !* n, Z willbe found to be as close approximation to practical averages as the’nature of 
wei^h- admas with our present imperfect knowledge. We suppose a good average trained horse 
we.gb.ng not less than about * a ton, well fed and treated. Such a one, when actually wafkjng for 
10 houis a day, at the rate of 2% miles per hour, on a good level road, such as the tow-path of a canal 

or a circular horse-path,* can exert a continuous pull, draught, power 
or traction, of lOO lbs. B '* wer ’ 

hi sdlv-s of fllu’ ‘t 220 ft pe . r mi , n - or 3 % ft P er sec l and si uoe 10 hours contain 600 min, 
nis day s work of actual hauling on a level, at that speed, amounts to 

min ft ft>s 

600 X 220 X 100 rr 13 200000 ft-fts per day. 

9 r ’ n °J “S* . ft ‘ Ib3 per sec ’ f Which means that he exerts force enough during the 

day to lift 13200000 fi>3 1 foot high ; or 1 320000 lbs 10 feet high ; or 132000 lbs 100 ft high" &c. He mav 
exert this force either in traction (hauling) or iu lifting loads. If he has to raise a small load to a 
great height, the machinery through which he does it must be so geared as to gain speed at the loss 
(commonly but improperly so expressed) of power. Whether be lifts the great weight through a 
small height, or the small weight through a great height, he exerts precisely the same amount of 
force or power. J 

Kxjierience shows that within the limits of 5 and 10 hours per day, (the speed remaining the same,) 

tne draft of a horse may be increased in about the same pro* 

portion as the time is diminished ; so that when working from 5 to 10 houn 
per day, it will be about as shown in the following table. Hence, the total amount of 13 200 000 ft lb# 
per day may be accomplished, whether the horse is at work 5, 0, or 8. &c, hours per day 1 This of 
course supposes him to be actually lifting or hauling all the time; and makes no allowance for swd- 
pages for any purpose. * 

Table of draft of a horse, at 2% miles per hour, on a level. 

Hours per day. Lbs. Hours per day. Lbs. 

. 100 7 . 142 2 - 

7 


9 

8 


111 


125 


.9 


6 

5 


166% 

200 


Experience also shows that at speeds between % and 4 
miles an hour, his force or draught will be inversely in pro¬ 
portion to his speed. Thus, at 2 miles an hour, for 10 hours of the day hi# 
draught will be *' ’ 

miles miles lbs lbs 
2 : 2% : : 100 : 125 draught. 

| w °uld be 166% Bis; at 3 miles, 83% lbs; and at 4 miles, 62% ffis; as per table I# 

Therefore, in this case also, the entire amount of his day’s work remains the sanie;§ and within 


I * To enable a horse to work with ease in a circular horse-walk, its diam 

j should not be less than 25 ft; 30 or 35 would be still better. 

t A nominal horse-power is 33000 ft-lbs per minute; this being the rato 

assumed by Boulton and Watt in selling their engines; so that purchasers wishing to substitute 
> steam for horses, should not be disappointed. Their assumption can be carried out by a very strong 
horse day after day for 8 or 10 hours; but as the engine can work day and night for months without 
stopping, which a horse cannot, it is plain that a one-horse engine can do much more work than any 
one such horse. Hence many object to the term horse-power as applied to engines; but since every¬ 
body understands its plain meaning, and such a term is convenient, it is not in fact objectionable. 
Boulton and Watt meant that a one-horse engine would at any moment perform the work of a very 
strong horse. An average horse will do bat 22000 ft-lbs per min. 
t It is plain that although the day's labor wiil be the same, that of an hour, or of a min, will vary 
I with the number of hours taken as a day’s work. It must, be remembered that a working day of a 
given number of hours, by no means implies, in every case, that number of hours of actual work • 
but includes intermissions and rests. 

? This remark about speed will not apply to loads towed 

through the water. Tims, if his draught at 2. miles an hour he 125 lbs ; and 

at 4 miles, 62% lbs; he will on land draw loads in these proportions ; but in hauling a boat through 
the water at the greater speed, he has to encounter the increased resistance of the water itself; which 
resistance at 4 miles is much more than twice as great as at 2 miles; probably 4 times as great. 
Therefore, at 4 miles on a canal, his draught of 62% lbs would not suffice for a load half as great os 
he could tow with his draft of 125 fi>s at 2 miles. 
















378 


ANIMAL POWER 


all the foregoing limits of hours anil speed, may be practically takea to be about 13 200 000 ft-lbs per i 
day ; or 22000 ft-lbs per min of a day of 10 hours. Hut it does not follow that the horse can always , 

in practice actually lift, loads at that rate ; because generally a part of his power is expended iu # 

overcoming the friction of the machinery which he puts in motion ; and moreover, the uature of tha 
work may require him to stop frequently ; so that iu a working day of 8 or 10 hours, the horse may 
not actually be at work more than 5, 0, or 7 hours. i |o 

As a rough approximation, to allow for the waste of force in overcoming the friction of hoisting 
machinerv, and the weight of the hoisting chains, buckets, &o. we may say that tile ll&cilll 

or paying daily net work of a horse, in hoisting- by a coin-, 11 

moil gin, is about 10 000000 ft-Bbs. That is, he will raise equivalent to 10000000 lbs net of* t . 
water, or ore, &c, 1 foot. The load which he can raise at once, including chaius, bucket, and an 
allowance for friction, will be as much greater thau his own direct force, as the diam of the horse- • 
walk is greater than that of the winding drum; nud it will move that much slower than be does. * 

His own direct force will vary according to the number of hours per day that he may be required to 
work, as in the foregoing table. With these data, the size of the buckets can be decided on ; and of q 
these there should be at least two, so that the empty one at the bottom may be filled while the full one „ 
at top is being emptied ; so as to save time. The same when the work is done by men. 

Art. 2. A practised laborer hauling along a level road, by “ 
a rope over his shoulders; or in a circular path, pushing before him a 1* 
hor lever, at a speed of from 1% to 3 miles per hour, exerts about % part as much force as a horse; 
or 2 200 000 ft-lbs per day; or 3666% ft-lbs per min of a day of 10 hours of actual hauling or pushing. 

But laborers frequently have to work under circumstances less advantageous for the exertiou of k 
their force thau when haulingor pushing in the manner just alluded to ; aud iu such cases they cauuot * 
do as much per day. Thus iu turning a winch or crank like that of a grindstone, or of a crane, the I tl 
continual bending of the body, and motion of the arms, is more fatiguing. The size Ol *1 
winch should not exceed IS ins, or the rad of acircle of 3 ft diam; and agaiust " 
it a laborer can exert a force of about 16 fils, at a vel of 2% ft per sec, or 150 ft per min, making very 1 
nearly 16 turns per min; for 8 hours per day. To these 8 hours au addition must be made of about 

% part, for short rests. Or if a working day is taken at 8, or 10, &c, hours, ^ part must generally be 
taken from it for such rests. On the foregoing data an hour’s work of 60 min of actual hoisting 
would be 

lbs ft min 

16 X 150 X 60 = 144000 ft lbs: 

. i it 

or, deducting part for rests, 115200 ft-fiis per hour of time , including rests. In practice, however, n 
a further deduction must be made for tte fric of the machine, and for the wt of the hoisting chains; K 
and in case of raising water, stone, ore, &c. from pits, for the wt of the buckets also. As a rough It 
average we may assume that these will leave but 100000 ft-lbs of payiug. or useful work per hour; 

that is, that h man at a wiucli will actually lift equivalent to ^ 
lOOOOO lbs of water, ore, Ac, 1 foot Iiigli'per hour’s time, in- ti 

Clu<lill<? rests. This is equal to 166G% ft-lbs per min of a day of 10 hours, including rests, to 
Therefore, in a day of 10 working hours he would raise l 000000 lbs net, 1 foot high; Or just J__ * ! 

pari of what a horse would do with a £'ill in the same time. We have up 

before seen that in hauling along a level road, he can at a slow pace perform about % of the daily lb 
duty of a horse. He may also work the winch with greater force, say up to 30 or even 40 Tbs; but lion 
he will do it at a proportionately slower rate; thus, accomplishing only the same daily duty. * 
\Vith a £ 111 , like those for horses, but lighter, with 2 or more buckets, a prac- 
tised laborer will in a working day of 10 hours, raise from 1 200000 to l 400 000 ft-lbs uet of water, ore, i Ci 
&c. With a shallow well or pit, more time is lost in emptying buckets than in a deep one; but the 

Wl ^ require a greater wtof rope. To save time in all such operations on a large scale, there & 
should be at least two buckets; the empty one to be tilled while the full one is being emptied. It. is 
em pl°y 2 or more men to hoist at the same time, by winches, at both ends of the axis; I 
and the men will work with more ease if the winches are at right angles to each other. Each winch 
handle mav he long enough for 2 or 3 men. An extra man should be employed to empty the buckets, to 
He may take turns with the hoisters. The same remarks apply in some of the following cases. 

4b u 2 t I read wh eel a practised laborer will do about 40 por cent more daily 
duty than at a winch ; or iu a working day * of 10 hours, including rests, he will do about 1 400000 ft.- ho 
lbs. And he can do this whether he works at the outer circumf of the wheel, stepping upon foot- (tfa> 
boards, or tread-boards, on a level with its axis; or walks inside of it, near its bottom. In both cases dr 
he acts by his wt. usually about 130 to 140 lbs; and not by the muscular strength of his arms. When,! 1 fc 
at the level of the axis, his wt acts more directly than when he walks on the bottom of the wheel; w 
but IQ the nr <t case he has to perform a slow and fatiguing duty resembling that of walking up a 
continuous flight of steps; while in the second he has as it were merely to ascend a very slightly in¬ 
clined plane; which he can do much more rapidly for hours, with comparatively little fatigue: and, j! 
this rapidity compensates for the less direct action of his wt. Therefore, in either case, as experience ! 
lias shown, he accomplishes about the same amount of daily duty. Treadwheels may be from 5 to 25 
it in diam, according to the nature of the work. They are generally worked by several men at once, 
and may at times be advantageously used in pile-driving, as well as in hoisting water, stone, &c. 

,O.V a tfood common pump, properly proportioned, a practised laborers 

will in a Jay of 10 working hour*!, raise about 10000D0 ft-lbs of water, net.t 

Hailing- witli a light bucket or scoot*, he can accomplish about \ 
200 000 ft-lbs net of water. By H bucket ami SWapfi. (along lever rocking vertically 
and weighted at one end so as to balance the full bucket hung from the other; often seen at country to 

* The working day must he understood to include necessary rests, and such intermissions as tht ’ 
nature of the work demands; but does not include time Inst at meals. A working dag of 10 hnrirJ 
may, therefore, have hut 8, 7, or 6, &c hours of actual labor . This will be understood when we here < 
after speak of a working dav, or simply a day. 

t Desagulier’s estimates of daily work of men and horses exceed the above, but are entirely too great! 





ANIMAL POWER 


379 


»ells,) 600 000 to 800000. In the last he has only to pull down the empty bucket, and thereby raise the 

iounterweight By 2 buckets at tlie ends of a rope suspended over 

a pulley . 500000 to OOOOOO. Here he works the buckets by pulling the rope by hand. 

By a tympaii, or tympanum,* worked by a treadwkeel, about 1 200000 

to 1100000 . 


By a Persian wheel.f a cliain-pump, a chain of buckets.:}: or 
an Archimedes screw, all worked by a treadwheel, from 800 000 to 1000 000 

ft-tts. Of these four, the first three lose useful effect by either spilling, leakiug, or the necessity for 
raising the water to a level somewhat higher than that at which it is discharged. 

When any of the five foregoing machines are worked by men at winches, the result will be about 
% less than by treadwheels. They are all frequently worked also by either steam,water, or horse-power. 

By walking' backward ami forward, on a lever which rocks 
on its center, a man may, according to Robison’s Mecli Philosophy, perform a 
much greater duty than by any of the preceding modes. He states that a young man weighing 135 
fl>s, and loaded with 30 tts in addition, worked in this manner for 10 hours a day without fatigue; 
and raised 9J4 cubic feet of water, 11% ft high per min. This is equal to 3 984 000 ft-9>s per day of 10 

hours; or 6640 ft-B>s per min; or nearly of the net daily work of a horse in a gin. 

A laborer standing 1 still, can barely sustain for a few min. a load of 100 
lbs, by a rope over his shoulder, and thence passing off hor over a pulley. And scarcely as much, 
when (facing the load and pulley) he holds the end of a hor rope with his hands before him. He can¬ 
not push hor with his hands at the height of his shoulders, with more than about 30 tt>s force. 

Weisbach states from his own observation, that 4practised men raised a dolly (a wooden beetle 
or rammer, of wood; with 4 hor projecting round bars for handles) weighing 120 lbs, 4 ft high, at the 
rate of 34 times per min, for i% min ; and then rested for i% min ; and so on alternately through 
the 10 hours of their working day. Therefore, 5 of these hours were lost in rests; and the duty per¬ 
formed by each man during the other 5 hours, or 300 mins, was 


120 X 4 X 34 X 300 

4 


• — 1 224000 ft-lbs. 


Ill the old mode of driving' piles, where the ram of 400 to 1200 lbs 

suspended from a pulley, was raised by 10 to 40 men pulling at separate cords, from 35 to 40 lbs of the 
1 n ram were allotted to each man, to be lifted from 12 to 18 times per min, to a height of 3% to feet 
each time, for about 3 min at a spell, and then 3 min rest. It was very laborious ; and the gangs had 
to be changed about hourly, after performing but % an hour’s actual labor. 


Hauling’ by horses. See Traction. When working all day, say 10 working 

hours, the average rate at which a horse walks while hauling a full load, and while returning with 
the empty vehicle, is about 2 to 2% miles per hour; but to allow for stoppages to rest, &c, it is safest 
to take it at but about 1.8 miles per hour, or 160 ft per min. The time lost on each trip, in loading 
and unloading, may usually be taken at about 15 min. Therefore, to find the number of loads that cau 
be hauled to any given dist in a day, first find the time in min reqd in hauling one load, and return¬ 
ing empty. Thus: div twice the di'st in ft to which the load is to be hauled; or in other words, div 
the length in ft, of the round trip, by 160 ft. The quot is the number of min that the horse is in mo¬ 
tion during each round trip. To this quot add 15 min lost each trip while loading and unloading ; the 
sum is the total time in min occupied by each round trip. Div the number of miu in a working day 
(600 min in a day of 10 working hours) by this number of min reqd for each trip; the quot will he the 
number of trips, or of loads hauled per day. 

Kx How many loads will a horse haul to a dist of 960 ft. in a day of 10 working hours, or 600 min ? 

1920 

Here. 960 X 2 = 1920 ft of round trip at each load. And - — = 12 min, occupied in walking. And 

160 600 min in 10 hours 


12 + 15 in loading, &c) —27 min reqd for each load. Finally, —- 


min per trip 


— 22.2. or 


*ay 22 trips; or loads hauled per day. 


Table of number of loads hauled per day of 10 working- 
hours. The first col is the distance to which the load is actually hauled; or half 
the length of the round trip. The cost of hauling per load, is supposed to be for otie-horse carts ; the 
driver doing the loading and unloading; rating the expense of horse, cart, and driver at $2 per day. 
See Cost of Earthwork, page 742. 


# The tympan revolves on a hor shaft: and is a kind of large wheel, the spokes, arms, or radii of 
which are gutters, troughs, or pipes, which at their outer ends terminate in scoops, whicli dip into 
I the wat°r. As the water is gradually raised, it flows along the arms of the wheel to its axis, where 
' it is dischd. The scoop wheel is a modification of it. It is an admirable machine for raising large 
(j quantities of water to moderate heights. We cannot go into any detail respecting this and other 

hydraulic machines. .... 

\ kind of large wheel with buckets or pots at the ends of its radiating arms ; revolves on a hor 
j axis; discharges at top. The buckets are attached loosely, so as to hang vert, and thus avoid spill- 
(i lng until they arrive at the proper point, where they come into contact with a contrivance for tilting 
r and emptying them. The noria is similar, except that the buckets are firmly held in place, and thus 
spill much water. It is therefore inferior to the Persian wheel. 

j An endless revolving vert chain of buckets. D’Aubuisson and some others erroneously call this 
the noria. It is an affective machine. 









380 


SPECIFIC GRAVITY 




Dist. 

Feet. 

No. of 
Loads. 

Cost per 
Load. 

Dist. 

Feet. 

No. of 
Loads. 

Cost per 
Load. 

Dist. 

Miles. 

No. of 
Loads. 

Cost per 
Load. 

50 

38 

Cts. 

5.26 

1500 

18 

Cts. 

31.11 

1 

7 

Cts. 

28.57 

100 

37 

5.A1 

2000 

15 

13.33 


6 

33.33 

2'K) 

34 

5.88 

2500 

13 

15.39 

114 

5 

40.00 

300 

32 

6.25 

3000 

11 

18.18 

2 

4 

50.00 

400 

30 

6.67 

3500 

10 

20.00 

3 

3 

66.67 

600 

27 

7.41 

4000 

9 

22.22 

4 

2 

100.00 

1000 

22 

9.09 

5000 

7 

28.57 

9 

1 

200.00 


If the loading and unloading is such as cannot be done by the driver alone; but requires the help 
of cranes, or other machinery, au addition of from 10 to 50 cts per load may become necessary. Haul¬ 
ing can generally be more cheaply done by using 2 or 3 horses, and one driver, to a vehicle. The neat 
load per horse, iu addition to the vehicle, will usually be from % to 1 ton, depending on the condition, 
aud grades of the road. From 13 to 15 cub ft of solid stone; or from 23 to 27 cub feet of broken stone, 

make i ton. In estimating lor hauling rough quarry stone for 
<1 rains, culverts, etc, bear in mind that each cub yard of common scabbled rubble- 
masonry, requires the hauling of about 1.2 cub yds of the stone"as usually piled up for sale in the 
quarry ; or about % of a cub yd of the original rock In place. A Cllll yd Ol solid Stone, 

when broken into pieces, usually occupies about 1.9 cub yds 
perfectly loose ; or about 1% when piled up. A strong cart for stone hauling, will weigh 
about % ton ; or 1500 lbs ; and will hold stone enough for a perch of rubble masonry ; or say 1.2 pers 
of the rough stone in piles. The average weight of a good working horse is about a ton. 

Morin gives the following results from careful experiments made by 
him for the French Government. The draft of the same wheeled vehicle on a road, may iu practice 
be considered to be, 

1st. On hard turnpikes, and pavements; in proportion to the 

loads ; inversely as the diams of the wheels; aud nearly independent of the width of tire. It increases 
to uncertain extents with the inequalities of the road; the stiffness (want of spring) of the vehicle; 
and the speed; (considerably less than as the square roots of the last.) 

2d. On soft roads, the draft is less with wide tires than 
with narrower ones; and for farming purposes lie recommends a width of 
4 ins. With speeds from a walk to a fast trot, the draft does not’ vary sensibly. 


SPECIFIC GRAVITY, 

The sp grav of a body, is its weight as compared with that of an equal bulk of 
some other body, which is adopted as a standard of comparison. For other substances 
than air and gases generally,pure water is the usual standard ; and since the weight 
of a given bulk of water varies somewhat with its temperature; and also with the state 
of the air, t)ie former is assumed to be 62° Fall; and the latter at 30 ins, at sea-level. 
On the continent of Europe, water at its greatest density, or at a temperature of 1° 
Centigrade, = 39.2° Fahrenheit, is taken as the standard. But where extreme 
scientific accuracy is not aimed at, all these considerations may be neglected; aud 
any clear fresh water, at any ordinary temperature, say from 60° to 80°, may he 
used. At 70°, the resulting sp gr is but 1 part in 1176 greater than at 62° F; at 75°, 
1 to 670; at 80°, 1 in 454; at 85°, 1 in 336; at 90°, 1 in 264. At 62° pure water 
weighs 62.355 lbs avoir per cub ft. 

To fiml the sp grav of a body, heavier than water. Weigh 

it first in the air; and then in water; and find the diff. The diff is what the body 
loses in water; and is the weight of a bulk of water equal to the bulk of the body. 
Then say, Diff: wt in air : : 1 ; sp grav of body. 




in 




; Ai 

In 


n 

Ai 

•it 

Aii 

At 

Ad 


Bi¬ 
ll His 


fc 
,j l&i. 

I M 

i Erii 

h 

% 

u 

let- 

jci, 

fat 


far 

far 

Co* 





























SPECIFIC GRAVITY, 


381 


inferred from it^n lr ti k f * Snbstance , wh ' c h is either porous, or absorbent of water, cannot be 
a sMid enh ff «f g , Tb - t PUre r . lVer ? and>13 pure <l uartz ! an<1 of course has the same sp gr ; j et, 

of the latter A UrirU 8 “T ? tW1Ce / S mUCh 38 a Cub flof sand ' oa account of the Amices 
weiVht sandstones, &c, absorb water; so that their sp gr will not furnish the 

e a dry mass of the same. In such cases, the engineer will generally first measure the con- 
t 0f u VT° th - 8ubsta “ ce - \ r a so ' id • a '*d then weigh it; thuslscertaining its weight per cub 
ft, &c. If it is in grains, or dust, he will measure, and then weigh, a cub ft of it. Footnote p 384. 

To the sp grav Ol a liquid. First carefully weigh some solid body as a 

jpiece of metal, in the air. Then weigh it in water, and note the loss, say L. Then weigh itVn the 
pother liquid: and note the loss, say l. Then as loss L, is to loss l, so is 1, or the sp gr of water to the 

it P in r thl n be -H qU T; 0r ’ - f thesp g - r ’ a ? d . we '8 ht of the solid body, are already known, merelv'weigh 
11 A? tbe *'quid. Then as its weight in air, is to its loss in the liquid, so is its sp gr, to that of the liquid. 
Timber, when first purchased from lumber yards, even under shelter, is rarely if ever perfectly 

rbout t ir lf gr t een rably SeaSOned> wi “ be about « part Boater than given in our’tables, or 

Table of specific gravities, and weights. 

In this table, the sp gr of air, and gases also, are compared with that of water 
instead of that of air; which last is usual. 


J he specific gravity of any substance is = its weight) 
in grams per cubic centimetre. 


Air, atmospheric; at60° Fah, and under the pressure of one atmosphere or 

14.7 fts per sq inch, weighs part as much as water at 60°. 

Alcohol, pure.1. 

*• of commerce." ’ 

“ proof spirit. 

Ash, perfectly dry. (See footnote, p 383.) ...................... average.. 

1000 ft board measure weighs 1.748 tons. 

Ash, American white, dry. «i 

1000 ft board measure weighs 1.414 tons. 

Alabaster, falsely so called; but really Marbles. 

real; a compact white plaster of Paris.average .. 

Aluminium. 

Antimony, cast, 6.66 to 6.74."average*!! 

“ native. •* 

Anthracite, i.3 to 1.84. Of Penna, 1.3 to 1.7.usually .. 

A cubic yard solid, averages about 1.75 cub yds, when broken to any mar¬ 
ket size ; and loose. 

Anthracite, broken, of any size. Loose.average.. 

“ moderately shaken. “ .. 

“ heaped bushel, loose, 77 to 83. 

A ton, loose, averages from 40 to 43 cub ft. “ .. 

at 54 lbs per cub ft, a cub yard weighs .651 ton. 

Asphaltum, 1 to 1.8. “ .. 

Basalt. See Limestones, quarried. “ 

Bath Stone, Oolite. »• .. 

Bismuth, cast. Also native. “ .. 

Bitumen, solid. See Asphaltum. 

Brass, (Copper and Zinc,) cast, 7.8 to 8.4. “ .. 

“ rolled. “ .. 

Bronze. Copper 8 parts; Tin 1.. (Gun metal.) 8.4 to 8.6. *• .. 

Brick, best pressed. 

“ common hard. 

“ soft, inferior. 

Brickwork. See Masonry. 

Boxwood, dry. •* .. 

Calcite, transparent. “ ,. 

Carbonic Acid Gas, is 1}4 times as heavy as air. “ .. 

Charcoal, of pines and oaks. “ .. 

Chalk, 2.2 to 2.8. See Limestones, quarried. “ .. 

Clay, potter’s, dry, 1.8 to 2.1. “ .. 

“ dry, in lump, loose. “ .. 

Coke, loose, of good coal. “ .. 

“ a heaped bushel, loose, 35 to 42 lbs. “ .. 

“ a ton occupies 80 to 97 cub ft. “ .. 

In coking, coals sw<“ll from 25 to 50 per cent. 

Equal weights of coke and coal, evaporate about equal wts of 
water: and each abt twice as much as the same wtof dry wood. 

Corundum, pure, 3.8 to 4. 

Cherry, perfectly dry.average 

1000 ft board measure weighs 1.562 tons. 

Coal, bituminous. 1.2 to 1.5.■. . “ . 

“ broken, of any size; loose. " . 

“ “ moderately shaken. “ . 

“ “ a heaped bushel, loose, 70 to 78 lbs. 

“ “ a ton occupies 43 to 48 cub ft. 

A cubic yard solid, averages about 1.75 yards when broken to any 
market size, aud loose. 


Average 
Sp Gr. 


.00123 

.793 

.834 

.916 

.752 

.61 

2.7 

2.31 

2.6 

6.70 

6.67 

1.5 


1.4 
2.9 
2.1 
9.74 

8.1 

8.4 

8.5 


.96 

2.722 

.00187 


2.5 

1.9 


3.9 

.672 


1.35 


Average 
Wt of a 
Cub Ft. 
Lb*. 


.0765 

49.43 

52.1 

57.2 
47. 

38. 

168. 

144. 

162. 

418. 

416. 

93.5 


52 to 56 
56 to 60 


87.3 

181. 

131. 

607. 

504. 

524. 

529. 

150. 

125. 

100 . 

60 

169.9 

15 to 30 
156. 

119. 

63. 

23 to 32 


42. 

84. 

47 to 52 
51 to 56 





































































382 


SPECIFIC GRAVITY, 


Table of specific gravities, and weights — (Continued.) 


The specific gravity of any substance is = its weight 
in grains per cubic centimetre. 


Chestnut, perfectly dry. (See footnote, p 3S3.).average. 

1000 feet board measure weighs 1.525 tons. 

Cement, hydraulic. American, Rosendale ; ground, loose.T.... average. 

“ “ “ “ U S. Struck bush, 70 lbs. 

“ “ 11 Louisville, “ “ 62. 

“ “ “ Copley, “ “ 67. 

“ English Portland. U.S. struck bush, by Gillmore, 100 to 128 

“ “ “ “ Various, weighed by writer, 95 to 102. 

“ “ “ “ a barrel 400 to 430 lbs. 

“ French Boulogne Portland, struck bush, 95 to 110. 

Differences of 4 or 5 pounds either more or less than we here give per 
loose struck U S. bush, often occur in the cement from the same 
manufactory, owing not only to the difficulty of measuring exactly, 
but to the want of uniformity in the composition of the stone, de¬ 
gree of burning, grinding, dryness, <fec. Moreover, the term “loose” 
is indefinite. We mean by it the average looseness which it has 
when thrown by a scoop into a half bushel when measuring that 
quantity for sale. By shaking it may easily be compacted about % 
part, so as to weigh 4- more per bush, or cub ft. And by ramming, 

about part, so as to weigh about y more. So with lime, plas¬ 
ter, &c. 

Copper, cast.8.6 to 8.8. 

“ rolled,.8.8 to 9.0. 

Crystal, pure Quartz. See Quartz. 

Cork. 

Diamond, 3.44 to 3.55 ; usually 3.51 to 3.55. 

Earth; common loam, perfectly dry. loose. 

“ ** “ “ “ shaken. 

“ “ “ “ “ moderately rammed. 

“ “ “ slightly moist, loose. 

“ “ “ more moist, “ . 

** “ “ “ shaken. 

“ “ “ “ moderately packed. 

“ “ “ as a soft flowing mud. 

“ “ “ as a soft mud, well pressed into a box. 

Ether. 

Elm, perfectly dry. (See footnote, p 383.).average.. 

1000 ft board measure weighs 1.302 tons. 

Ebony, dry. “ 

Emerald, 2.63 to 2.76. “ 

Fat. “ 

Flint. “ 

Feldspar, 2.5 to 2.8. “ 

Garnet, 3.5 to 4.3; Precious, 4.1 to 4.3. “ 

Glass, 2.5 to 3.45. « 

“ common window. “ 

“ Millville, New Jersey. Thick flooring glass. “ 

Granite, 2.56 to 2 . 88 . See Limestone, 160 to 180. “ 

Gueiss, common, 2.62 to 2.76. . “ 

“ in loose piles. 

“ Horublendic. «• 

“ “ quarried, in loose piles. “ 

Gypsum, Plaster of Paris, 2.24 to 2.30.. “ 

“ in irregular lumps.. “ 

“ grouud, loose, per struck bushel, 70. “ 

“ “ well shaken. “ “ 80. “ 

“ “ Calcined, loose, per struck bush, 65 to 75-. “ 

Greenstone, trap, 2.8 to 3 2. •< 

“ “ quarried, in loose piles. “ 

Gravel, about the same as sand, which see. 

Gold, cast, pure, or 24 carat.. “ 

“ native, pure, 19.3 to 19.34. “ ” 

“ frequently containing silver, 15.6 to 19.3. “ 

“ pure, hammered, 19.4 to 19.6. “ 

Gutta Percha. « 

Hornblende, black, 3.1 to 3.4. *« * ’ 

Hydrogen Gas, is 14)$ times lighter than air; and 16 times lighter than 
oxygen.average.. 

Hemlock, perfectly dry. (Footnote, p 383.)..-... << 

1000 feet board measure weighs .930 ton. 

Hickory, perfectly dry. (See footnote, p 383.). “ 

1000 feet board measure weighs 1.971 tons. 

Iron, oast, 6.9 to 7.4. <* 

“ “ usually assumed at... “ 

At 450 lbs. a cub iuch weighs .2604 lb ; 8601 6 cub inches a ton : arid 
a lb = 3.8400 cub inches ; cast iron gun metal. 



.66 


8.7 

8.9 

.25 

3.53 


* See tables, pp. 398 aud 399. 



75 to 


90 to 


104 to 


110 to 

.716 

44.6 

.56 

35. 

1.22 

2.7 

76.1 

.93 

58. 

2.6 

162. 

2.65 

4.2 

166. 

2.98 

186. 

2.52 

157. 

2.53 

158. 

2.72 

170. 

2.69 

168. 


96. 

2.8 

175. 


100. 

2.27 

141.6 


82. 


56. 


64. 


52 to 

3. 

187. 

107. 

19.258 

1204. 

19.32 

1206. 

19.5 

1217. 

.98 

61.1 

3.25 

203. 


.Of 

.4 

25. 

.85 

53, 

7.15 

446. 

7.21 

450. 

7.48 

4C7. 


Average 
Wt of a 
Cub FL 
Lbs. 


41. 

56. 

49.6 

53.6 

81 to 102 
76 to 81.6 

76 to 88 


542. 

555. 

15.6 

72 to 
82 to 
90 to 100 
70 to 76 
68 
90 


80 

92 


f * 
























































































SPECIFIC GRAVITY 


383 


Table of specific gravities, and weights — (Continued.) 


Jhe specific gravity of any substance is = its weight 
in grams per cubic centimetre. 


Iron, wrought, 7.6 to 7.9; the purest has the greatest sp gr.average.. 

large rolled bars... > „ b 

usually assumed at ( ® ee PP’ to “ • • 

“ sheet. (See pp. 410, 411). 

At 480 lbs, a cub inch weighs .2778 fl>; and a lb = 3.U000 cub ins. 
Light iron indicates impurity. 


Ivory. 


average. 


About -I of the mass 


Ice, .917 to .922 

India rubber.u 

Lignum vitae, dry.! . •< 

Lard.... ..!.!!!!!!".l!".!".".".”!!.".l “ 

Lead, of commerce, 11.30 to 11.47; either rolled or cast. “ 

Limestones and Marbles,2.4 to 2.86, 150 to 178.8.. 

ordinarily about. 

quarried in irregular fragments, 1 cub yard solid, 
makes about 1.9 cub yds perfectly loose; or about 
}% yds piled. In this last case’, .571 of the pile 
is solid; and the remaining .429 part of it is 

voids.piled.. 

Lime, quick, of ordinary limestone and marbles 92 to 98 !bs per cub ft..!! 

“ “< either in small irregular lumps; or ground, loose 50 to 58_ 

In either case 1 solid measure makes about 1.8 meas loose; and then 
■555 of the mass i3 solid, and .445 is voids. 

To measure correctly, none of the lumps should exceed about % or 
of the smallest dimension of the vessel used for measuring. 

Lime, quick, ground, loose, per struck bushel 62 to 70 lbs. 

“ “ “ well shaken, •* *• ....80 “ . 

“ “ “ thoroughly shaken, “ ....93% “ . 

Mahogany, Spanish, dry*.average.. 

*f Honduras, dry. •• . [ 

Maple, dry*. »< 

Marbles, see Limestones. 

Masonry, of granite or limestones, well dressed throughout 
“ ** “ well-scabbled mortar rubble. 

will be mortar_ __ 

“ " " well-scabbled dry rubble. 

“ “ roughly scabbled mortar rubble. About % to % part 

will be mortar. 

“ “ “ roughly scahbled dry rubble. 

At 155 lbs per cub ft, a cub yard weighs 1.868 tons; and 14.45 cub ft, 
1 ton. 

i Masonry of sandstone; about % part less than the foregoing. 

. “ “ brickwork, pressed brick, fine joints.average.. 

“ “ “ medium quality. “ 

“ “ “ coarse; inferior soft bricks. “ 

At 125 lbs per cub ft, a cub yard weighs 1.507 tons; and 17.92 cub 
ft. 1 ton. 

Mercury, at 32° Fah. 

•• 60° “ . 

I “ 212° “..!!!!!!!!!!!!!! 

lifica. 2.75 to 3.1. 

‘Mortar, hardened, 1.4 to 1.9. 

Mud, dry, close. 

“ wet, moderately pressed. 

“ wet. fluid. 

Yaphtha. 

Vitrogen Gas is about part lighter than air. 

Dak, live, perfectly dry, .88 to 1.02*.average.. 

“ white, “ “ .66 to .88. “ 

J “ red, black. &o*. “ 

Oils, whale; olive. “ 

*• of turpentine. “ 

Oolites, or Roestones, 1.9 to 2.5. “ 

Oxygen Gas, a little more than yL part heavier than air. 

Petroleum. 

Peat, dry, unpressed. 

;Piue, white, perfectly dry, .35 to .45*. 

1000 ft board measure weighs .930 ton.* 

yellow, Northern, .48 to .62. 

1000 ft board measure weighs 1.276 tons.* 

“ Southern, .64 to .80.......... 

1000 ft board measure weighs 1.674 tons.* 


A verage 
Sp Gr. 


7.77 
j 7.6 
< 7.69 


1.82 

.92 

.93 

1.33 

.95 

11.38 

2.6 

2.7 


1.5 


.85 

.56 

.79 


13.62 

13.58 

13.38 

2.93 

1.65 


.848 


.95 

.77 


.92 

.87 

2.2 

.00136 

.878 


.40 

.55 

.72 


A verage 
Wt of a 
Cub Ft. 
Lbs. 


485. 

474. 

480. 

485. 


114. 

57.4 

58. 

83. 

59.3 

709.6 

164.4 

168. 


96. 

95. 

53. 


53. 

64. 

75. 

53. 

35. 

49. 

165. 

154. 

138. 

150. 

125. 


140. 

125. 

100 . 


849. 

846. 

836. 

183. 

103. 

80 to 110 
110 to 130 
104 to 120 
52.-) 
.0744 

59.3 
48. 

32 to 45 

57.3 

54.3 
137. 

.0846 

54.8 
20 to 30 
25. 

34.3 
45. 


* Greon timbers usually weigh from one-fifth to nearly one-half more than 

Irj ; and ordinary building timbui s when tolerably scasoued about oue-sixth more than perfectly dry. 


























































































384 


SPECIFIC GRAVITY. 


Table of specific gravities, and weights — (Continued.) 


The specific gravity of any substance is = its weight 
in grains per cubic centimetre. 


Pine, heart of long-leafed Southern yellow, nnseas. (Footnote, p 383.)... 

1000 ft hoard measure weighs 2.418 tons. 

Pitch... 

Plaster of Paris ; see Gypsum. 

Powder, slightly shaken. 

Porphyry, 2.60'to 2.8 . ..!!!...!.!!!!!!!. 

Platinum. 21 to 22 . 

“ native, in grains. 16 to 13. 

Quartz, common, pure.2.64 to 2.67. 

finely pulverized, loose... . 

“ “ “ “ well shaken. 

“ “ “ well packed... 

“ quarried, loose. One measure solid, makes full \% broken and 

piled. 

Ruby and Sapphire, 3.8 to 4.0.. 

Rosin. 

Salt, coarse, per struck bushel; Syracuse, N. York.!!.!"!!!!! ”.56 lbs !! 

“ Turk’s Island; Cadiz; Lisbon. 76 to 80 .. 

“ ‘ ‘I “ St.Barts.84 tow.. 

“ some well-dried West India_90 to 96 .. 

" “ “ Liverpool.50 to 55.. 

Liverpool fine. Tor table use.60 to 62 .. 

Sand, of pure quartz, perfectly dried, and loose, usually 112 to 133 lbs per 

struck bushel. 

At the average of 98 lbs per cub ft, a struck bushel weighs 122)4 lbs ; 
and 18.29 bushels, 1 ton ; a cub yd = 1.181 tons; 22.86 cub ft, 1 'ton. 
Slight shaking compacts it about 2 to 3 per ct; and ramming about 
12 per ct when dry. 

“ perfectly wet, voids full of water. 

“ at the mean of 124 tbs, a cub yard weighs 1.495 tons ; 
and 18.06 cubic feet — 1 ton. 

“ sharp angular sand of pure quartz with very large and very small 

grains dry may weigh. 

If any ordinary pure natural sand be sifted into 2 or 3 or more parcels 
of differently sized grains, a measure of any of these parcels will 
weigh considerably less than an equal measure of the original sand. 
Thus, a sand weighing 98 lbs per cub foot, may give others weighing 
not more than 70 to 80 lbs. At 98 lbs per cub ft. 1 bulk of pure quartz, 
has made 1.68 bulks of sand; of which the solid occupies .6 ; and the 
voids .4. But if this same sand be compacted to 110 lbs per cub ft, 
then 1 measure of solid quartz makes 1 >4 measures of sand ; of which 
% are solid, and % voids. Sand is very retentive of moisture ; and 
when in large bulks, is rarely as dry as that above in this table. But 
with its natural moisture, and loose, it is lighter than when dry, its 
average weight then not exceeding about 85 to 90 lbs per cub ft; or 
106)4 to 112)4 lbs per struck bushel. See Voids in Sand, p 678. 

Sandstones, fit for building, dry, 2.1 to 2.73,.131 to 171. 

“ quarried, and piled, 1 measure solid, makes about 1 J£ piled... 

Serpentines, good.2.5 to 2.65. 

Snow, fresh fallen.. 

“ moistened, and compacted by rain. 

Sycamore, perfectly dry. (See footnote, p 383.). 

1000 ft board measure weighs 1.376 tons. 

Shales, red or black.2.4 to 2.8.average.. 

’• quarried, in piles. •> 

Slate.2.7 to 2.9. <* ” 

Silver... <• 

Soapstone, or Steatite.2.65 to 2.8 .!."!!!.”! *• !! 

Steel, 7.7 to 7.9. The heaviest contains least carbon.. “ 

Steel is not heavier than the iron from which it is made; unless the 
iron had impurities which were expelled during its conversion iuto 
steel. 

Sulphur ..average.. 

Spruce, perfectly dry. Footnote, p 383.. 

1000 ft board measure weighs .930 ton. 

Spelter, or Zinc. 6.8 to 7.2. « 

Sapphire; and Ruby, 3.8 to 4.“ 

Tallow. __ . ,, 

Tar.** 

Trap, compact, 2.8 to 3.2.« 

“ quarried ; in piles.'....'.'. 1 '.!!*’ “ 

Topaz. 3.45 to 3.65. u 


Average 
Sp Gr. 

Average 
Wt of a 
Cub Ft. 
Lbs. 

1.04 

65. 

1.15 

71.T 

1 . 

62.3 

2.73 

170. 

21.5 

1342. 

17.5 


2,65 

165. 

90. 

105. 

112. 

94. 

3.9 


1.1 

68.6 

45. 

62. 

70. 

74. 

42. 

49. 

• 

90 to 106 


2.41 

151. 


86. 

2.6 

162. 

5 to 12 
15 to 50 

.59 

37. 

2.6 

162. 


92. 

2.8 

175. 

10.5 

655. 

2.73 

170. 

7.86 

490. 

2. 

125. 

.4 

*25. 

7.00 

3.9 

437.5 

.94 

58.6 

1 . 

62.4 

3. 

187. 

3.55 

107. 


118 to 129 


117. 


* The sp gr of pure quartz sand found as directed near foot of p 380. is of course the same as tin 
of puie quartz, or 2 6o. But a cub ft of dry saud weighs, as above, from 90 to 106 lbs or otilv fro, 

gr U See iC^ q "“ U * ° f M ° Sl aUthorities «»*• “>ou’t 1-5 £ ihe°s 





















































































WEIGHTS AND MEASURES. 


385 


Table of specific gravities, and weights — (Continued.) 


The specific gravity of any substance is = its weight 
in grams per cubic centimetre. 


Tin,cast,, 7.2 to 7.5.average.. 

Turf, or Peat, dry, unpressed. 

Water, pure rain, or distilled, at H2° Fah. Barorn 30 ins. 

“ “ “ “ “ 62° “ “ '• “. 

«< « •• 44 44 212 ° 11 ** 11 **.. .......... .... 

( At 60°, a cub inch weighs .03607 lb; or .57712 oz avoir. And a lb con¬ 
tains 27.724 cub ins; equal to a cube of 3.0263 inches on each edge. 

Water, sea. 1.026 to 1.030 .average .. 

Although the wt of fresh water is generally assumed as sixty two 
and one-third lbs per cub ft, yet 62% would be nearer the truth, at 
ordinary temperatures of about 70°; or a lb = 27.759 cub ins; and a 
cub in = .5764 oz avoir; or .4323 oz troy; or 252.175 grains. The grain 
is the same in troy, avoir, and apoth. 

Wax, bees.average.. 

Wines, .993 to 1.04. “ •• 

: Walnut, black, perfectly dry. (See footnote, p 383.) . “ 

| 1000 ft board measure weighs 1.414 tons. 

Zinc, or Spelter, 6.8 to 7.2. “ 

Zircon, 4.0 to 4.9. “ 


Average 

Sp Ur. 

Average 
Wt of a 
Cub Ft. 
Lbs. 

7.35 

459. 


20 to 30 


62.417 

1 . 

62.355 


59.7 

1.028 

64.08 

.97 

60.5 

.998 

62.3 

.61 

38. 

7.00 

437.5 

4.45 



WEIGHTS AND MEASURES. 

United States and British measures of length and weight, 

of t he same denomination, may, for all ordinary purposes, be considered as equal; 
but the liquid and dry measures of the same denomination differ widely 
in the two countries. The standard measure of length of both coun¬ 
tries is theoretically that of a pendulum vibrating seconds at the level of the 
sea, in the latitude of London, in a vacuum, with Fahrenheit’s thermometer at 
62°! The length of such a pendulum is supposed to be divided into 39.1393 
equal parts, called inches; and 36 of these inches were adopted as the standard 
yard of both countries. But the Parliamentary standard having been destroyed 
bv fire, in 1834, it was found to be impossible to restore it by measurement of a 
pendulum; and the present Britisli standard yard is, in consequence, shorter 
than that of the U. S. by the latest comparison, about 1 part in 40,000, or .03 inch 
in 100 feet; or 1.584 inches in a mile. But at a temperature of 62°.25 Fah for 
the British standard and 59°.62 for the U. S. one, the two are of the same length, 
and on this basis the U. S. government declares the measures of the two coun¬ 
tries to be the same; as in our tables. 

, Troy Weight. 

U. S. and British. 


24 grains. 1 pennyweight, dwt. 

20 pennyweights. 1 ounce — 480 grains. 

12 ounces. 1 pound = 240 dwts = 5760 grains. 


Troy weight is used for gold and silver. . ... _ 

A earat of the jewellers, for precious stones, is in the U. S. — 3.2 grs; in London. 3.17 grs; in 
Paris, 3.18 srs, divided into 4 jewellers 1 grs. In troy, apothecaries’, and avoirdupois, the 
grain is the same. 


Approximate Values of Foreign Coins, in U. S. Money. 

The references ( l , 2 , 3 and 4 ) are to foot-notes on next page. 


From Circular of U. .S. Treasury Department, Bureau of the Mint, Jan. 1,1887; 
from “Question Monetaire,” by H.Costes, Paris, 1884; and from our lOtb edition. 


Argentine Repub.—Peso = 100 Centavos, 96.5 cts. 23 Argentino = 5 Pesos, $4.82. 
Austria— Florin = 100 Kreutzer,47.7 cts. 2 35.9 cts. 3 Ducat, $2.29. Maria Theresa 
Thaler, or Levantin, 1780, $1.00. 2 Rix Thaler, 97 cts.* Souverain, $3.57. 4 
BeMum. 1 —Franc = 100 centimes, 17.9 cts., 2 19.3 cts. 3 ^ ^ „ 

Bolivia— Boliviano = 100 Centavos, 96.5 cts., 2 72.7 cts. 3 Once, $14.9o. Dollar, 
96 cts 4 

Brazil.—Milreis = 1000 Reis, 50.2 cts., 2 54.6 cts. 3 
Canada.—English and U. iS. coins. Also Pound, $4. 

Central America. 4 —Doubloon, $14.50 to $15.65. Reale, average 5 y£ cts. See 
Honduras. 

Ceylon.—Rupee, same as India. 

2o 


































386 


FOREIGN COINS. 


(Foreign Coins Continued. Small figures ( 1 , 2 , 3 , 4 ) refer to footnotes.) 


Chili.—Peso = 10 Dineros or Decimos = 100 Centavos, 96.5 cts., 2 91.2 cts. 3 Con¬ 
dor = 2 Doublo >ns = 5 Escudos = 10 Pesos. Dollar, 93 cts. 4 
Cuba.—Peso, 93.2 cts. 3 Doubloon, $5.02. 

Denmark.—Crown = 100 Ore, 25.7 cts., 2 26.8 cts. 3 Ducat, $1.81. 4 Skilling, % ct 4 
Ecuador.—Sucre, 72.7 cts. 3 Doubloon, $3.86. Condor, $9.65. Dollar, 93 cts. 4 

Heale, 9 cts. 4 # 

Egypt.—Pound = 100 Piastres — 4000 Paras, $4 94,3. 3 
Finland.—Markka =® 100 Penni, 19.1 cts 2 10 Markkaa, $1.93. 

France. 1 —Franc = 100 Centimes, 17.9 cts., 2 19.3 cts. 3 Napoleon, $3.84. 4 Livre, 
18.5 cts. 4 Sous, 1 ct. 4 

Germany.—Mark = 100 Pfennigs, 21.4 cts., 2 23.8 cts. 3 Augustus (Saxony), $3.98. 4 
Carolin (Bavaria), $4.93. 4 Crown (Baden, Bavaria, N. Germany), $1.06. 4 
Ducat (Hamburg, Hanover), $2.28. 4 Florin (Prussia, Hanover), 55 cts. 4 
Groschen, 2.4 cts. 4 Kreutzer (Prussia), .7 ct. Maximilian (Bavaria), $3.30. 4 
Rix Thaler (Hamburg, Hanover), $1.10 4 (Baden, Brunswick), $1.0U 4 (Prussia, 
N. Germany, Bremen, Saxony, Hanover), 69 cts. 4 
Great Britain.—Pound Sterling or Sovereign (£) = 20 Shillings = 240 re nee, 
$4.86.65. 3 Guinea = 21 Shillings Crown = 5 Shillings. Shilling ( s ), 22.4 
cts., 2 24.3 cts. (5^ pound sterling). Penny ( d ), 2 cts. 

Greece. 1 —Drachma = 100 Lepta, 17 cts., 2 19.3 cts. 3 
Hayti.—Gourde of 100 cents, 96.5 cts .2 3 

Honduras.—Dollar or Piastre of 100 cents, $1.01. See Central America. 

India.—Rupee = 16 Annas, 45.9 cts., 2 34.6 cts. 3 Mohur = 15 Rupees, $7.10. Star 
Pagoda (Madras), $1.81. 4 

Italy, etc. 1 —Lira = 100 Centesimi, 17.9 cts. 2 19.3 cts. 3 Carlin (Sardinia), $8.21. 4 
Crown (Sicily), 96 cts 4 Livre (Sardinia), 18.5 cts. 4 (Tuscany, Venice), 16 
cts. 4 Ounce (Sicily), $2.50. 4 Paolo (Rome), 10 cts. 4 Pistola (Rome), $3.37. 4 
Scudo 4 (Piedmont), $1.36 (Genoa), $1.28 (Rome), $1.00 (Naples; Sicily), 95 
cts. (Sardinia), 92 cts. Teston (Rome). 30 cts. 4 Zecchino (Rome), $2.27. 4 
Japan.—Yen = 100 Sen (gold), 99.7 cts. 3 (silver), $1.04 2 , 78.4 cts. 3 
Liberia.—Dollar, $1.00. 3 4 

Mexico.—Dollar, Peso, or Piastre — 100 Centavos (gold), 98.3 cts. (silver), $1.05 2 
79 cts. 3 Once or Doubloon = 16 Pesos, $15.74. 

Netherlands.—Florin of 100 cents, 40.5 cts. 2 40.2 cts. 3 Ducatoon, $1.32. 4 Guilder. 

40 cts. 4 Rix Dollar, $1.05. 4 Stiver, 2 cts. 4 
New Granada.—Doubloon, $15.34. 4 

Norway.—Crown = 100 Ore = 30 Skillings, 25.7 cts., 2 26.8 cts. 3 
Paraguay.—Piastre = 8 Reals, 90 cts. 

Persia.—Thoman — 5 Sachib-Kerans = 10 Banabats = 25 Abassis = 100 Scahis 
$2.29. ’ 


Peru.—Soi'= 10 Dineros = 100 Centavos, 96.5 cts. 2 72.7 cts. 8 Dollar, 93 cts. 4 
Portugal—Milreis = 10 Testoons = 1000 Reis, $1.08. 3 Crown = 10 Milreis. 
Moidore, $6.50. 4 • 


Russia.—Rouble = 2 Poltinniks = 4 Tchetvertaks = 5 Abassis = 10 Griviniks = 
20 Pietaks — 100 Kopecks, 77 cts. 2 58.2 cts. 3 Imperial = 10 Roubles, $7 72 
Ducat = 3 Roubles, $2.39. 

Sandwich Islands.—Dollar, $1.00. 4 

Sicily.—See Italy. 

Spain.—Peseta or Pistareen = 100 Centimes, 17.9 cts. 2 19.3 cts. 3 Doubloon (new) 
= 10 Escudos = 100 Reals, $5.02. Pure = 2 Escudos, 4 $1.00. 2 Doubloon (old) 
$15.65. 4 Pistole = 2 Crowns, $3.90 4 Piastre, $1.04. 4 Reale Plate, 10 cts 4 
Reale vellon, 5 cts 4 

Sweden.—Crown = 100 Ore, 25.7 cts., 2 26.8 cts. 3 Ducat, $2.20. 4 Rix Dollar, $1 05 4 

Switzerland. 1 —Franc = 100 Centimes, 17.9 cts., 2 19.3 cts. 3 

Tripoli.—Mahbub = 20 Piastres, 65.6 cts. 3 

Tunis.—Piastre = 16 Karobs, 12 cts 2 10 Piastres, $1.16.6. 

Turkey.—Piastre = 40 Paras, 4.4 cts. 3 Zecchin, $1.40. 4 

Dnited States of Colombia.—Peso = 10 Dineros or Decimos = 100 Centavos 96 5 
cts., 2 72.7 cts. 3 Condor = 10 Pesos, $9.65. Dollar, 93 5 cts. 4 

Uruguay —Peso = 100 Centavos or Centesimos (gold), $1.03 (silver), 96.5 cts 2 

Venezuela—Bolivar = 2 Decimos, 17.9 cts., 2 19.3 cts. 3 Venezolano = 5 Bolivars. 


\ 


1 

1 




I 

I 

II 

1 




1 France, Belgium, Italy, Switzerland, and Greeee form the Latin Union 
Their coins are alike in diameter, weight, and fineness. 

2 = 19.3 times the value of a single coin in francs as given by Costes 
Cirrida? f exchauge > or equivalent value in terms of U.S. gold'dollar.—Treasury 

4 From our 10th edition. 
















WEIGHTS AND MEASURES, 


387 


^7, lc U. 8. gold dollar weighs 25.8 grs; and contains 23.22 grs of pure gold. 

“ “ I® “ “ 258 grs; “ “ 232.20 |rs “ “ 

30 “ 516 grs; 11 *• 464.40 grs “ *• 

^ Pe j feC i ,y > PH re gold is worth Per 23.25 grs = $20.67183 per troy oz rr $18.84151 per avoir oz. 
Handard (U. S. coiu) is worth $18.6046o per troy oz = $16.95736 per avoir oz. It consists of 9 
iarts by wt of piire gold, to 1 part alloy. Its value is that of the pure gold only: the cost of the alloy 
u of the coinage being borne by Gov A cub foot of pure gold weighs about 1204 avoir lbs; 
no is worth *362963. A cuh inch weighs about 11.148 avoir oz; and is worth $210.04. 

I 1 V,r C ** called fine, or 24 carat gold ; and when alloyed, the alloy is supposed to be divided 

Qt v? in ! l' ts wt ! an< ^ accor( ling as 10, 15, or 20, &c, of these parts are pure gold, the alloy is said 
o he 10, 15. or 20, &c, carat. 

Pure silver fluctuates in value; thus during 1878. 1879, it ranged between $1.05 and $1.18 per 
roy °z. or $.957 and $1,076 per avoir oz. A cub inch weighs about 5.528 troy, or 6.065 avoir ounces. 
I he U. S. silver dollar weighs 412.5 grs troy; but its subdivisions weigh at the rate of about 8 
>e ct less. All consist of 9 parts silver to 1 part alloy. 

The average fineness of California native gold, by some thousands of assays at the U. 
. mint in Phiiada, is 88.5 parts goid, 11.5 silver. Some from Georgia, 99 per ct gold. 

Apothecaries* Weight. 

U. S. and British. 


20 grains. 1 scruple. 

3 scruples. 1 dram 60 grs. 

8 drams. 1 ounce = 24 scruples :r 480 grs. 

12 ounces. 1 pound = 96 drams = 288 scruples = 5760 grains. 


In troy and apoth weights, the grain, ounce, and pound are the same. 

Avoirdupois, or Commercial Weight. 

C. 8. and British. 

27.34375 grains. 1 dram. 

16 drams. 1 ounce = 43716 grains. 

16 ounces. 1 pound — 256 drams — 7000 grains. 

28 pounds. 1 quarter — 448 ounces. 

4 quarters. 1 hundredweight = 112 ibs. 

20 hundredweights. 1 ton — 80 quarters = 2240 fbs. 

A stone =r 14 pounds. A quintal — 100 pounds avoir. 

The standard of the avoir pound, which is the one in common commercial use, is the 
veigbt of 27.7015 cub ins of pure distilled water, at its maximum density at about 39°.2 Fahr, In 
.ltiiiide of London, at the level of the sea; barom at 30 ins. But this involves an error of 
ibout 1 part in 1362, for the 1 lb of water = 27.68122 cub ins. 

A troy ib ~ .82286 avoir fl>. Ad avoir lb rr 1.21528 troy lb, or apoth. 

A troy oz ^ 1.09714 avoir oz. An avoir oz = .911458 troy oz. or apoth. 

Long Measure. 

U. 8. and British. 


By law, the U. 8. standards of length, as well as of weight, are made the same as the British 


12 inches.... . 1 foot — .3047973 metre. 

3 feet. 1 yard — 36 ins — .9143919 metre. 

516 yards. 1 rod, poie, or perch = 1616 feet = 198 ins. 

40 rods. 1 furlong ~ 220 yards z 660 feet. 

8 furlongs. 1 statute, or land mile — 320 rods = 1760 yds = 5280 ft = 63360 ins. 

3 miles. 1 league = 24 furlongs = 960 rods — 5280 yds = 15840 ft. 


I A point — inch. A line = 6 points =r inch. A palm =: 3 ins. A hand — 4 ins. A 
i span = 9 ins. A fathom = 6 feet. A cable’s length ^ 120 fathoms = 720 feet. A Gunter’s 
[ surveying chain is 66 feet, or 4 rods long. It is divided into 100 links of 7.92 ins long. 80 Gun- 
f ler’s chains — 1 mile. 

? A nautical mile, jgeog-rapiiical mile, sea mile, or knot, is 

variously defined as being = the length of 

metres feet statute miles 


1 min of longitude at the equator 


latitude 

U 


“ pole 

“ “ at lat 45° 

“ a great circle of a true') 
sphere whose surface area is > 

equal to that of the earth J 
British Admiralty knot 

The above lengths of minutes, in metres and feet, are those published by the U. S. 
I Coast and Geodetic Survey in Appendix No 12, Report for 1881, and are calculated 
from Clarke’s spheroid, which is now the standard of that Survey. 


1855.345 6087.15 1.15287 

1842.787 6045.95 1.14507 

1861.655 6107.85 1.15679 

1852.181 6076.76 1.15090 

( value adopted by IT. S. Coast 
- and Geodetic Survey 
11853.248 6080.27 1.15157 

1853.169 6080.00 1.15152 


At the equator 1° of lat = 68.70 land miles 5 at lat 20° — 68.18 5 at 40° — 
69.00 ; at 60° = 69.23 , at 80° = 69.39 ; at 90° = 69.41. 























388 


WEIGHTS AND MEASURES. 


Lengths of a Decree of Longitude in different Latitudes, 

and at the level of tile Sea. These lengths are in common land or statute miles, 
of 5280 ft. Since the figure of the earth has never been precisely ascertained, these are but close ap 
proximations. Intermediate ones maybe found correctly by simple proportion. 1° of longitude 
corresponds to 4 mins of civil or clock time; 1 min of longitude to 4 secs of time. 


Deg of 
I,at. 

Miles. 

Deg of 

Lat. 

Miles. 

Deg of 
Lat. 

Miles. 

Deg of 
Lat. 

Miles. 

Deg of 
Lat. 

Miles. 

Deg of 
Lat. 

Miles. 

0 

69.16 

14 

67.12 

28 

61.11 

42 

51.47 

56 

38.76 

70 

23.72 

2 

69.12 

16 

66.50 

30 

59.94 

44 

49.83 

58 

36.74 

72 

21.43 

4 

68.99 

18 

65.80 

32 

58.70 

46 

48.12 

60 

34.67 

74 

19.12 

6 

68.78 

20 

65.02 

34 

57.39 

48 

46.36 

62 

32.55 

76 

16.78 

8 

68.49 

22 

64.15 

36 

56.01 

50 

44.54 

64 

30.40 

78 

14.42 

10 

68.12 

24 

63.21 

38 

54.56 

52 

42.67 

66 

28.21 

80 

12.05 

12 

67.66 

26 

62.20 

40 

53.05 

54 

40.74 

68 

25.98 

82 

9.66 

See p 34. 

Inches reduced to Decimals of a Foot. 

No errors. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

0 

.0000 

2 

.1667 

4 

.3333 

6 

.5000 

8 

.6667 

10 

.8333 

1-32 

.0026 


.1693 


.3359 


.5026 


.6693 

.8359 

1 16 

.0052 


.1719 


.3385 


.5052 


.6719 


.8385 

3 32 

.0078 


.1745 


.3411 


.5078 


.6745 


.8411 

X 

.0101 

X 

.1771 

X 

.3438 

X 

.5104 

X 

.6771 

X 

.8438 

5 32 

.0130 

.1797 


.3464 


.5130 


.6797 

.8464 

316 

.0156 


.1823 


.3490 


.5156 


.6823 


.8490 

7-32 

.0182 


.1849 


.3516 


.5182 


.6849 


.8516 

x 

.0208 

. X 

.1875 

X 

.3542 

X 

.5208 

X 

.6875 

X 

.8542 

9 32 

.0234 


.1901 


.3568 


.5234 

.6901 

.8568 

5-16 

.0260 


.1927 


.3594 


.5260 


.6927 


.8594 

11 32 

.0286 


.1953 


.3620 


.5286 


.6953 


.8620 

X 

.0313 

% 

.1979 

X 

.3646 

X 

.5313 

X 

.6979 

X 

.8646 

13 32 

.0339 


.2005 


.3672 

.5339 

.7005 

.8672 

7-16 

.0365 


.2031 


.3698 


.5365 


.7031 


.8698 

15-32 

.0391 


.2057 


.3724 


.5391 


.7057 


.8724 

X 

.0417 

X 

.2083 

X 

.3750 

X 

.5417 

X 

.7083 

X 

.8750 

17 32 

.0443 


.2109 


.3776 

.5443 

.7109 

.8776 

9 16 

.0469 


.2135 


.3802 


.5469 


.7135 


.8802 

19-32 

.0495 


.2161 


.3828 


.5495 


.7161 


.8828 

% 

.0521 

% 

.2188 

X 

.3854 

% 

.5521 

% 

.7188 

X 

.8854 

21 32 

.0547 


.2214 


.3880 

.5547 

.7214 

.8880 

11 16 

.0573 


.2240 


.3906 


.5573 


.7240 


.8906 

23 32 

.0599 


.2266 


.3932 


.5599 


.7266 


.8932 

X 

.0625 

X 

.2292 

X 

.3958 

X 

.5625 

X 

.7292 

X 

.8958 

25-32 

.0651 


.2318 


.3984 

.5651 

.7318 

.8984 

13 16 

.0677 


.2344 


.4010 


.5677 


.7344 


.9010 

27-32 

.0703 


.2370 


.4036 


.5703 


.7370 


.9036 

% 

.0729 

X 

.2396 

X 

.4063 

X 

.5729 

X 

.73ifc 

X 

.9063 

29 32 

.0755 


.2422 


.4089 

.5755 

.7422 

.9089 

15-16 

.0781 


.2448 


.4115 


.5781 


.7448 


.9115 

31-32 

.0807 


.2474 


.4141 


.5807 


.7474 


.9141 

1 

.0833 

3 

.2500 

5 

.4167 

7 

.5833 

9 

.7500 

11 

.9167 


.0859 


.2526 


.4193 


.5859 

.7526 

.9193 


.0885 


.2552 


.4219 


.5885 


.7552 


.9219 


.0911 


.2578 


.4245 


.5911 


.7578 


.9245 

X 

.0938 

X 

.2604 

X 

.4271 

X 

.5938 

X 

.7604 

X 

.9271 


.0964 


.2630 


.4297 

.5964 

.7630 

.9297 


.0990 


.2656 


.4323 


.5990 


.7656 


.9323 


.1016 


.2682 


.4349 


.6016 


.7682 


.9349 

X 

.1042 

X 

.2708 

X 

.4375 

X 

.6042 

X 

.7708 

X 

.9375 


.1068 


.2734 


.4401 

.6068 

.7734 

.9401 


.1094 


.2760 


.4427 


.6094 


.7760 


.9427 


.1120 


.2786 


.4453 


.6120 


.7786 


.9453 

% 

.1146 

% 

.2813 

% 

.4479 

X 

.6146 

X 

.7813 

X 

.9479 


.1172 


.2839 


.4505 

.6172 

.7839 

.9505 


.1198 


.2865 


.4531 


.6198 


.7865 


.9531 


.1224 

X 

.2891 


.4557 


.6224 


.7891 


.9557 

X 

.1250 

.2917 

X 

.4583 

X 

.6250 

X 

.7917 

X 

.9583 


.1276 


.2943 


.4609 

.6276 

.7943 

.9609 


.1302 


.2969 


.4635 


.6302 


.7969 


.9635 


.1328 


.2995 


.4661 


.6328 


.7995 


.9661 

% 

.1354 

% 

.3021 

X 

.4688 

% 

.6354 

X 

.8021 

X 

.9688 


.1380 


.3047 


.4714 

.6380 

.8047 

.9714 


.1406 


.3073 


.4740 


.6106 


.8073 


.9740 

X 

.1432 


.3099 


.4766 


.6432 


.8099 


.9766 

.1458 

X 

.3125 

X 

.4792 

X 

.6458 

X 

.8125 

X 

.9792 


.1484 


.3151 


.4818 


.6484 

.8151 

.9818 


.1510 


.3177 


.4844 


.6510 


.8177 


.9844 

X 

.1536 


.3203 


.4870 


.6536 


.8203 


.9870 

.1563 

X 

.3229 

X 

.4896 

X 

.6563 

X 

.8229 

X 

.9896 


.1589 


.3255 


.4922 

.6589 

.8255 

.9922 


.1615 


.3281 


.4948 


.6615 


.8281 


.9948 


.1641 


.3307 


.4974 


.6641 


.8307 


.9974 
























































WEIGHTS AND MEASURES, 


389 


Square, or Land Measure. 

U. 8. and British. 

144 square inches. 1 S q foot. 100 sq ft = 1 square. 

*’1 feet -”. 1 sq yard - 1296 sq ins. 

3° y< sq yards. 1 sq rod - 272)4 sq feet. 

40 sq !; ods . 1 rood — 1210 sq yds = 10890 sq feet. 

4 roods ..1 acre = 100 rods — 4840 sq y ds = 43560 sq feet. 

Iand K l - mile i q ' ° r 27878400 sq ft 5 nr 3097600 sq yds; or 640 acres. An acre 
ouattlr q in t l^-“‘r. 3 chalas ,: A “«*« ls 2 08.710 feet; a sq half acre, 147.581 ft; and a sq 
’ 40 M o:> 4t on euch s ’de. A circular acre is 235.504 feet: a circular half acre = 
li tn. -. s; f, r - a iK‘ t r ^ ar q. uarter acre = 117.752 ft diam. A circular Inch is a circle of 1 inch 
diam, a sq it _ 183.346 cir ins. Also 1 sq Inch = 1.27324 cir ius ; and 1 


iuch. 


cir inch = .7854 of a sq 


Cubic, or Solid Measure. 


U. 8. and British. 

1728 cubic inches. 1 cubic, or solid foot. 

27 cubic feet... 1 cubic, or solid yard. 

A °T.^ 0 i 0d = !*L ing 4 ft X 4 ft X 8 ft. A perch of masonry actually con¬ 

tains cub It; being 16)4 ft X 1)4 ft X 1 ft. It is generally taken at 25 cub ft; but by some at 22 . 
. I and there is every probability that a payer will be cheated unless the number of cubic ft be dis¬ 
tinctly agreed upon iu his contract. It is gradually falling into disuse among engineers; and the cub 
yd is very properly taking its place. To reduee cub yds to perches of 25 cub ft, mult by 1.080; 

re duce perches to cub yds, mult by .926. The Brit rod of brickwork, of house builders, is 
16^ feet square, by 14 inches (1)4 English bricks) thick = 272)4 sq ft of 14 inch wall, it is conven¬ 
tionally taken at 272 sq ft; which gives 317cub ft. In Brit engineering works the rod is 306 cub 
ft, or 1134 cub yds. The Montreal, (Canada,) tolse — 261)4 cub ft; or 9.6852 cub yds, or 10.46 
perches of 25 cub ft. The Canadian chaldron = 58.64 cub ft. A ton (2240 lbs) of Pennsylvania 
anthracite, when broken for domestic use, occupies from 41 to 43 cub ft of space; the mean of which 
is equal to 1.556 cub yds: or a cube of 3.476 ft on each edge. Bituminous coal 44 to 48 cub ft; mean 
equal to 1.704 cub yd; or a cube of 3.583 ft on each edge. Coke 80 cub ft. 

A cubic loot is equal to 

1728 cub ins, or 3300.23 spherical ins. 

.037037 cub yard, or 1.90985 spherical ft. 

.002832 myriolitre, or decastere. 

.028316 kilolitre, or cubic metre, or stere. 

.283161 hectolitre, or decistere. 

2.83161 decalitres, or centisteres. 

28.3161 litres, or cub decimetres. 

283.161 decilitres. 

2831.61 centilitres. 

28316.1 millilitres, or cub centimetres. 

.803564 U.S. struck bushel of 2150.42 cub ins, or 
1.24445 cub ft. 

.779013 Brit bushel of 2218.191 cub ins, or 1.28368 
cub ft. 

S. 21426 U. S. pecks. • 

A cubic iuch is equal to 

16.38663 millilitres; or 1.638663 centilitres; or .1638663 decilitre; or .01638663 litre; or to .0005787 
cub ft; or to .138528 U. S. gill; or 1.90985 spherical ins. 

A cubic yard is equal to 

S. galls. 


3.11605 Brit pecks. 

7.48052 U. S. liquid galls of 231 cub ins. 
6.42851 U. S. dry galls. 

6.23210 Brit galls of 277.274 cub ins. 
29.92208 U. S. liquid quarts. 

25.71405 U. S. dry quarts. 

24.92842 Brit quarts. 

59.84416 U. S. liquid pints. 

51.42809 U. 8. dry pints. 

49.85684 Brit pints. 

239.37662 U. S. gills. 

199.42737 Brit gills. 

.26667 Hour barrel of 3 struck bushels. 
.23748 U. S. liquid barrel of 31)4 galls. 


27 cub feet, or to 201.974 U 
46656 cub ins. 

.0764534 myriolitre. 

.764534 kilolitre, or cub metre. 
7.64534 heciolitres. 

7.2 flour barrels of 3 struck bushels. 


76.4534 decalitres. 

764.534 litres, or cub decimetres. 
7645.34 decilitres. 

21.69623 U. S. bushels (struck). 
21.03336 Brit bushels. 


A sphere 1 foot in 

.01939 cub yard. 

.5236 cub foot. 

904.781 cub inches. 

.42075 U. S. bushel. 

1.6830 U. S. pecks. 

13.4639 U. S. dry quarts. 

26.9278 U. 8. dry flints. 

3.9168 U. S. liquid gallons. 

15.6672 U. S. liquid quarts. 

A sphere 1 inch in 

.000303 cub foot. 

.5236 cub iuch. 

.07253 U. S. gill. 


diameter, contains 

31.3344 U. S. liquid pints. 
125.3376 U. S. liquid gills. 
3.2631 Brit imp gallons. 
13.6525 Brit imp quarts. 
26.1050 Brit imp pints. 
104.4201 Brit imp gills. 
14.8263 litres. 

1.48263 decalitres. 

.148263 hectolitres. 

diameter, contains 

.06043 Brit gill. 

8.580 millilitre. 

.8580 ceutilitre. 

.08580 decilitre. 
















390 


WEIGHTS AND MEASURES, 


A cylinder 1 foot, in diameter, 

.02909 cub yard. 

.7854 cub foot. 

1357.1712 cub inches. 

.63112 U. S. dry bushels. 

2.5245 U. S. dry pecks. 

20.1958 U. S. dry quarts. 

40.3916 U. S. dry piuts. 

5.8752 U. S. liquid gallons. 

23.5008 U. S. liquid quarts. 


and 1 foot tiigrti, contains 

47.0016 U. S. liquid piuts. 

188.0064 U. 8. liquid gills. 

4.8947 Brit imp gallons. 

19.5788 Brit imp quarts. 

39.1575 Brit imp pints. 

156.6302 Brit imp gills. 

222.395 decilitres. 

22.2395 litres. 

2.22395 decalitres. 

.222395 hectolitre. 

and 1 foot hit'll, contains 

.2719 Brit imp pint. 

1.0877 Brit imp gill. 

15.4441 centilitres. 

1.54441 decilitres. 

.154441 litres. 


A cylinder 1 inch in diameter, 

.005454 cub foot. 

9.4248 cub inches. 

.2805 U. S. dry pint. 

.3264 liquid pint. 

1.3056 U. S. gill. 


s. 

A cylinder 7 ins 


For others, see below; also p 157. 


ILiquhl Measure, c. 8. only. 

The bnsla of this measure in the U. S. is the old Brit wine gallon of 231 cub ins - or 8 33888 
avoir of pure water, at its max density of about 39°.2 Fahr; the barom at 30 ius. A cylinder 7 

6 ius high, contains 230.904 cub ins, or almost precisely a gallon ; as does a’lso a cube of 
6.13o8 ins on an edge. Also a gallon — .13368 of a cub ft,; and a cub ft contains 7.48052 galls ; nearly 
ally weighs 8 34500?it> “ owever h»T°lve« an error of about 1 part in 1362, for the water actu- 
cub ins. 

4 *! lls .* P'n‘ = 28.875. | 63 gallons.1 hogshead. 

'j P ,nts .} quart — 57..7a0 — 8 gills. I 2 hogsheads. 1 pipe, or butt. 

4 quarts.1 gallon — 231. =8 pints = 32 gills. | 2 pipes. l tun. 

In the U. S. and Great Brit. 1 barrel of wine or brandv = 31*4 galls : in Pennsylvania, a hair 
ga i S: f o,^ ubl n barre1 ’. « 4 galls; a puncheon, 84 galls; a tierce, 42 galls, a liquid 
measure bai re 11 of 31*4 galls contains 4.211 cub ft = a cube of 1.615 ft on au edge ; or 3.384 U. S. struck 
bushels. A gill — 7.21875 cub ius. The following cylinders contain some of these measures 
very approximately. For others, see above; also p 157. 


Diam. Height, 

cub ins. Ins. Ins. 

Gill (7.21875). 1 % . 3 

H Pint. 2*4 . 3H 

Pint. 314 . 3 

Quart. . 6 


Diam. 
Ins. 

Gallon. 7 . 

2 gallons. 7 . 

8 gallons. 14 . 

10 gallons. 14 . 


Height. 

Ins. 

. 6 
. 12 
. 12 
. 15 


^ s * measures to Brit ones of the same denomina 

tion, dmde by 1.20032; or near enough tor commou use, by 1.2; or to reduce Brit to U. S. multi pi 

I>ry measure. 

U. S. only. 

ins T !^ < 77 V-£nV 4 wf is the old British Winchester struck bushel of 2150.42 cul 

ins , 0^77.621413 pounds avoir of pure water at its max density. Its dimensions by law are 18*^ in 

L ° Ute r dia ™ : aud ® ius de ^ ; aud wheu Soaped, the cone is uo{ to be Jess thaL , 

ius high , which makes a heaped busbel equal to 1*4 struck ones ; or to 1.56556 cub it. “ 

Edge of a cube of 

2 pints 1 quart, = 67.2006 cub ins = 1.16365 liquid qt. ^““^066 mT 

4 quarts 1 gallon, = 8 pints, = 268.8025 cub ius, = i.16365 liq gal'.!!!’.".’. K 454 “ 

2 gallons peck = 16 p.nts, = 8 quarts, = 537.6050 cub insIf.. !.. ..! ...!'" 81*1 » 

4 pecks 1 struck bushel, - 64 piuts, = 32 quarts, = 8 gals, = 2150.4200 cub ins. 12.908 “ 

* tr, * ck l>«sl*el = 1.24445 cub ft. A cub ft = .80856 of a struck bushel 
The dry flour barrel = 3.75 cub ft; =3 struck bushels The v Un!il 

amount of its contents should be speciLd in pou^s nr gaUs ’ Ug by lbt 

hv 1 n 3 ififi <lu !f. e S * measures to Brit imp ones of the same name div 

by 1.031516, and to reduce Brit ones to U. S. mult by 1.031516 ; or for common purposesuse P032 







































WEIGHTS AND MEASURES. 


39T 


British Imperial Measure, both liquid and dry. 

This system is established throughout Great Britain, to the exclusion of the old ones. Its basis ia 
the imperial gallon of 277.274 cub ins, or 10 lbs avoir of pure water at the temp of 62° Fahr, when 

the barom is at 30 ins. This basis involves an error of about 1 part in 

1836, for 10 lbs of the water = only 277.123 cub ins. 



Avoir lbs. 

Of water. 

Cub. ins. 

Cub. ft. 

Edge of a cube of 

equal capacity. 
Inches. 

4 gills 1 pint. 

2 pints 1 quart. 

2 quarts 1 pottle. 

2 pottles 1 gallon. 

i 2 gallons 1 peck. 

4 pecks 1 bushel. 

4 bushelsl coomb. 

j 2 coombs 1 quarter. 

1.25 

2.50 

5. 

10. 

20. 1 

80. 1 Dry 
320. ( meas. 
640. J 

34.6592 

69.3185 

138.637 

277.274 

554.548 

2218.192 

8872.768 

17745.536 

1.2837 

5.1347 

10.2694 

3.2605 

4.1079 

5.1756 

6.5208 

8.2157 

13.0417 


The imp gall = .16046 cub ft; and 1 cub ft= 6.23210 galls. The imp gal = 1.20032, or very nearly 


TJ. S. liquid galls. 


The weight of water affords an easy way to find the cubic contents of a vessel. First weigh the ves¬ 
sel by itself; and then full of water. The diff will be the weight of the water; and this divided by 
62.3 or by the number in the table opp the temp of water, will be the contents in cub ft. 


To obtain the size of commercial measures by means of the 

weiii'ht of water. 



An inch square’, and one foot long, .432292 ft. Also 1 B = 27.75903 cub ins, or a cube of 3.028 ins on an 
edge. An ounce, 1.735 cub ins ; a ton, 35.984 cub ft, all near enough for common nse. 

Original. 


Liquid Measures. 


XT. S* Gill. 

i U. S. Pint. 

U. S. Quart. 

TJ. S. Gallon 8 lbs 5^ oz. 

U. S. Wine Barrel, 3l}4 Gall. 


Lbs Avoir, 
of Water. 
.26005* 
1.0402 
2.0804 
8.3216 
262.1310 


Liquid and Dry. Lbs Avoir. 

1 J of Water. 


British Imp Gill.31214* 

“ “ Pint. 1.24858 

“ “ Quart. 2.49715 

“ “ Gallon. 9.9886 

“ “ Peck. 19.9772 

“ “ Bushel... 79.9088 


Dry Measures. 


* 4.9942; or very nearly 5 ounces. 


U. S. Pint. 1.2104 

U. S. Quart. 2.4208 

TJ. S. Gallon. 9.6834 

U. S. Peck. 19.3668 

TJ. S. Bushel, struck. 77.4670 


* Or 4 ounces; 2 drams; 15.6625 grs. 


French Measures. 


Centilitre.. .021987 

Decilitre. .21984 

Litre. 2.1981 

Decalitre, or Centistere. 21.9808 

Metre, or Stere.... 2198.0786 


t Or 5.6271 drams; or 153.866 grs. 
4 3.5169 ounces. 


-- - -- 

METRIC WEIGHTS AND MEASURES. 


The French Metre. 

The French metre was intended to be the one ten-millionth part of the dist from either pole of the 
|, earth to the equator: but after it had been introduced into use, errors were discovered iu the calcu- 
s lations employed for ascertaining that dist; so that the French metre, like the Brit standard yard, 
is not what it was intended to be. 

e The IT. S. Govt adopts for its length 1.093623 yds = 3.280869 ft = 39.370432 
ins U.S. or British measure. But in ordinary business transactions 
^ 39.37 ius are a legal metre. At 3 ft 3% ins, the length h> part in 8616 too great. 
































































WEIGHTS AND MEASURES. 




French Measures of Length. 
By U. 8. and British Standard. 


Millimetre*. 

Centimetref. 


Decimetre 

Metre % . 

Decametre.. 

Hectometre 

Kilometre... 

Myriametre 


Ins. 

Ft. 

.039370 

.003281 

.39370428 

.032809 


.3280869 

39.370428 

3.280869 

393.70428 

32.80869 

Road 

328.0869 

measures. 

3280.869 

32808.69 


Yds. 


.1093623 

1.093623 

10.93623 

109.3623 

1093.623 

10936.23 


Miles. 


.0621375 

.6213750 

6.213750 


t Full % inch. 


* Nearly the A- part of an inch. 

1 Very nearly 3 ft, 3% ins, which is too long by only 1 part in 8616. 

French Square Measure. 

By U. 8. and British Standard. 



Sq. Ins. 

Sq. Feet. 

Sq. Yds. 

Sq Millimetre. 

.001550 

.00001076 

.0000012 

Sq Centimetre. 

.155003 

.00107641 

.0001196 

Sq Decimetre. 

15.5003 

.10764101 

.0119601 

Sq Metre, or Centiare. 

1550.03 

10.764101 

1.19601 

Sq Decametre, or Are. 

Decare (not used). 

155003 

1076.4101 

10764.101 

i<V7«di m 

119.6011 

11 QA A11 

Hectare. 


1 10AA 1 1 

Sq Kilometre. 

3861090 sq miles. 

10764101 

1196011. 

Sq Myriametre. 

38.61090 “ 



Acres. 


.000247 

.024711 

.247110 

2.47110 

247.110 

24711.0 


French Cubic, or Solid Measure. 
According to C. 8. Standard. 

Only those marked “Brit” are British. 


Millilitre, or cub 
Centimetre.... 


Centilitre. 


Decilitre. 


Litre, or cubic 
Decimetre...;.. 


Cub Ins. 

.0610254 


.610254 


6.10254 


61.0254 


Decalitre, 

Centistere. 


or 


Hectolitre, 
Decistere. 


or 


Kilolitre, or 
Cubic Metre, 
or Stere. 


Myriolitre, 
Dc 


or 


)ecastere. 


610.254 

Cub Ft. 

.353156 


3.53156 


35.3156 


(Liquid. .0084537 gill. 

< “ .0070428 Brit gill. 

(Dry. .0018162 dry pint. 


(Liquid. .084537 gill. 

< “ .070428 Brit gill. 
(Dry. .018162 dry pint. 


(Liquid. .84537 gill = .21134 pint. 
i “ .70428 Brit gill = .17607 Brit pint. 

(Dry. .18162 dry pint. 


(Liquid. 1.05671 quart = 2.1134 pints. 

1 “ -88036 Brit quart = 1.7607 Brit pints. 

(P noir. , -’ - 


( Liquid. 2.64179 U. S. liquid gal. 
] “ 2.20090 Brit gal. 

'-Dry. 


.283783 bush = 1.1351 peck = 9.081 dry qts. 


(Liquid. 26.4179 U. S. liquid gal. 
< “ 22.0090 Brit gal. 

(Dry. 2.83783 bush. 


353.156 


| Liquid. 264.179 U. S. liquid gal 1 

W SE !i J Cub yds, 1.3080. 


( Liquid. 2641.79 U. S. liquid gal. 1 ~ J 
(Dry- 283.783 bush. } Cub yds, 13.080. 


.oowou Din quart = l.iou/ rsrit pints. 

. Dry. .11351 peck = .9081 dry qt = 1.8162 dry pt. 















































































WEIGHTS AND MEASURES 


393 


French Weights, reduced to common Commercial or Avoir 
Weight, of 1 pound - 16 ounces, or 7000 grains. 



Grains. 

Milligramme. 

.015432 

.15432 

Centigramme. 

Decigramme. 

1.5432 

Gramme. 

15.432 

By law a 5-cent nickel = 5 grammes. 

Pounds av. 

Decagramme. 

.022046 

Hectogramme. 

.22046 

Kilogramme. 

2.2046 

Myriogramme. 

22.046 

Quintal*. 

220.46 

Tonneau; Millier; or Tonne. 

2204.6 



The gramme is the basis of French weights; and is the weight of a cub centimetre of distilled 
water at its max density, at sea level, in lat of Paris ; barom 29.922 ins. 

French Measures of the “ System® "Usuel.” 

This system was in use from about 1812 to 1840, when it was forbidden by law to use even its names. 
This was done in order to expedite the general use of the tables which we have before given. But as 
the Systems Usuel appears in books published during the above interval, we add a table of some of its 
values. 

Measures of Length. 


I.igne usuel, or line. 

Pouce usuel, or inch, = 12 lignes 
Pied usuel, or foot, = 12 pouces . 

.Aune usuel, or ell. 

Toise usuel, = 6pieds. 


Weights, Usuel. 


Grain usuel. 

.8375 grains. 

Gros usuel. 

60.297 ,4 

Once usuel. 

1.10258 avoir oz. 

Marc usuel. 

.55129 avoir lb. 

Livre usuel, > 
or pound, y * 

1.10258 avoir lb. 


Yards. 

Feet. 

Inches. 



.09113 


.09113 

1.09362 

.36454 

1.09362 

13.12344 

1.31236 

3.93708 

47.245 

2.18727 

6.56181 

78.74172 


Cubic, or Solid, Usuel. 


Litron usuel, or 1 litre = 1.7608 British pint. 
Boisseau usuel. 2.7512 British gals. 


Before 1812, or before the “Systeme usuel," the Old System, “ Systeme Ancien,” was in use. 


French Measures of the “Systeme Ancien.” 


Lineal. 


Square. 


Cubic. 


Point ancien, .0148 ins. 

Ligne ancien, .0888 ins. 

Pouce ancien, 1.06577 ins = .0888 ft. 

Pied ancien, 12.7892 ins = 1.06577 ft. 

Aune ancien, 46.8939 ins = 3.90782 ft= 1.30261 yds 

Toise ancien. = 6.3946 ft= 2.1315 yds. 

League = 2282 toises = 2.7637 miles. 


Sq. ins. 

.00789 

1.1359 


Sq. ft. Sq. yds. 
i.1359 

40.8908 4.5434 


C. ins. 

C. ft. 

C. yds. 

.0007 


1.2106 



. 

1.2106 



261.482 

9.6845 


There is, however, much confusion about these old measures. Different measures bud tue auine 
name in different provinces. 


* The avoirdupois quintal is 100 avoirdupois pounds. 















































































394 


WEIGHTS AND MEASURES, 


Russian. 

Foot; same as U. S. or British foot. Sachine = 7 feet. Verst = 500 
sachine — 3500 feet = 1166% yards = .6629 mile. Pood = 36.114 lbs avoirdupois. 


Spanish. 

The eastellano of Spain and New Granada, for weighing gold, is variously 
estimated, from 71.07 to 71.04 grains. At 71.055 grains, (the mean between the 
two,) an avoirdupois, or common commercial ounce contains 6.1572 eastellano; 
and a lb avoirdupois contains 98.515. Also a troy ounce = 6.7553 eastellano ; and 
a troy lb = 81.064 eastellano. Three U. S. gold dollars weigh about 1.1 eastellano. 

The Spanish mark, or marco, for precious metals, in South America, 
may be talcen in practice, as .5065 of a lb avoirdupois. In Spain, .5076 lb. In 
other parts of Europe, it has a great number of values; most of them, howevpr, 
being between .5 and .54 of a pound avoirdupois. The .5065 of a lb = 3545% 
grains; and .5076 lb = 3553.2 grains. 1 marco = 50 Castellanos = 400 tomine = 
4800 Spanish ^oM-grains. 

The arroba has various values in different parts of Spain. That of Cas¬ 
tile, or Madrid, is 25.4025 lbs avoirdupois; the tonelada of Castile = 203^2 
lbs avoirdupois; the quintal = 101.61 lbs avoirdupois; the libra = 1 0161 
lbs avoirdupois; the cantara of wine, Ac, of Castile = 4.263 U. S. gallons; 
that of Havana = 4.1 gallons. 

The vara of Casiile = 32.8748 inches, or almost precisely 32% inches; or 2 
feet 8% inches. The fanegada of land since 1801 = 1.5871 acres = 69134 08 
square feet. The fanega of corn. Ac = 1.59914 U. S. struck bushels. In 
California, the vara by law = 33.372 U. S. inches; and the legua = 5000 
varas; or 2.6335 U. S. miles. 







TIME 


395 


Civil, or Common Clock. Time. 


60 thirds, marked 

1 second, marked ". 

60 seconds 

1 minute 

60 minutes 

1 hour, rr 3600 sec. 

24 hours 

1 civil day, = 1440 min. =: 86400 sec. 

7 days 

1 week, ~ 168 hours 10080 min. 

4 weeks 

1 civil month, = 28 days = 672 hours. 


For Standard Railway Time, see p 396. 

IS civil months, (or 52 weeks.) 1 day, 5 hours, 48 min, 49sec; or 365 days,5 hours,48 min, 49^- 

•ec, — 1 civil year. A solar day is the time between two successive solar noons, or transits of the 
j sun over the meridian of a place. These intervals are not of equal lengths all the year round. The 
average length of all the solar days is called the mean solar day; and is the same as the common 
civil day of 24 hours of clock time. Civil noon is at 12 o'clock ; but solar, or apparent noon, may be 
about li% min before; or min after 12 of correct clock time. A sidereal day is the interval 
between two passages of the same star past the range of two fixed objects; aud is the precise time 
reqd for one complete rev of the earth on its axis. The sidereal day never varies; but is always equal 
to 23 hours, 56 min, 409 sec;* so that a star will on any night appear to set, or to pass the range of 
any two fixed objects, 3 min, 55.91 sec earlier by the clock, than it did on the night before,t so that 
the number of sidereal days in a civil year is 1 greater than that of the civil days. 

An astronomical day begins at noon, and its hours are counted from 0 to 24. In comparing it 
with the civil day, the last is supposed to begin at the midnight before the noon at which the first began. 

Astronomers are now (1884-5) taking measures to make their “day” correspond 
with the civil day. 


TABLE showing how much earlier a star passes a given 
range, on each succeeding night. — (Original.) 


Nights. 

Min. 

Sec. 


Nights. 

H. 

Min. 

Sec. 


Nights. 

H. 

Min. 

Sec. 

1 

3 

55.91 


11 


43 

15.01 


21 

1 

22 

34.11 

2 

7 

51.82 


12 


47 

10.92 


22 

1 

26 

30.02 

3 

11 

47.73 


13 


51 

C.83 


23 

1 

30 

25.93 

4 

15 

43.64 


14 


55 

2.74 


24 

I 

34 

21.84. 

5 

19 

39.55 


15 


58 

58.65 


25 

1 

38 

17.75 

6 

23 

35.46 


16 

1 

2 

54.56 


26 

1 

42 

13.66 

7 

27 

31.37 


17 

1 

6 

50.47 


27 

1 

46 

9.57 

8 

31 

27.28 


18 

1 

10 

46.38 


28 

1 

50 

5.48 

9 

35 

23.19 


19 

1 

14 

42.29 


29 

1 

54 

1.39 

10 

39 

19.10 


20 

1 

18 

38.20 


30 

1 

57 

57.30 










31 

2 

1 

53.21 


* This gives a means of regulating a watch with much accuracy and 

by a very simple process. The writer, after having regulated his chronometer watch for a year by 
this method only, differed but a few seconds from the actual time as deduced from careful solar obser¬ 
vations. Even a person not accustomed to ranging objects very accurately, need scarcely err a min¬ 
ute in a period of any number of years. It having occurred to him that the motion of a star in a 
second or two might be visible to the naked eye. he stuck a pin horizontally into a window-jamb ; and 
placing his eye close to it, sighted along one side of it, at a large star setting behind the top of a roof 
about 100 feet distant, and found that his conjecture was correct. Those stars which are farthest 
from the poles appear to move the fastest, and are therefore the best. Those less than of the second mag¬ 
nitude are uot satisfactory. If the first observations of a given star be made as late as midnight, that 
same star will answer for about three months, until at last it will begin to pass.the range in daylight. 
Before this happens, the observer must transfer the time to another star which sets later; if near 
midnight, the better, as it will serve for a longer time A window looking west is the best. The 
longer the range, the greater will be the apparent motion of the star; and, consequently, the obser- 
vatTons will be more correct. If such a range can be secured as will strike the heavens at an angle 
of at least 40° above the horizon, the error from refraction will not appreciably affect an observation ; 
at a much less angle it may do so to the extent of three or four seconds. A candle must be so placed 
as to reader the pin and the watch visible at the same time. A little practice will render the process 
very easy, and supersede the necessity for more remarks on the subject. Of course, a memorandum 
must be made and preserved of the date, hour, minute, and (approximately) second, at which the 
first passage of the star took place. Subsequent passages will occur earlier, a,s shown in the forego¬ 
ing table. The watch must be previously known to be right, when taking the first observation, it we 
require afterward to keep the correct time. Any person who will take the trouble thus to observe, 
and note down throughout a year, about half a dozen stars following each other at tolerably equal 
intervals of time, will on almost any clear night afterward be able, after a short calculation, to ascer¬ 
tain the correct clock time. The writer observed the passages of two or three stars behind different 
ranges, on the same nights, in order to obtain a mean of several observations; his object being to 
ascertain how pocket chronometers of the best makers would keep time utider the vicissitudes of tem¬ 
perature. railroad travelling, &c, &c, to which they are ordinarily exposed. He used two of the best 
for this purpose, and the result was that their changes of rate were at times asS r . eat as f ™“ 1 * 
eight seconds per day. For ordinary purposes, therefore, they are of but little, if any, more service 
than a good common watch, of one-fourth the cost* 

T More accurately 3 znin # 55.90944 sec. 


















396 


STANDARD RAILWAY TIME. 


STANDARD RAILWAY TIME, ADOPTED 1883. 

The following arrangement of standard time was recommended by the General 
and Southern Time Conventions of the railroads of the United States and Canada 
held respectively in St. Louis, Mo., and New York city, April, 1883, and in Chicago, 
Ill., and New York city, in October, 1883, and went into effect on most of the rail- 
roads of the United States and Canada, November 18th, 1883. Most of the principal 
cities of the United States have made their respective local times to correspond with 
it. This system was proposed by Mr. W. F. Allen, Secretary of the Time Conven¬ 
tions, and its adoption was largely due to his efforts. We are indebted to Mr. Allen 
for documents from which the following has been condensed. Five standards of time, 
or five‘•times,” have been adopted for the United States and Canada These are 
respectively the mean times of the 60th, 75th, 90th, 105th, and 120th meridians west 
of Greenwich, England. As each of these meridians, in the above order is 15° west 
of its predecessor, its time is one hour slower. Thus, when it is noon on the 90th 
meridian, it is 1 p. m. on the 75th, and 11 a. m. on the 105th. The following gives 
the name adopted for the standard time of each meridian, and the conventional 
color adopted, and uniformly adhered to, by Mr. Allen, for the purpose of designat¬ 
ing it and its time, Ac, on the maps published under his auspices: 


Longitude west 

Name of 

Conventional 

from Greenwich. 

Standard Time. 

color. 

60° 

Intercolonial. 

Brown. 

75° 

Eastern. 

Red. 

90° 

Central. 

Blue. 

105° 

Mountain. 

Green. 

120° 

Pacific. 

Yellow. 


Theoretically, each meridian may be said to give the time for a strip of country 
lo wide running north and south, and having the meridian for its center. Thus 
the meridian on which the change of time between two standard meridians is snu- 
posed to take place, lies lialf-way between them. But it would, of course, not be 
f.™ c ‘ ca , b e for the railroads to use an imaginary line in passing from one time 
standard to another. The changes are made at prominent stations forming the ter- 
mini of two or more lines; or, as in the case of the long Pacific roads, at the ends 
of divisions. As far as practicable, points at which changes of time had previously 
been made were selected as the changing points under the new system Detroit 
Mich. Pittsburgh, Pa, Wheeling and Parkersburg, W. Ya, and Augusta, Ga„ al¬ 
though not situated upon the same meridian, are points of change between eastern 
a d central standard times. A train arriving at Pittsburgh frona the east at noon 
and leaving for the west 10 minutes after its arrival, leaves (by the figures shown 

^ mT P ; Thus, most of the roads between Buffalo and Detroit, on the 

Idrfenftlf °! t ak l I i| rie ’ r ",> eastern,” or “red,” time; while those on the south 
side of the Lake, between Buffalo and Toledo, immediately opposite to and directlv 

south of them, run by “central” or “blue” time. oirectiy 

If the changes of time were made at the meridians midway between the standard 

ZVt t e u e< ; e8fiary for t0Wn to chan S e its time more than 30 min- 
utes. As it is, somewhat greater changes had to be made at a few points Thus 

mean locaV tinier D<3tr01t 18 32 miuute8 ahead > aud at Savannah 36 minutes back, of 

In most cases the necessary change was made upon the railways by simply setting 

,he —»-iuti.-S’-TSSS 

Halifax, and a few adjacent cities, use the time of the 60th meridian that being 
the nearest one to them; but the railroads in the same district have adopted the 

Q n I aV 873,her « 71 tim ® standftl 'ds in use on the railroads of the United States 
and Canada At the time of the adoption of the present system this number had 
been reduced, by consolidation of roads, Ac, to 53. By its adoptionthe number 1 e 

roS»,‘° ' 1 " i “ dc,1 ’ li0, ‘ w * stern ,im ” G l»t.«oloutal 













DIALS 


397 


DIALLING. 


u 


h 


To make a horizontal Sun-dial, 

Draw a line a b ; and at right angles to it, draw GG. From any convenient point, as c, 
in a b, draw the perp c o. Make the angle can equal to the lat of the place; also 
the angle c o e equal to the same ; join o e. Make e n equal to o e; and from n as a 
center, with the rad e«, describe a quadrant e s; and div it into 6 equal parts. Draw e 
y, parallel to 6, 6; and 
from n, through the 5 
points on the quadrant, 
draw lines n t, n i, &c, 
terminating in ey. From 
a draw lines a 5, a 4, &c, 
passing through t, i, &c. 

From any convenient 
point, as c, describe an 
arc r vi h, as a kind of fin¬ 
ish or border to half the 
dial. All the lines may 
now be effaced, except 
the hour lines a 6, a 5, 
a 4, &c, to a 12, or ah; 
unless, as is generally 
the case, the dial is to 
be divided to quarters 
of an hour at least. In 
this case each of the 
divisions on the quad¬ 
rant e s, must be subdivided into 4 equal parts; and lines drawn from n, through 
the points of subdivision, terminating in ey. The quarter-hour lines must be drawn 
from a, as were the hour lines. Subdivisions of 5 min maybe made in the same 
way; but these, as well as single min, may usually be laid off around the border, by 
eye. About 8 or 10 times the size of our Fig will be a convenient one for an ordi¬ 
nary dial. To draw the other half of the Fig, make a d equal to the intended thick¬ 
ness of the gnomon, or style, of the dial; and draw of 12, parallel, and equal to a 12; and 
draw the arc xg w, precisely similar to the arc r m h. Between x and w, on the arc xg to, 
space off divisions equal to those on the arc mi h; and number them for the hours, 
as in the Fig. The style F, of metal or stone, (wood is too liable to warp,) will be 
triangular; its thickness must throughout be equal to a of or hw; its base must 
cover the space adhw; its point will be at ad; and its perp height h u, over hiv, 
must be such that lines vd,ua, drawn from its top, down to a and d, will make the 
angles u a //, vd w. each equal to the lat of the place. Its thickness, if of metal, may 
conveniently be from % to % inch; or if of stone, an inch or two, or more, according 
to the size of the dial. Usually, for neatness of appearance, the back huvw of the 
style is hollowed inward. The upper edges, ua, vd, which cast the shadows, must 
be sharp and straight. The dial must be fixed in place hor, or perfectly level; ah 
and d w must be placed truly north and south ; ad being south,and hw north. The 
dial gives only sun or solar time; but clock time can be found by means of the “ fast 
or slow of the sun,” as given by all almanacs. If by the almanac the sun is 5 min, 
<fcc, fast, the dial will be the same; and the clock or watch, to be correct, must be 5 
min slower than it; and vice versa. 



To make a Vertical Sun-Dial. 

Proceed as directed above, except that the angles cao and coe on the drawing, 
and the angle uah or vdw of the style, must, he equal to the co-latitude (== dif¬ 
ference between the latitude and 90°) of the place, and the hours must, be num¬ 
bered the opposite way from those in the above figure; i e. from h to y number 
12, 11, 10, 9, 8, 7; and from w to g number 12,1, 2, 3, 4, 5. The dial plate must be 
placed vertically, in the position shown in the figure, facing exactly south, and 
with ah and dw vertical. 





















398 


WEIGHT OF CAST-IRON, 


TABLE OF WEIGHT OF CAST IRON.* 

The weight of a pattern of perfectly dry white pine, if mult 

by 20, will give approximately the wt of the casting. If well seasoned, but still not 
perfectly dry, mult by 19, or by 18. 

Assuming 450 lbs to a cub ft, a pound contains 3.8400 cubic inches; a ton 5 cub ft; 
and a cubic inch weighs .2604 lbs. 


I Thickness 

1 or Diameter 

1 in Inches. 

Thick¬ 
ness or 
Diam. 
in deci¬ 
mals of 
a foot. 

Wt. of a 
Square 
Foot. 
Lbs. 

Wt. of a 
Square 
bar. 1 ft.' 
long. 
Lbs. 

Wt. of 
Round 
bar, 1 ft. 
long. 
Lbs. 

Wt. of 
Balls. 
Lbs. 

t 

Thickness 

or Diameter 

in Inches. 

Thick¬ 
ness or 
Diam. 
in deci¬ 
mals of 
a foot. 

Wt. of a 
Square 
Foot. 
Lbs. 

Wt. of a 
Square 
bar. 1 ft. 
long. 
Lbs. 

Wt. of a 
Round 
bar, 1 ft. 
long. 
Lbs. 

Wt. of 
Ralls. 
Lbs. 
t 

1-32 

.0026 

1.173 

.003 

.002 


3 X 

.2604 

117.3 

30.52 

23.97 

4.162 

1-16 

.0052 

2.344 

.012 

.010 


M 

.2708 

121.8 

33.01 

25.93 

4.681 

3-32 

.0078 

3.516 

.027 

.021 

.0001 

X 

.2813 

126.5 

35.60 

27.95 

5.243 

h 

.0104 

4.687 

.048 

.038 

.0003 

X 

.2917 

131.2 

38.28 

30.07 

5.846 

5 32 

.0130 

5.861 

.076 

.060 

.0005 

% 

.3021 

135.9 

41.07 

32.25 

6.498 

3-16 

.0156 

7.032 

.110 

.086 

.0009 

X 

.3125 

140.6 

43.95 

34.51 

7.193 

7-32 

.0182 

8.203 

.150 

.118 

.0014 

Vs 

.3229 

145.3 

46.93 

36.85 

7.934 


.0208 

9.375 

.195 

.154 

.0021 

4. 

.3333 

150.0 

50.01 

39.27 

8.726 

y 32 

.0234 

10.54 

.247 

.194 

.0030 

X 

.3438 

154.7 

53.18 

41.77 

9.572 

5-16 

.0260 

11.73 

.305 

.240 

.0042 

Y\ 

.3542 

159.3 

56.46 

44.33 

10.47 

11-32 

.0287 

12.89 

.370 

.290 

.0056 

% 

.3646 

164.0 

59.82 

46.99 

11.42 

% 

.0313 

14.06 

.440 

.346 

.0072 

X 

.3750 

168.7 

63.33 

49.71 

12.43 

13-32 

.0339 

15.24 

.516 

.400 

.0092 

% 

.3854 

173.4 

66.86 

52.52 

13.49 

7-16 

.0365 

16.41 

.598 

.470 

.0114 

X 

.3958 

178.1 

70.52 

55.39 

14.62 

15 32 

.0391 

17.56 

.687 

.540 

.0140 

X 

.4063 

182.8 

74.28 

58.34 

15.81 

j? 

.0417 

18.75 

.781 

.610 

.0170 

5. 

.4167 

187.5 

78.12 

61.37 

17.05 

9 16 

.0469 

21.10 

.989 

.777 

.0243 

X 

.4271 

192.2 

82.10 

64.47 

18.35 

% 

.0521 

23.44 

1.221 

.959 

.0334 

y* 

.4375 

196.9 

86.14 

67.65 

19.73 

11-1-6 

.0573 

25.79 

1.478 

1.161 

.0444 

h 

.4479 

201.6 

90.29 

70.52 

21.18 

X 

.0625 

28.12 

1.758 

1.381 

.0575 

X 

.4583 

206.2 

94.54 

74.26 

22.68 

13-16 

.0677 

30.47 

2.064 

1.621 

.0732 

% 

.4688 

210.9 

98.89 

77.66 

24.27 


.0729 

32.81 

2.393 

1.880 

.0913 

X 

.4792 

215.6 

103.3 

81.16 

25.93 

15 16 

.0781 

35.16 

2.747 

2.158 

.1124 

y» 

.4896 

220.3 

107.9 

84.72 

27.41 

1. 

.0833 

37.50 

3.125 

2.455 

.1363 

6. 

.5000 

225.0 

112.5 

88.36 

29.44 

1-16 

.0885 

39.84 

3.528 

2.771 

.1636 

H 

.5208 

234.4 

122.1 

95.89 

33.28 

x 

.0938 

42.19 

3.955 

3.107 

.1942 

X 

.5417 

243.8 

132.0 

103.7 

37.44 

3-16 

.0990 

44.53 

4.407 

3.461 

.2284 

X 

.5625 

253.1 

142.4 

111.9 

41.94 

V\ 

.1042 

46.87 

4.883 

3.835 

.2664 

7. 

.5833 

262.5 

153.2 

120.2 

46.77 

5-16 

.1094 

49.22 

5.384 

4.229 

.3084 

X 

.6042 

271.9 

164.2 

129.0 

51.97 

% 

1146 

51.57 

5.909 

4.640 

.3546 

X 

.6250 

281.3 

175.8 

138.1 

57.54 

7-16 

.1198 

53.91 

6.461 

5.073 

.4058 

X 

.6458 

290.7 

187.7 

147.4 

63.47 

X 

.1250 

56.26 

7.033 

5.523 

.4603 

8. 

.6667 

300.0 

200.1 

157.0 

69.82 

9 16 

.1302 

58.60 

7.632 

5.993 

.5204 

X 

.6875 

309.4 

212.7 

167.0 

76.58 

% 

.1354 

60.94 

8.253 

6.484 

.5852 

X 

.7083 

318.8 

225.8 

177.3 

83.74 

11-16 

.1406 

63.28 

8.900 

6.991 

.6555 

X 

.7292 

328.2 

239.3 

187.9 

91.35 

X 

.1458 

65.63 

9.572 

7.518 

.7310 

9. 

.7500 

337.4 

253.1 

198.8 

99.42 

13-16 

.1510 

67.97 

10.27 

8.064 

.8122 

X 

,7708 

346.8 

267.4 

210.0 

107.9 

75 

.1563 

70.32 

10.99 

8.630 

.8991 

X 

.7917 

356.2 

282.1 

221.5 

116.8 

15-16 

.1615 

72.66 

11.73 

9.215 

,9920 

X 

.8125 

865.6 

297.0 

233.8 

126.3 

2. 

.1667 

75.01 

12.50 

9.821 

1.073 

10. 

.8333 

875.0 

312.5 

245.5 

136.3 

H 

.1771 

79.70 

14.11 

11.09 

1.308 

X 

.8542 

384.4 

328.4 

257.8 

146.8 

h 

.1875 

84.40 

15.83 

12.43 

1,554 

X 

.8750 

393.7 

344.5 

270.6 

157.9 

h 

.1979 

89.07 

17.63 

13.85 

1.827 

X 

.8958 

403.1 

361.2 

283.7 

169.3 


.2083 

93.75 

19.54 

15.34 

2.131 

11. 

.9167 

412.5 

878.2 

297.0 

181.5 

H 

.2188 

98.44 

21.54 

16.56 

2.467 

X 

.9375 

421.9 

395.5 

319<6 

194.2 

X 

.2292 

103.2 

23.64 

18.56 

2.&'i5 

X 

.9583 

431.2 

413.3 

324.6 

207.3 

X 

, 2396 

107.8 

25.84 

20.29 

3.241 

X 

.9792 

440.6 

431.4 

338.8 

‘219.2 

3. 

.2500 

112.6 

28.13 

22.10 

3.682 

12. 

1 Foot. 

450. 

450. 

353.4 

235.6 


t Wts of balls are as the cubes of their diams. See table, p 416. 

To fiiul the weight of a spherical shell. From the weight of a ball 
which has the outer diam of the shell, take the wt of one which has its inner diarn. 


* For Copper, mult by 1.2; Lead, mult by 1.6; Brass, add l-7th • ZincTmult 
by .97. All approximate. See table, 415. ’ ’ 1 u > mult 






























































WEIGHT OF CAST-IRON PIPES 


399 


WEIGHT OF CAST-IRON PIPES per running foot. 

Ilssuming the weight of cast-iron at 450 fos per cub ft, or ,2604 lb per cub inch. No 
j allowance is here made for the spigot and faucet-joints used in water-pipes. As 
these are now commonly made, (see Fig 38, page 295,) they add to the weight of 
,, each length or section of pipe of any size, about as much as that of 8 inches iji 
' length of the plain pipe as given in the table. 

For load-pipe mult by 1.6; copper, mult by 1.2; brass, add l-7th ; 
welded iron, mult by 1.0667, or add one fifteenth part. 

>- 1 --—--■ 


THICKNESS OF PIPE IN INCHES. 


f 

j °.2 

H 

% | 


% 

% 

% 

1 

ihi 

!/•£ 

| 

| 

I 

2 

••V, 


Wt ia 

Wt ia 

W t ia 

Wt ia 

Wt ia 

Wt ia 

Wt ia 

Wt ia 

Wt ia 

Wt ia 

Wt ia 

Wt in ! 

Wt in 

■ 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 


L. 

3.07 

5.07 

7.38 

9.99 

12.9 

16.2 

19.7 

23.5 

27.7 

32.1 

36.9 

47.4 

59.1 


% 

3.69 

6.00 

8.61 

11.5 

14.8 

18.3 

22 2 

26.3 

30.8 

35.5 

40.6 

51.7 

64.0 



4.30 

6.92 

9.84 

13.1 

16.6 

20.5 

24.6 

29.1 

33.8 

38.9 

44.3 

56.0 

68.9 

(J 

% 

4.92 

7.84 

11.1 

14.6 

18.5 

22.6 

27.1 

31.8 

36.9 

42.3 

48.0 

60.3 

73.8 

5 

2. 

5.53 

8.76 

12.3 

16.2 

20.3 

24.8 

29.5 

34.6 

40.0 

45.7 

51.7 

64.6 

78.7 

g 


6.15 

9.69 

13.5 

17.7 

22.2 

26.9 

32.0 

37.4 

43.1 

49.0 

55.4 

68.9 

86.7 

$ 

1 34 

6.76 

10.6 

14.8 

19.2 

24.0 

29.1 

34.5 

40.1 

46.1 

52.4 

59.1 

73.2 

88.6 

■ft 

H 

7.37 

11.5 

16.0 

20.8 

25.9 

31.2 

36.9 

42.9 

49.2 

55.8 

62 7 

77.5 

93.5 


3. 

7.98 

12.5 

17.2 

22.3 

27.7 

33.4 

39.4 

45 7 

52.3 

59.2 

66 4 

81.8 

98.4 

u 

>4 

8.60 

13.4 

18.5 

23.8 

29.5 

35.5 

41.8 

48.4 

55.4 

62.3 

70.1 

86.1 

103. 

•j 

• 34 

9.21 

14.3 

19.7 

25.4 

31.4 

37.7 

44.3 

51.2 

58.4 

65.9 

73.8 

90.4 

108. 


y* 

9.83 

15.2 

20.9 

26 9 

33.2 

39 8 

46.8 

54.0 

61.5 

69.3 

77.5 

94.7 

113. 


4. 

10.3 

16.1 

22.2 

28.5 

35.1 

42.0 

49.2 

56.7 

64.6 

72.7 

81.2 

99.0 

118. 

* 

i 34 

11.1 

17.1 

23.4 

30.0 

36 9 

44.1 

51.7 

59.5 

67.7 

76.1 

84.9 

103. 

123. 



11.7 

18.0 

24.6 

31.5 

38.8 

46.3 

54.1 

62.3 

70.7 

79.5 

88.6 

108. 

128. 


< y* 

12.3 

18.9 

25.8 

33.1 

40.6 

48.5 

56.6 

65.0 

73.8 

83.9 

92.3 

11*. 

133. 


5 

12.9 

19.8 

27.1 

34.6 

42.5 

50.6 

59.1 

67.8 

76.9 

87.2 

96.0 

116. 

138. 


■ H 

13.5 

20.8 

28.3 

36.1 

44.3 

52.8 

61.5 

70.6 

80.0 

90.6 

99.6 

121. 

143. 

t 

34 

14.2 

21.7 

29.5 

37.7 

46.1 

54.9 

64.0 

73.3 

83.0 

94.0 

103. 

125. 

148. 


H 

14.8 

22.6 

30.8 

39.2 

48.0 

57.1 

66.4 

76.1 

86.1 

97.4 

107. 

129. 

153. 


6. 

15.4 

23.5 

32.0 

40.8 

49.8 

59.2 

68.9 

78.9 

89.2 

99.8 

111. 

134. 

158. 


34 

16.6 

25.4 

34.5 

43.8 

53.5 

63.5 

73.8 

84.4 

95.3 

107. 

118. 

142. 

167. 


'Y. 

17.8 

27.2 

36.9 

46.9 

57.2 

67.8 

78.7 

89.4 

102. 

113. 

126. 

151. 

177. 


34 

19.1 

29.1 

39.4 

50.0 

60.9 

72.1 

83.7 

95 5 

108. 

120. 

133. 

159. 

187. 


8. 

•20.3 

30.9 

41.8 

53.1 

64 6 

76.4 

88.6 

101. 

114. 

127. 

140. 

168. 

197. 


* 

21.5 

32.8 

44.3 

56.1 

68.3 

80.7 

93.5 

107. 

120. 

134. 

148. 

177. 

207. 


9. 

22.8 

34.6 

46.8 

59.2 

72.0 

85.1 

98.4 

112. 

126. 

140. 

155. 

185. 

217. 


34 

24 0 

36.4 

49.2 

62.3 

75.7 

89.3 

103. 

118. 

132. 

147- 

163. 

194. 

226. 

■■ 

10. 

25.1 

38.3 

51.7 

65.3 

79.4 

93.6 

108. 

123. 

138. 

154. 

170. 

202. 

235. 

* 


26.4 

40 1 

54.1 

68.4 

83.0 

97.9 

113.2 

129. 

145. 

161. 

177. 

211. 

245. 


ii. 

27.6 

42.0 

56 6 

71.5 

86.7 

102. 

118. 

134. 

151. 

168. 

185. 

220. 

255. 


% 

28.8 

43.8 

59.1 

74 6 

90.4 

107. 

123. 

140. 

157. 

174. 

192. 

228. 

265. 


12 . 

30.0 

45.7 

61.5 

77.7 

94.1 

111. 

128. 

145. 

163. 

181. 

199. 

237. 

275. 


13. 

32.5 

49.4 

66.4 

83.8 

102. 

120. 

138. 

156. 

175. 

195. 

214. 

254. 

294. 


14. 

35.0 

53.1 

71.4 

89.4 

109. 

128. 

148. 

168. 

188. 

208. 

229. 

271. 

314. 

! 

15. 

37.4 

56.7 

76.3 

96.1 

116. 

137. 

158. 

179. 

200. 

222. 

244. 

289. 

334. 


ia. 

39.1 

60.4 

81.2 

102. 

124. 

145. 

167. 

190. 

212. 

235. 

258. 

306. 

353. 


17. 

42.3 

64.1 

86.1 

108. 

131. 

154. 

177. 

201. 

225. 

249. 

273. 

323. 

3 <3. 


18. 

44.8 

67.8 

91.0 

115. 

139. 

163. 

187. 

212. 

237. 

262. 

288. 

340. 

393. 


19. 

47.3 

71.5 

96.0 

121. 

146. 

171. 

197. 

223. 

249. 

276. 

303. 

357. 

412. 


20. 

49.7 

75.2 

101. 

127. 

153. 

180. 

207. 

234. 

261. 

289. 

317. 

375. 

432. 


21. 

52.2 

78.9 

106. 

133. 

161. 

188. 

217. 

245. 

274. 

303. 

332. 

392. 

452. 


22. 

51.6 

82.6 

111. 

139. 

168. 

196. 

227. 

256. 

286. 

316. 

347. 

409. 

471. 


23. 

57.1 

86.3 

116. 

145. 

175. 

206. 

236. 

267. 

298. 

830. 

362. 

4‘26. 

491. 


24. 

59.6 

89.9 

121. 

152. 

183. 

214. 

246. 

278. 

311. 

343. 

375. 

441. 

511. 


25, 

62.0 

93.6 

126. 

158. 

190. 

223. 

256. 

289. 

823. 

357. 

391. 

461. 

I}31. 


26, 

61.5 

97.3 

131. 

164. 

198. 

231. 

266. 

300. 

335. 

370. 

406. 

478. 

550. 


27. 

66.9 

101. 

135. 

170. 

205. 

240. 

276. 

311. 

348. 

384. 

4*21. 

495. 

570. 


28. 

69.4 

105. 

140. 

176. 

212. 

219. 

286. 

323. 

360. 

397. 

436. 


590. 


29. 

71.8 

109. 

145. 

182. 

220. 

257. 

295. 

334. 

372. 

411. 

450, 

530. 

609. 


HO. 

74.2 

*112. 

150. 

188. 

227. 

266. 

305. 

345. 

384. 

424. 

465. 

547. 

629. 


31. 

76.7 

116. 

155. 

195. 

234. 

275. 

315. 

356. 

397. 

438. 

480. 

564. 

649. 


32. 

79.1 

1'20. 

160. 

201. 

242. 

283. 

325. 

367. 

409. 

451. 

495. 

581. 

668. 


33. 

81 6 

123. 

165. 

207. 

249. 

292. 

335. 

378. 

421. 

465. 

509. 

598. 

688. 


31 

84 1 

127. 

170. 

213. 

257. 

300. 

345. 

389. 

434. 

479. 

524. 

616. 

708. 


35. 

86.5 

131. 

175. 

219. 

264. 

309. 

354. 

400. 

446. 

492. 

539. 

633. 

726. 


33 

89 0 

134. 

180. 

225. 

271. 

318. 

364. 

411. 

458. 

506. 

554. 

650. 

746. 


42. 

48. 

104. 

119. 

156. 

178. 

210. 

239. 

262. 

298. 

315. 

359. 

370. 

422. 

423. 

482. 

478. 

544. 

532. 

605. 

588. 

669. 

644. 

733. 

1 53. 
856. 

864. 

982. 


For warming building's by steam it usually suffices to allow lsq ft 
of cast or wrought pipe surface for each 120 cub It of space to be warmed ; uud 1 cub 
ft of boiler for each 2000 cub ft of such space. 























































400 WEIGHT OF WROUGHT IRON AND STEEL, 


Table of Weight of WROUGHT IROJT* and STEEL. 

Wrought iron is here taken at 485 lbs per cub ft; or a sp gr of 7 77 Verv nu 
soft wrought iron weighs from 488 to 492 lbs per cubic foot. Light weight indicat 
impurities, and weakness. Steel weighs about 490 lbs per cubic loot; therefore f« 

r^vT , x«a , ; , .: , ,rri s be nuu,t *• -.—«■■■<•-■**,: 

At 485 lbs per cub ft a cubic inch weighs .28067 ft>; a fb contains 3.5629 cub in, 
and a ton, 4.bl86 cub It; and this is about the average of hammered iron The usu 

Tf S VJ,?n!l Jn 18 \ 8 ° ftS ? er c " b . ft is nearer the average of ordinary rolled iro 
At 480 lbs, a cubic inch weighs .2< <8 of a lb; a lb contains 3.6 cub ins • a ton 4 66i 
cub ft; a rod of 1 sq inch area, weighs 10 lbs per yard; or 3^ Cper foot exactly 


3. 


Thickness 
or Diameter 
in Inches. 

Thick 
ness oi 
Diam. 
indeci 
mals o 
a foot. 

Wt. of 
Square 
Boot. 

Lbs. 

Jwt. of a 

Square 
bar, 1 ft. 
long. 

Lbs. 

1-35 

.0026 

1.263 

.0033 

l-ll 

.0052 

2.526 

.0132 

3-32 

.0078 

3.789 

.0296 

X 

.0104 

5.052 

.0526 

5-32 

.0130 

6.315 

.0823 

3-16 

0156 

7.578 

.1184 

7-32 

.#182 

8.«41 

.1612 

X 

.0208 

10.10 

.2105 

9 32 

.0234 

11.37 

.2665 

5 16 

.0260 

12.63 

.3290 

11-32 

.0287 

13.89 

.3980 

X 

.0313 

15.16 

.4736 

13-32 

.0339 

16.42 

.5558 

7-16 

.0365 

17.68 

.6446 

15-32 

.0391 

18.95 

.7400 

X 

.0417 

20.21 

.8420 

9-16 

.0469 

22.73 

1.066 

% 

.0521 

25.26 

1.316 

11-16 

.0573 

27.79 

1.592 

X 

.0625 

30 31 

1.895 

13-16 

.0677 

32.84 

2.223 

7 A 

.0729 

35.37 

2.579 

15-16 

.0781 

37.89 

2.960 

1 . 

.0833 

40.42 

3.368 

1-16 

.0885 

42.94 

3.803 

X 

.0938 

45.47 

4.263 

3-16 

.0990 

48.00 

4.750 

X 

.1012 

50.52 

5.263 

5-16 

.1094 

53 05 

5.802 

% 

.1146 

55.57 

6.368 

7-16 

.1198 

58.10 

6.960 

X 

.1250 

60.63 

7.578 

9-16 

.1302 

63.15 

8.223 

H 

.1354 

65.68 

8.893 

11-16 

.1406 

68.20 

9.591 

X 

.1458 

70.73 

10.31 

13-16 

.1510 

73.26 

11.07 

y» 

.1563 

75.78 

11.84 

15-16 

.1615 

78.31 

12.64 

. 

.1667 

80.83 

13.47 1 

X 

.1771 

85.89 

15.21 1 

X 

.1875 

90.94 

17.05 1 

% 

.1979 

95.99 

19.00 1 

X 

.2083 

101.0 

21.05 I 

X 

.2188 

106.1 

23.21 1 

X 

.2292 

111.2 

25.47 2 

X 

.2396 

116.2 

27.84 2 

• 

.2500 

121.3 

30.31 2 


Wt. of a 
Round 
bar, 1 ft. 
long. 

Lbs. 


.00 26 
.0104 
.0233 
.0414 
.0646 
.0930 
.1266 
.1653 
.2093 
.2583 
.3126 
.3720 
.4365 
.5063 
.5813 
.6613 
.8370 
1.033 
1.250 
1.488 
1.746 
2.025 
2.325 
2.645 
2.986 
3.348 
3.730 
4.133 
4.557 
5.001 
5.466 
5.952 
6.458 
6.985 
7.533 
8.101 
8.6S0 
9.300 
9.930 


Wt. of 
Balls. 

Lbs. 


.0001 
.0003 
.0005 
.0009 
.0015 
.0023 
.0033 
.0045 
.0060 
.0078 
.0098 
.0123 
.0151 
.0184 
.0262 
.0359 
.0478 
.0620 
.0788 
.0985 
.1211 
.1470 
.1763 
.2093 
.2461 
.2870 
.3323 
.3820 
.4365 
.4960 
.5606 
.6306 
.7062 
.7876 
.8750 
.9688 
1.069 
1.176 
1.410 
1.674 
1.969 
2.296 
2.658 
3.056 
3.492 
3.968 


® 01 ^ 
ii C o 
a g s 

* 5 u 
y .5 £ 

o 


3 x 

X 

X 

X 


4. 


X 

A 

x 

% 

x 

% 

x 

y» 

5. 

X 

H 

% 

X 

X 

% 

'A 

6 . 

x 

x 

x 

’x 

x 

x 


11 . 


X 

X 

% 

X 

X 

X 

I. 

X 

X 

X 


10 . 


X 

X 

X 


Thick¬ 
ness or 
Diam. 
indeci¬ 
mals o 
a foot 

Wt. of a 
Square 
Foot. 

Lbs. 

.2604 

126.3 

.2708 

131.4 

.2813 

136.4 

.2917 

141.5 

.3021 

146.5 

.3125 

151.6 

.3229 

156.6 

.3333 

161.7 

.3438 

166.7 

.3542 

171.8 

.3646 

176.8 

.3750 

181.9 

.3854 

186.9 

.3958 

192.0 

.4063 

197.0 

.4167 

202.1 

.4271 

207.1 

.4375 

212.2 

.4479 

217.2 

.4583 

222.3 

.4688 

227.3 

.4792 

232.4 

.4896 

237.5 

.5000 

242.5 

.5208 

252.6 

.5417 

262.7 

.5625 

272.8 

.5833 

282.9 

.6042 

293.0 

.6250 

303.1 

.6458 

313.2 

.6667 

323.3 

.6875 

333.4 

.7083 

343.5 

.7292 

353.6 

.7500 

363.8 

.7708 

373.9 

.7917 

384.0 

.8125 

394.1 

.8333 

404.2 

.8542 

414.3 

.8750 

424.4 

.8958 

434.5 

.9167 

444.6 

.9375 

454.7 

.95K3 

464.8 

9792 

474.9 

Foot. 

485. 


Wt. of ahrt. of a 
Square Round 
bar. 1 ft. bar. 1 ft. 

long. 


long. 

Lbs. 


32.89 

35.57 

38.37 

41.26 

44.26 

47.37 

50.57 

53.89 

57.31 
60.84 

64.47 
68.20 
72.05 
75.99 
80.05 
84.20 

88.47 
92.83 

97.31 

101.9 
106.6 

111.4 

116.3 

121.3 
131.6 

142.3 

153.5 
165.0 
177.0 

189.5 

202.3 

215.6 

229.3 

243.4 

247.9 

272.8 
288.2 
304.0 

320.2 

336.8 

353.9 

371.3 

389.2 

407.5 

426.3 

445.4 
465.0 
485. 


Lbs. 

25.83 

27.94 

30.13 
32.41 
34.76 
37.20 
39.72 
42.33 
45.01 
47.78 
50.63 
53.57 
56.59 

59.69 
62.87 

66.13 
69.48 
72.91 
76.43 
80.02 

83.70 
87.46 
91.31 
95.23 

103.3 
111.8 

120.5 

129.6 
139.0 

148.8 

158.9 

169.3 
180.1 

191.1 

202.5 

214.3 

226.3 

238.7 

251.5 

264.5 

277.9 
2*>1.6 

305.7 

320.1 

334.8 

349.5 

365.2 

380.9 


W t. c 
Balls 

Lbs. 


4.48 

5.04 

5.64 

6.3C 

7.0C 

7.75 

8.55 

9.40 

10.32 

11.28 

12.31 
13.39 
14.54 
15.75 
17.03 
18.37 
19.78 
21.26 
22.82 

24.45 
26.16 
27.94 
29.80 
31.74 
35.88 

40.31 
45.lt ( 
50.41 
56.00 
62.00 
68.40 

75.24 
82.52 

90.25 

98.45 

107.1 

116.3 
126.0 

136.2 

146.9 

158.2 

170.1 
182.6 
195.6 

209.2 
223.5 

238.4 

253.9 








































































WEIGHT OF FLAT IRON, 


401 


Weight of 1 ft in length of FIAT ROLLED IRON, at 480 tbs per 
cubic foot. For cast iron, deduct part; for steel, add ^L; for copper, add 
y; for cast brass, add yy; for lead, add for zinc, deduct yy. 


THICKNESS IN INCHES. 


fcs.2 

1-16 

X 

| 3-16 

X 

5-16 

% 

7-16 

X 

% 

X 

X 

1 

1 . 

.2083 

.4166 

.6250 

.8333 

1.042 

1.25C 

1.458 

1.666 

2.083 

2.500 

2.916 

3.333 

X 

.2341 

.4688 

.7033 

.9375 

1.172 

1.406 

1.640 

1.875 

2.344 

2.812 

3.280 

3.75 

x 

.260.) 

.5210 

.7810 

1.042 

1.303 

1.563 

1.823 

2.083 

2.605 

3.125 

3.646 

4.166 

x 

.2863 

.5730 

.8595 

1.146 

1.432 

1.719 

2.006 

2.292 

2.864 

3.438 

4.012 

4.583 

X 

.3125 

.6250 

.9375 

1.250 

1.562 

1.875 

2.188 

2.500 

3.125 

3.750 

4.375 

5.000 

% 

.3385 

.6771 

1.015 

1.354 

1.692 

2.031 

2.370 

2.708 

3.384 

4.062 

4.740 

5.416 

X 

.3616 

.7292 

1.094 

1.458 

1.823 

2.188 

2.550 

2.916 

3.646 

4.375 

5.105 

5.833 

X 

.3906 

.7812 

1.172 

1.562 

1.953 

2.344 

2.735 

3.125 

3.906 

4.688 

5.470 

6.25 

2. 

.4166 

.8333 

1.25 

1.666 

2.083 

2.500 

2.916 

3.333 

4.166 

5.000 

5.833 

6.666 

% 

.4427 

.8855 

1.328 

1.771 

2.214 

2.656 

3.098 

3.542 

4.428 

5.312 

6.196 

7.083 

X 

.4688 

.9375 

1.406 

1.875 

2.344 

2.812 

3.281 

3.750 

4.688 

5.624 

6.562 

7.500 

% 

.4948 

.9895 

1.484 

1.979 

2.474 

2.968 

3.463 

3.958 

4.948 

5.936 

6.926 

7.916 

X 

.5210 

1.042 

1.562 

2.083 

2.605 

3.125 

3.646 

4.166 

5.210 

6.250 

7.291 

8.333 

X 

.5470 

1.094 

1 641 

2.187 

2.735 

3.282 

3.829 

4.375 

5.470 

6.564 

7.658 

8.750 

H 

.5730 

1.146 

1.719 

2.292 

2.865 

3.438 

4.011 

4.583 

5.730 

6.876 

8.022 

9.166 

X 

.5990 

1.198 

1.797 

2.396 

2.995 

3.594 

4.193 

4.792 

5.990 

7.188 

8.386 

9.583 

3. 

.625 

1.250 

1.875 

2 500 

3.125 

3.750 

4.375 

5.000 

6.250 

7.500 

8.750 

10.00 

H 

.6515 

1.303 

1 954 

2.605 

3.257 

3.908 

4.560 

5.210 

6.514 

7.816 

9.120 

10.42 

x 

.6770 

1.354 

2.031 

2.708 

3.385 

4.062 

4.739 

5.416 

6.770 

8.124 

9.478 

10.83 

% 

.7031 

1.406 

2.109 

2.812 

3.516 

4.218 

4.921 

5.625 

7.032 

8.436 

9.842 

11.25 

X 

.7291 

1.458 

2.188 

2.916 

3.646 

4.375 

5.105 

5.833 

7.291 

8.750 

10.21 

11.66 

% 

.7555 

1.511 

2.266 

3.021 

3.777 

4.533 

5.288 

6.042 

7.554 

9.066 

10.58 

12.08 

X 

.7812 

1.562 

2.343 

.3.125 

3.906 

4.686 

5.468 

6.25 

7.812 

9.372 

10.94 

12.50 

X 

.8070 

1.614 

2.421 

3.229 

4.035 

4.842 

5.65 

6.458 

8.070 

9.684 

11.30 

12.92 

i. 

.8333 

1.666 

2.500 

3.333 

4.166 

5.000 

5.833 

6.666 

8.333 

10.00 

11.66 

13.33 

X 

.8595 

1.719 

2.578 

.3.438 

4.297 

5.156 

6.016 

6.875 

8.594 

10.31 

12.03 

13.75 

X 

.8855 

1.771 

2.656 

3.542 

4.427 

5.312 

6.198 

7.083 

8.854 

10.62 

12.40 

14.16 

X 

.9115 

1.823 

2.734 

3.646 

4.557 

5.468 

6.380 

7.291 

9.114 

10.94 

12.76 

14.58 

X 

.9375 

1.875 

2.812 

3.750 

4.687 

5.624 

6.562 

7.500 

9.374 

11.25 

13.12 

15.00 

% 

.9616 

1.927 

2.891 

3.854 

4.818 

5.782 

6.745 

7.708 

9.636 

11.56 

13.49 

15.42 

X 

.9895 

1.979 

2.968 

3.958 

4.947 

5.936 

6.926 

7.917 

9.894 

11.87 

13.85 

15.83 

X 

1.016 

2.031 

3.048 

4 062 

5.080 

6.096 

7.112 

8.125 

10.16 

12.19 

14.22 

16.25 

1 . 

1.042 

2.083 

3.125 

4.166 

5.210 

6.25 

7.291 

8 333 

10.42 

12.50 

14.58 

16.66 

X 

1.068 

2.136 

3.204 

4.271 

5.340 

6.408 

7.476 

8.542 

10.68 

12.81 

14.95 

17.08 

X 

1.094 

2.188 

3.282 

4.375 

5.470 

6.564 

7.658 

8.750 

10.94 

13.13 

15.31 

17.50 

X 

1.120 

2.240 

3.360 

4.479 

5.600 

6.720 

7.840 

8.958 

11.20 

13.44 

15.68 

17.92 

X 

1.146 

2.292 

3.438 

4.584 

5.730 

6.876 

8.022 

9.167 

11.46 

13.75 

16.04 

18.33 

% 

1.172 

2.344 

3.516 

4.687 

5.860 

7.032 

8.204 

9.375 

11.72 

14.06 

16.40 

18.75 

H 

1.198 

2 396 

3.594 

4.791 

5.990 

7.188 

8.386 

9.583 

11.98 

14.37 

16.77 

19.16 

X 

1.224 

2.4 48 

3.672 

4.896 

6.120 

7.344 

8.568 

9.792 

12.24 

14.69 

17.13 

19.58 


1.250 

2.500 

3.750 

5.000 

6.250 

7.500 

8.750 

10.00 

12.50 

15.00 

17.50 

20.00 

X 

1.276 

2.552 

3.828 

5.104 

6.380 

7.656 

8.932 

10.21 

12.76 

15.31 

17.86 

20.42 

X 

1.302 

2.604 

3.906 

5.208 

6.510 

7.812 

9.114 

10.42 

13.02 

15.62 

18.23 

20.83 

% 

1.328 

2.657 

3.984 

5.31.3 

6 640 

7.968 

9.297 

10.63 

13.28 

15.93 

18.59 

21.25 

X 

1.354 

2.708 

4.063 

5.417 

6.770 

8.126 

9.480 

10.83 

13.54 

16.25 

18.96 

21.66 

% 

1.381 

2.761 

4.143 

5 521 

6.906 

8.286 

9.668 

11.04 

13.81 

16.57 

19.33 

22.08 

H 

1.406 

2.813 

4.218 

5.625 

7.030 

8.436 

9.843 

11.25 

14.06 

16.87 

19.69 

22.50 

X 

1.432 

2.864 

4.298 

5.729 

7.160 

8.592 

10.02 

11.46 

14.32 

17.18 

20.04 

22.92 

• 

1.458 

2.916 

4.375 

5.833 

7.291 

8.750 

10.20 

11.66 

14.58 

17.50 

20.42 

23.33 

X 

1.484 

2.969 

4.452 

5.938 

7.420 

8.904 

10.39 

11.87 

14.84 

17.81 

20.78 

23.75 

X 

1.511 

3.021 

4.533 

6.042 

7.555 

9.066 

10.58 

12.08 

15.11 

18.13 

21.16 

24.16 

X 

1.536 

3.073 

4.608 

6.146 

■ 7.680 

9.216 

10.75 

12.29 

15.36 

18.43 

21.50 

24.58 

X 

1.562 

3.125 

4.686 

6.250 

7.810 

9.372 

10.93 

12 50 

15.62 

18.74 

21.86 

25.00 

X 

1.588 

3.177 

4.764 

6.354 

7.940 

9.528 

11.12 

12.71 

15.88 

19.05 

22.24 

25.42 ■ 

X 

1.615 

3.229 

4.845 

6.458 

8.075 

9.690 

11.31 

12.92 

16.15 

19.38 

22.62 

25.83 

X 

1.641 

3.281 

4.923 

6.562 

8.205 

9.846 

11.48 

13.13 

16.41 

19.69 

22.96 

26.25 


1.666 

3.333 

5.000 

6.666 

8.33.3 

10.00 

11.66 

13.33 

16.66 

20.00 

23.33 

26.66 

X 

1.693 

3.386 

5.079 

6.771 

8.455 

10.15 

11.85 

13.54 

16.91 

20.30 

23.70 

27.08 

X 

1.719 

3.438 

5.157 

6.875 

8.595 

10.31 

12.03 

13.75 

17.19 

20.61 

24.06 

27.50 

X 

1.745 

3. <89 

5.235 

6.979 

8.725 

10.47 

12.21 

13.96 

17.45 

20.94 

24.42 

27.92 

X 

1.771 

3.542 

5.313 

7.083 

8.855 

10.63 

12.40 

14.17 

17.71 

21.26 

24.80 

28.33 

% 

1.797 

3.594 

5.391 

7.188 

8.985 

10.78 

12.58 

14.37 

17.97 

21.56 

25.16 

28.75 

X 

1.823 

3.616 

5.469 

7.292 

9.115 

10.94 

12.76 

14.58 

18.23 

21.88 

25.52 

29.17 

X 

1.849 

3.698 

5.547 

7.396 

9.245 

11.09 

12.94 

14.79 

18.49 

22.18 

25.88 

29.58 


1.875 

3.750 

5.625 

7.500 

9.375 

11.25 

13.12 

15.00 

18.75 

22.50 

26.24 

30.00 

X 

1.901 

3.802 

5.703 

7.604 

9.505 

11.41 

13.31 

15.21 . 

19.00 

22.81 

26.62 

30.42 

X 

1.927 

3.854 

5.781 

7.708 

9.635 

11.56 

13.49 

15.42 

19.27 

23.12 

26.98 

30.83 

X 

1.953 

3.906 

5.859 

7.812 

9.765 

11.72 

13.67 

15.62 

19.53 

23.44 

27.34 

31.25 

- X 

1.979 

3.958 

5.9.37 

7.916 

9.895 

11.87 

13.85 

15.84 

19.79 

23.74 

27.70 

31.67 

% 

2.005 

4.010 

6.015 

8.021 

10.02 

12.03 

14.04 

16.04 

20.04 

24.06 

28.08 

32.08 

1 X 

2.031 

4.062 

6.093 

8.125 

10.16 

12.18 

14.21 

16.25 

20.32 

24.36 

28.42 

32.50 

X 

2.057 

4.114 

6.171 

8.229 

10.29 

12.34 

14.40 

16.46 

20.58 

24.68 

28.80 

32.92 


2.083 

4.166 

6.250 

8.33.3 

10.41 

12.50 

14.58 

16.66 

20.82 

25.00 

29.16 

33.33 

x 1 

2.109 

4.219 

6.327 

8.438 

10.55 

12.65 

14.76 

16.87 

21.10 

25.30 

29.52 

33.75 

1 X 

2.135 

4.270 

6.405 

8.541 

10.67 

12.81 

14.94 

17.08 

21.34 

25.62 

29.88 

34.17 


29 





























































402 


IRON AND STEEL. 


Weight of 1 ft in length of FLAT ROLLED IRON, at 480 It* 

per cubic foot —(Continued.) 


J2 09 

«-> r-» 

rs .2 




THICKNESS 

IN INCHES 

• 




ts 

1-16 

X 

3-16 

H 

5-16 

% 

7-16 

X 1 

% 

* 1 

X ! 

1 

10 % 

2.162 

4.323 

6.486 

8.646 

10.81 

12.97 

15.13 

17.29 

21.62 

25.94 

30.26 

34.58 


2.188 

4.375 

6.564 

8.750 

10.94 

13.13 

15.31 

17.50 

21.88 

26.26 

30.62 

35.00 

% 

2.214 

4.427 

6.642 

8.854 

11.07 

13.28 

15.5Q 

17.71 

22.14 

26.56 

31.00 

35.42 

M 

2.239 

4.479 

6.717 

8.958 

11.20 

13.43 

15.67 

17.92 

22.40 

26.86 

31.34 

,^5.83 

% 

2.266 

4.531 

6.798 

9.062 

11.33 

13.59 

15.86 

18.12 

22.66 

27.18 

31.72 

36.25 

ii. 

2.291 

4.583 

6.873 

9.166 

11.46 

13.75 

16.04 

18.33 

22.90 

27.50 

32.08 

36.66 

% 

2.318 

4.636 

6.954 

9.271 

11.59 

13.91 

16/22 

18.54 

23.18 

27.82 

32.44 

37.08 


2.344 

4.688 

7.032 

9.375 

11.72 

14.06 

16.40 

18.75 

23.44 

28.12 

32.80 

37.50 

% 

2.370 

4.740 

7.110 

9.479 

11.85 

14.22 

16.59 

18.96 

23.70 

28.44 

33.18 

37.92 

X 

2.895 

4.791 

7.185 

9.582 

11.97 

14.37 

16.76 

19.16 

23.94 

28.74 

33.52 

38.33 

% 

2.422 

4.844 

7.266 

9.688 

12.11 

14.53 

16.95 

19.37 

24.22 

29.06 

33.90 

38.75 

% 

2.448 

4.896 

7.344 

9.792 

12.24 

14.68 

17.13 

19.58 

24.48 

29.36 

34.26 

39.16 

% 

2.474 

4.948 

7.422 

9.896 

12.37 

14.84 

17.32 

19.79 

24.74 

29.68 

34.64 

39.58 

12. 

2.500 

5.000 

7.500 

10.00 

12.50 

15.00 

17.50 

20.00 

25.00 

30.00 

35.00 

40.00 


ROLLED IRON AND STEEL.* AVERAGE PRICES, Phila, 1888. 
Iron bars. Base price, or price for “ordinary sizes”; i e, from % inch to 2 ins 
diameter, round and square; and from 1 X % inch to 6 X 1 inch, fiat: Ordinary 
merchant quality, called “refined ”, 2 ets per lb; “ Extra refined ” and rivet iron, 3 
cts per lb. Sizes larger or smaller than “ ordinary ” bring higher prices per lb. The 
“ extra”, or charge in addition to the above base price, increases gradually up to .7 
ct per lb for % inch round and square; 2.5 cts for 7 inches round and square; 1.2 cts 
for X tb inctl flat '> and 11 cts for 12 X 2 ins flat. Swedish and Norway 
iron : y X % inch and % inch square, and larger; in the original bar as imported 
(called “Swedish”), 4 cts per tb; re-rolled after importation (called “Norway”), 4 y 
cts per lb. Hoop iron : from iy inch X No 17, 2]4 cts per lit; to ^£inch X No 
22, 5 cts per lb. .Short iron: see page 403. Plato iron. Rectangular 
plates, T 3 g inch thick and heavier; and say from 2 to 5 feet wide, and from 4 to 12 feet 
long. “ Common or “ puddled ”, for bridge plates, sheathing of ships etc, where little 
or no bending is required, 2 cts per lb; “ Shell ” ; for shells of boilers etc, to stand 
bending cold with the grain to cylinders with radii of say one or two feet, but not 
flanging, 2y cts per lb; “ Best flange ”, cts per lb. Fire-box plates of shell or 
flange iron, 1 ct per lb extra. Extra charges for plates of unusually great or small 
widths or lengths. Angle and T iron: see pages 52-5 etc. I beams, 
channel beams, deck beams; see pages 521 etc. Pig iron. American 
foundry, $20 per ton of 2240 tbs; Forge, for conversion into wrought iron by pud¬ 
dling e,tc, ,§17 per ton of 2240 lbs; Charcoal foundry and forge, §22 per ton of 2240 lbs. 
4'astWteel For tools. Best American. Ordinary sizes, %y cts per lb; Ma¬ 
chinery steel ; for shafting etc, ordinary sizes, 3 cts per lb. 'The range of “or¬ 
dinary ” sizes is nearly the same as given above for iron. For larger and smaller 
sizes, extras running up to from 50 to 100 per cent. Steel plates for boilers and 
fire-boxes, 4 cts per lb. Tire and spring steel. Bessemer, 3 cts per lb. 
Sheet steel. American.! Not lighter than No 17. Cast, 7 cts per lb; Bessemer, 
4 y cts per lb. For each number lighter than No 17, y ct per lb extra. 

Rolled Star Iron. Standard sizes. Carnegie Bros. & Co., Limited, Pittsburgh. 
The thicknesses are those at the end and at the root of one of the four arms, iu 
inches. Rolled in lengths of 20 to 25 feet. Area iu Bquare inches. Weight in lbs. 
per foot run. 


Ins. 

Thick. 

Area. 

Weight. 

In 8. 

Thick. 

Area. 

Weight. 

4X4 

56-9-16 

3.54 

11.8 

2.5 X 2.5 

5-16 — 13-32 

1.67 

5.6 

3.5 X 3.5 

%-% 

■ 2.87 

9.6 

2 X 2 

% —13-32 

1.21 

4.0 

3X3 

5-16 — 15-32 

2.19 

7.3 

1.5 X 1.5 

3-16— 5-1G 

0.69 

2.3 


* Morris, Wheeler & Co., 16th and Market Streets, Philadelphia, 
f William & Harvey Rowland, Frankford, Philadelphia. 

























































SHEET-IRON. 


403 


lo***fl4‘to°";*28 18s !i >» <*«<• 24 to 82 inche, wide, 6 to 8 feet 

S“ «> & - K 

,do £df the Sgg^ a'aa kISuUTp" 1, £ n “ ,is * 


No. 

1 ~- 

Ounces 
avoir 
per sq ft 

Sq ft 
per 

2240 lbs. 

No. 

Ounces 
avoir 
per sq rt. 

Sq ft 
per 

2240 lbs. 

No. 

Ounces 
avoir 
per sq ft. 

Sq ft 
per 

2240 lbs. 

29 

28 

27 

26 

25 

12 

13 

14 

15 

16 

2987 

2757 

2560 

2389 

2240 

24 

23 

22 

21 

20 

17 

19 

21 

24 

28 

2108 

18S6 

1706 

1493 

1280 

19 

18 

17 

16 

14 

33 

38 

43 

48 

60 

1086 

943 

833 

746 

597 


mssss^^^ms 

w p te^rrr^r ( ^ lle W C ^ ,RON - for Unction) is J^TeZetatinThr roof !?“ 

SSSS2SSS 

Paint for roofs should not have much dryer. See Painting, p 429 tmcKness ol the sheet. 

The sulphurous fumes from coal are very corrosive of 

roofed w?tl| 1 'eUher?°i/efficientmeans' are n^proviXd & ftSt b ° U8 . fiS h * 

other metals. The acid■ or oak txm BER is safi to Sest^/K^cof galvTnlz^ Zn %eeZVlS 
!h!'t»t F th ° n is usually nailed upon a sheeting of boards; but the strength of corrugated iron 
obviates the necessity for this, and enables it to stretch 5 or 6 ft from pnrli n to p i.Hin withou i nter. 
mediate support. The corrugated sheets are riveted together on the roofbv vet’s of gah-anited 
wire about one-eighth inch thick, 300 to a pound, well driven (so as to evHoril £ al .'? n, . zed 

apart, all around the edges. The Vivet holes are firstpunchedby,machine™!^o as to insure cofncT 
rnnf CC Ji'" Jw se 1 ' < : ral sheets , : and the rivets are driven bv two men, one above, and one beneath the 
roof. I or black iron, ungalvanized nails, boiled in linseed oil as a partial preservative from rust ire 
used = as also ^ shingling or slating. Galvanized ones, however,™„idUbebSteHn* U 
these cases, or even copper ones for slating because good slate endures much longer than either 
shingles or iron, and therefore it becomes true economy to use durable metals for fastening it In 
none of these cases, however, are the nails fully exposed to the weather fastening it. In 

I lie sheets of flat iron are put together by overlapping and 

folding THE edges n.uehthe same as shown by the fig page 418, head Tin ; the joints wmich run 

- “P “, d ,u ^“ the ro ° f bel “^ ,he same as at « a. and the horizontal ones as at t t ; 

except that inasmuch as these are not soldered in the iron sheets the joint is made 
about X to 1 inch wide, instead of H inch, the better to provide against leaking? 
Cleats are used as in tin, with 2 nails to a cleat. The iron plates are best laid on 
sheeting boards ; but tn sheds. Ac, are sometimes laid directly on rafters not. more 
fh . than about 18 ins apart in the clear; the plates being allowed to sag a little between 

slightly? ali h? Iu 8UCh cascs >* is wel1 *bevel off the to'ps of the rLftm-s 

Hon A ,r er, f OI,s objection to iron as a roof covering, is its rapid con- 

tocemn«°fl^ Phe ^ C , m ° StU 'u : Which falls fr0,n the iron in drops rain, and may do injury 
snnrlllt hi’ fio . or ?> ? r a, '‘'. cles . >“ the apartments immediately beneath the roof. Painting does not 

of Trusses y n'^83 n ' Sh ‘r ' S: U "’Yi howev e r - be obviated by plastering, as shown at 11, or Pigs 21^? 
o rrusses, p o«3. Corrugated iron makes an excellent permanent street or other awning* 

Sheet iron. The size of sheets generally used for corrugating 

T ,<le b T ,DChes lo “K* Corrugation reduces the width to 27J^ inches. When the cor- 

t r n6he1alon e g thefr Jnd d u ( j , . on . tbethe overlapping of about 2 M inches along the sides, and of 4 
inches along their ends, diminishes the area or roof covered by a sheet, to about seven-eighths of that 
,of the enure corrugated sheet itself; or, the weight per s,,uare foot of roof covert, will be about 
lrm.’ S ner' b tban tbat P er s 3 uare foot of the corrugated sheet; or. the weight of corrugated 

it?s made qUarC f ° 0t ° f r °° f C0Vered 18 about one - firth greater than that of the flat sheets from which 

About 6 Inches are usually allowed for the extension over the eaves. 

....rv LYllh , Pe M q '. 1 . are J f Y t corresponding to the different numbers of the Birmingham wire gauge, 

5 x \V andU ‘ ?i ff " rent makers - The two s ^ les °f corrugation given in the table below, 

5X U <iu d t are those most frequently used. 



' _* Marshall, Bros A Co, Front St and Girard Ave, Phila. 
Washington Ave, Phila. 


McDaniel A Haivey Co, 16th St and 







































404 


CORRUGATED SHEET IRON 


No. 

Bmghrn 
wire ga. 

Thick¬ 

ness 

in ins. 

Wt in tbs per 
sq ft of sheets. 

Wt in lbs per 
sq ft of roof. 

Approx prices in cts 
per lb, Phila, 1888. 

Black 

Black 

Black 

Lead coatedt 
or galv'd 

Black 

Lead coatedt 
or galv’d 

Black 

Lead coatedt 
or galv'd 

20 

.035 

1.84 

2. 

2.12 

2.3 


5 H 

22 

.028 

1.50 

1.6 

1.73 

1.84 



24 

.022 

1.20 

1.25 

1.38 

1.44 

3\4 

3/4 

26 

.018 

1.00 

1.12 

1.15 

1.29 

3 % 

5% 


Strength of Corrugated Iron. Experiments by the author. 





First. A sheet d rf, of No. 16 iron, 

(about T U inch thick,) 27 ins wide, by 4 ft long, 
with five complete corrugations of 5 ins by 1 inch, 
was laid on supports 3 ft 9 ins apart. A block of 
wood c, 9 ins wide, by 7 ins thick, and 30 ins long, 
was placed across the center, and gradually loaded 
with castings weighing 1600 lbs. 

This caused a deflection at the center of precisely M an 
inch. On the removal of the load after an hour, no perma¬ 
nent set was appreciable. The severity of the test was pur¬ 
posely increased by applying the several castings very 
roughly, jolting the whole as much as possible.* The sus- 






if 


W/X 



T#' 




pended area of the sheet was 8.44 sq ft; and since the actual center load of 1600 lbs Is about equtva- 
1 3000 

lent to 3000 lbs equally distributed, it amounts to — 355 lbs per sq ft distributed. But 3000 lbs 

distributed would produce a deflection of but about full y A of an inch. Again, 355 lbs per sq ft 
is about 4 times the weight of the greatest crowd that could well congregate upon a floor. Conse¬ 
quently this irou, at 3' 9" span, is safe in practice for any ordiuary crowd. Moreover, such a crowd 
would produce a center deflection of only the }ilh part of % of an inch ; or yU of an inch ; or 


of the clear span ; which is but two-thirds of Tredgold's limit of yy -q of the span. 


1 

72(1 


In one experiment the ends of the sheets rested upon supports dressed so as to preseut undulations 
corresponding tolerably closely with the shape of the corrugations; but in the other the supports 
were flat, and each end of the sheet rested only upon the lower points of the corrugations. No ap¬ 
preciable difference was observed in the results 


Second. An areli of No. 18 (wg- 
inch) iron, corrugated like the foregoing, 
but the depth of corrugation increased to 
V/± ins by the process of arching the sheet; 
clear span 6 ft 1 inch ; rise 10 ins; breadth 27 
ins, (of which, however, only 25 ins bore 
against the abutments ) 

Each foot o of the arch abutted upon a casting j, 
the inner portion t of which was undulated on top, to 
correspond with the corrugations of the arch, which 
rested upon it. At y, (one-fourth of the span,) tw-o 
wooden blocks were placed, occupying a width of 9 
inches, and extending across the arch ; on them was 
piled a load, l , of castings, to the extent of 4480 lbs, 
or 2 tons. Under this load the arch descended about 
half an inch at y. becoming Hatter ou that side, and 

slightly more curved upward along the unloaded side n. Two similar blocks were then placed at n, 
and two tons of load, *. were piled upon them, in addition to the 2 tons at l ; making a total of 8960 
lbs, or 4 tons. This brought the arch more nearly back to its original shape; but still slightly 
straightened at both n and y. and a little more curved in the center. The load was then increased to 
10000 lbs. and left standing for several days. Two iron ties, each M by 1%, which were used for pre¬ 
venting the abutment castings j from spreading, were found to have stretched nearly y of an inch. 
Additional ones were inserted, and the load increased to a total of 6 tons, or 13440 lbs; parts of it on 
s and (, and part in the shape of long broad bars of iron at the center of the arch, below the loads s 
and l, and between n and y. So far as could be judged by eye. the shape of the arch was now almost 
perfect.. The loads s and 1 did not touch each other. After standing more than a week, the load 
was accidentally overturned, crippling the arch. The load was equal to about 1000 lbs per sq ft. of 

the arch. Such arches have since come into common use instead of brick, lor 

fireproof floors. 

Curved roofs of 25 to 30 ft span, rising about % span, may be made 

of ordinary corrugated iron of Nos 16 to 13, riveted as usual; and having no aeces- I 
eories except tie-rods a few feet apart; continuous angle-iron skewbacks; and thin 1 
vertical rods to prevent the ties from sagging. 



* Without letting the deflection exceed H inch ; which was prevented by a stop under the sheet, 
t Marshall, Bros & Co, Front St and Girard Ave, Phila. 













































WEIGHT OF METALS. 




Outer 

Dia.t 

Ths 

Ths. 

Wt 
per ft. 

Price) 

1 per ft. 

Outer 

Dia.J 

Ths 

Ths. 

Wt 
per ft. 

Price) 
per ft. 

Outer 

Dia.J 

Ths 

| Ths. 

1 

Wt 
per ft. 

Price) 
per ft. 

I us. 

fas. 

B’gm 
w. ga. 

Lbs. 

$ 

Ins. 

Ins. 

B’gni 
w. ga. 

Lbs. 

$ 

Ins. 

Ins. 

B’gm 
w. ga. 

Lbs. 

$ 

1 

.072 

15 

.70 

.23 

3 

.109 

12 

3.33 

.34 

8 

.165 

8 

13.65 

L£5 


.072 

15 

.90 

.23 

3* 

.120 

11 

3.96 

.38 

9 

.180 

7 

16.76 

2.25 

IVj 

.083 

14 

1.24 

.23 

3* 

.120 

11 

4.28 

.43 

10 

.203 

6 

21.00 

2 75 


.095 

13 

1.66 

.22 

3% 

120 

11 

4.60 

.45 

11 

.220 

5 

25.00 

3.25 

2 

.095 

13 

1 91 

.22 

4 

.134 

10 

5.47 

.52 

12 

.229 

4* 

28.50 


*Y* 

.095 

13 

2.16 

.25 

4* 

.134 

10 

6.17 

.60 

13 

.238 

4 

32.06 

4.20 

'1% 

.109 

12 

2.75 

.28 

5 

.148 

9 

7.58 

.72 

14 

.248 

3* 

36.00 

4.75 

2% ’ 

.109 

12 

3.04 

.31 

6 

.165 

8 

10.16 

1.00 

15 

.259 

3 

40.60 

5.75 






7 

.165 

8 

11.90 

1.45 

16 

.270 

2* 1 

45.20 

6 75 


* The Allison Manufacturing Co, 32d and Walnut Sts, Phila, manufacture pipes, tubes 
fittings, railroad cars and furnish appliances for roofs, buildings, and bridges, rnilroai 

supplies, &c, <vc. 

Morris, Tanker & Co, Limited, Pascal Iron Works, Phila, manufacture pipes and tubes 
and their fittings, iron and brass valves and cocks, machines and tools for manipulation of pipes Ac 
f In ordering pipes, give the •• nominal ” inner diameter. It is merely an arbitrary name for th< 
pipe, and, in some cases, tends to mislead. Thus, the pipe whose “ nominal ' inner diam is one 
eighth inch, has an “ actual ” inner diam of full quarter inch. 

1 In orderiug boiler tubes, give the outer diameter. 

) Adopted June 11, 1881. 


Dimensions, weights, and list-prices of standard sizes of lap-welded wroHg'lit* 
irou boiler tubes,* in lengths up to 20 ft. Other sizes and lengths made to 
order, at extra prices. Discount, 1888, ab«ut 50 to 55 per cent. 


and tist-prices of standard sizes of welded u ro 11 glit-i rou 
fn llSth^of ahmit^S^, anti water; (for boiler tubes, see lower table) usually 

. 0th . er 81Ze f. >e»gtbs made to order, at extra prices. 
« <?, ^ 1 H 1US luuer dlum (“nominal”) are usually “6utf-welded 

cem eaW’d i lm - on butt-welded, black, 45 to 50 per 

cent., ga lv d, o5 to 40; on lap-welded, black, 55 to 60; galv’d, 40 to 45. 


Inner Diam. 

CO 

ce 

4> 

a 

^3 

V 

’-3 

H 

Wt per foot. 

Threads per 
inch of screw. 

Price per ft run. 

Adopted Mar. 23, 
1887. 

Inner Diam. 

Thickness. 

3 

1 

t. 

0> 

c, 

fcs 

Threads per 

inch of screw. 

! Price per ft run. 

Adopted Mar. 23, 
1887. 

Nominal.t 

Actual. 

| Nominal.’ 

*3 

a 

*-» 

< 

Plain. 

Galv’d. 

Plain. 

Galv’d. 

lus. 

Ids. 

Ins. 

Lbs 


$ 

$ 

Ins. 

lus. 

Ins. 

Lbs. 



$ 

% 

.270 

.068 

.24 

27 

.04 

.05 

4 

4.026 

.237 

10.66 

8 

.85 

1 no 

V\ 

.364 

.088 

.42 

18 

.04 

.05 

4* 

4.508 

.246 

12.34 

8 

1.00 

1.25 

% 

.494 

.091 

.56 

18 

.04 

.05* 

5 

5.045 

.259 

14.50 

8 

1.20 

1.50 

hi 

.623 

.109 

.84 

14 

.05 

.07 

6 

6 065 

.280 

18.76 

8 

1.65 

2.00 

h 

.824 

.113 

1.12 

14 

.07 

.09 

7 

7.023 

.301 

23.27 

8 

2 00 


1 

1.048 

.134 

1.67 

11^ 

.09* 

.12* 

8 

7.982 

.322 

28.18 

8 



*/4 

1.380 

.140 

2.24 

II* 

.12* 

.17 

9 

9.001 

.344 

33.70 

8 

3.70 


1 4 

1.611 

.145 

2.68 

11>6 

.22 

.25 

10 

10 019 

.366 

40.06 

8 

4.75 


2 

2.067 

.154 

3.61 

11« 

.28 

.32 

11 

11. 


45.02 

8 

5.75 


2^ 

2.468 

.204 

5 74 

8 

.44 

.49 

12 

12. 


49.00 

8 



3 

3.067 

.217 

7.54 

8 

.58 

.64 

13 

13.25 

.375 

54.00 

8 

7.75 


3>£ 

3.548 

.226 

9.00 

8 

.70 

.86 

14 

14.25 

.375 

58.00 

8 

9 00 









15 

15.25 

.375 

61.00 

8 

10.00 



Fittings for Wrougllt-irou Pipes. 1, Elbow. 2, Service Elbow, 
o, Elbow with side outlet. 4, Reducing T. 5. T. 6, Reducing Cross. 7, Reducing 
Coupling or Socket. 8, Return Rend with side outlet. 9, Return Bend with back 
outlet. 10, Cross. 11, Flange Union. 12, Oval Flange. 13, Plug. 






















































































































































406 


BOLTS, NUTS, WASHERS. 


Bolts, nuts, and washers. The following dimen¬ 
sions lor holts and nuts were proposed by Mr. Wm. Sellers 
and adopted by the Franklin Institute, of Philadelphia, as a 
standard, in 1864, and by the U. S. Navy Dept * in 1868. They 
have been adopted by the principal machinists of the country, 
and are known as “'Franklin Institute Standard,” 
or “ Sellers,” dimensions. The angle a , Fig 1, between 
the two sides of a thread, is 60°. w is the width, (measured 
lengthwise of the bolt) of the flat top and bottom of each 
thread. N is the number of threads per inch of length of 
screw. 



D 

ins 

d 

ins 

w 

ins 

N 

D 

ins 

d 

ins 

w 

ins 

N 

D 

ins 

d 

ins 

w 

ins 

N 

I) 

ins 

d 

ins 

w 

ins 

N 


.185 

.0062 

20 

1 

.837 

.0156 

8 

2 

1.712 

.0277 

4M 

4 

3.567 

.0413 

3 

5-16 

.240 

.0074 

18 

18 

.940 

.0178 

7 

2’4 

1.962 

.0277 

4^ 

4 X 

3.798 

.0435 


% 

.294 

.0078 

16 

1.065 

.0178 

7 

1 $ 

2.176 

.0312 

4 


4.028 

.0454 

m 

7-16 

.344 

.0089 

14 

Ip 

1.160 

.0208 

6 

2.426 

.0312 

4 

4% 

4.256 

.0476 

2% 


.400 

.0096 

13 

1.284 

.0208 

6 

3 

2.629 

.0357 

3K 
3 X 

5 

4.480 

.0500 

*X 

9-16 

.454 

.0104 

12 


1.389 

.0227 

5y 2 

3 m 

2.879 

.0357 


4.730 

.0500 


% 

l 

.507 

.0113 

11 


1.491 

.0250 

5 

?> X A 

3.100 

.0384 

3k 


4.953 

.0526 

m 

.620 

.731 

.0125 

.0138 

10 

9 

iVs 

1.616 

.0250 

5 

3% 

3.317 

.0413 

3 

'5 % 
6 

5.203 

5.423 

.0526 2% 
.0555 2% 


Dimensions of Heads and Nuts. 


X 

H (in head) 
H (in nut) 


Ron till. 

H inch - 


Finished. 

1 x / 2 D + 1-16 inch. 
~D—1-16 inch. 

t( a 



In the Whitworth (English') standard thread, the angle a, Fig 1, is 55°. 
The tops and bottoms of the threads are rounded, instead of flat as in the American 
standards. The number (N) of threads per inch is the same as above for diams of 
bolt up to three ins, except for D = % inch; where N = 12. 


Plate-iron washers. Standard sizes. Diameters of washers and bolt-holes 
in inches. Approximate thickness by Birmingham wire gauge, p. 410. Approxi¬ 
mate number in one lb. 


Diams. 

Tbs. 

No. 

Diaml. 

Ths. 

No. 

Diams. 

Ths. 

No. 

x 

X 

18 

450 

ix 

X 

14 

43 

2 Y± 

15-16 

9 

8.6 

% 

5-16 

16 

210 

m 

9-16 

12 

26 

2 x 

1 1-16 

9 

6.2 

% 

5-16 

16 

139 

ix 

% 

12 

22.5 

2 % 

l X 

9 

5.2 

% 

Vs 

16 

112 

m 

11-16 

10 

13 1 

3 


9 

4. 

1 

7-16 

14 

68 

2 

13-16 

10 

10.1 

3 X 

1 X 

9 

2.8 


Price in Philadelphia, 1888, about 4 to 4% cents per lb. net, for outer diameters 
from 1 % to 3% inches. 


* Except that the Navy Department uses for both rough and finished heads and nuts, 
the dimensions given above for rough heads and nuts. The standard of the Navy 
Department is known as ‘‘United States Standard.” 






































































407 


BOLTS, NUTS, WASHERS. 


A square head and nut together, weigh about as much as a length of the bolt equal to 7 or 8 times 
D. Hexagon, 6 or 7. 

With the above dimensions a bolt will generally fail by breaking 
off between the head and the nut, where the diameter is decreased 
by cutting the thread, rather than by stripping off its threads. 

The diani I> of the thread must of course be greater 

than that required to bear safely the proposed tensile strain, by an amount 
equal to twice the depth of the thread. The waste of iron, which would 
result from making the entire, bolt of this greater diam, is frequently 
avoided by making the bolt from a bar of only sufficient dimensions to bear 
the strain safely, and upsetting; its ends as in Fig 3, 
thus increasing their diam sufficiently to allow for the cutting of the 
threads. But see Rem, p 408. 

In carpentry, as well as in ties for masonry, washers , w w, of either cast 
or wrought iron, are placed between the timber, or stone, and the head 
and nut: in order to distribute the pressure over a greater surface, and 
thus prevent crushing; especially in timber. 

When much strained ag-ainst wood, the side 

of a square wrought-iron washer; or the diam w w of a circular one, should not be less than 4 dinms 
of the screw, as in the fig; and its thickness, tw, H diam at least. 

Two such square washers will together weigh as much as 18 diams in 
length of a round rod of the same diam as the screw. Two round 
-washers will weigh together as much as 14 diams of rod of same diam 
as screw. In either case, a square head and nut will weigh as much 
as 6 diameters. Cast-iron washers, being more apt to split under 
heavy strains, may be made about twice as thick as wrought ones. 

Wheu the strain is very great, the diam of the washer may be 5 or 
6 times that of the screw; and its thickness equal to diam ; but 4 
diams will suffice for most practical purposes, or even 2.5 when there 
is but little strain, and tbe thickness may then be but .1 or .2 diam of 
bolt. 

Table of machine and car bolts, with 

square and hexagou heads and nuts, Figs 4 and 5; made by Hoopes 
& Townsend, 1330 Buttonwood St, Phila. All their bolts 
are cut with U. S. Standard threads, as 
per table on p 406, unless otherwise ordered. For washers, see p 406. 


Diam D, Fig 1, of 
bolt, ins. 

Length 
ins ex¬ 
clusive 
of head. 

Weight of 
100 bolts 
with nuts, 
in B>s. 

List price* 
in dollars 
per 100. 

Size of sq 
head, ins. 

Size of hex 
head, ins. 

Size of 
nut, ins. 

Chamfered 

Square 

Nuts. 

Chamfered 

Hexagon 

Nuts. 

a 

Max. 

Min. 

Max. 

Min. 

Max. 

Width. 

Thickn's. 

‘WAi 1 

1 

Thickn's. 

Width. 

Thickn’s. 

No. per 

100 lbs. 

List pricet 
in cts per 
11>. 

No. per 

100 fts. 

V u 

£ CU 

. 

M 

y< 

1>£ 

8 

4 

13.75 

2.80 

4.10 

7-16 

3-16 

7-16 

Va 


Va 

7000 

20. 

8200 

27. • 

5-16 

( 4 

44 

7 

20.75 

3.20 

5.15 

A 

X 

• X 

5 16 

19-32 

5-16 

1300 

18. 

5100 

24. 

Vs 

44 

12 

10.5 

‘ 44.5 

3.60 

7.80 

% 

5-16 

X 

% 

11-16 

% 

2550 

14.5 

3000 

18.5 

7-16 

44 

44 

15.2 

61.4 

4.60 

10.90 

11-16 

% 

11-16 

7-16 

25-32 

7-16 

1770 

14. 

2030 

18. 


<i 

20 

22.5 

123.2 

5.00 

16.10 

13-16 

7-16 

13-16 

A 

% 

X 

1180 

11.3 

1400 

14. 

9-16 

14 

44 

B0 

159.5 

7.20 

25.75 

Vs 

X 

y» 

9-16 

31-32 

9-16 

920 

11.3 

1060 

14. 

y» 

it 

4 ( 

39.5 

196.5 

7 20 

25.75 

1 

X 

i 

% 

1 1-16 

% 

660 

10. 

780 

12.5 

H 

4 1 

44 

63 

286.8 

10.50 

32.70 

1 3-16 

Vs 

1 3-16 

H 


Va 

3 HO 

9.4 

470 

10.9 

% 

<4 

44 

100 

415.3 

14.90 

46.40 

1% 

H 

1% 

Vs 

1 7-16 

% 

260 

9.4 

308 

10.9 

i 

44 

44 

153 

558. 

22.00 

62.70 

IX 

% 

1 9-16 

‘ 


1 

172 

9.2 

212 

10.7 



Fig. 4. Fig. 5. 



Fig 3. 


# For bolts with square heads and nuts. Bolts with hex heads and nuts are about 20 per cent 
higher llisconnt, 1888, about 70 per cent. Bolts are also made (at extra prices) 
with hut ton-shaped and countersunk heads. The price per 100 varies 

with the length We give the extremes. The holts have finished points, and chamfered heads and 

nuts, as shown. From y 2 inch to 8 ins the lengths increase by ms; from 

8 ins to 20 ins, by ins. ^ -_ „ 

f Not tapped. Hisconnt, 1888, about 4.8 cts. per ID. 

X Not tapped. Iliscount, 1888, about 4.8 cts. per tt>. 

Expansion bolts, for fastening plates, timbers, 
etc;, to walls of brick or masonry. Steward & Remains 
Manufacturing Company, 6th and Cherry Sis., Philadel¬ 
phia. The wedge-shaped nut, traveling up the bolt, as 
the latter is turned, presses the wings against the 6ides 
of the hole, whi- h, in practice, is drilled just large enough 
to admit the nut and wings, so as to prevent the former 
from turning with the bolt. If the hole is made larger, 
as shown, the nut must bo held by a small wedge. 



































































































408 


BOLTS, NUTS, WASHERS, 


Lock-nut washers. When bolts are subjected to mnch rough 
jolting, as at rail-joints. &c, the nuts are liable to wear loose, and unscrew 
themselves. On railroads this is a source of great annoyance, and innumerable 
devices for preventing it have been tried. The Verona lock-nut 
washer isasiniplecirenlar washer made of steel; with a slits scut through it, leaving 
sharp edges. On one side, a, of the slit, the metal is pressed upward about ^ inch; 
and that on the other side, c, downward, the same distance; so that a perspective 
view would be somewhat as at f. Now, when the nut is screwed down over the ijCN . 

washer, in the direction of the arrow, the slit offers no obstruction ; but if the nut / ^ fiS £ 

afterward tends to unscrew itself, the sharp upper edge of the slit, along a, presents Ejte—Safjffl 

friction against the bottom of the nut, which tends to hold it in place. Besides, the L!llli!fl* ,p 

washer, by its elasticity, tends to resume its original shape, and thus presses the 

threads of the nut against those of the bolt; and the additional friction thus produced, also aids in 

holding the nut. The same principles are employed in a nut-lock recently (1884) introduced upon 

the Houston and Texas Central, where the lock-nut waslier is a long 

stripof steel, with two holes, each of which has its edges formed like those of a Verona washer, 
and through each of which passes one of the bolts of the rail joint. Another 
device is to cut at the end of the screw a few threads of a screw of less diam 
than the main one, and in the opposite direction. The nut is then screwed upon the larger diam; 
and after it the lock-nut is screwed in the other direction upon the smaller diam, until it comes 
into contact with the main nut. In the Smith lock-nut bolt, this second nut is 
only about % inch thick ; and after being driven home, one of its corners is bent over the edge of 
the main nut. These bolts cost, in 1884, about 5 cts each. 

See the Cambria nut-lock, p 765, 

The Atwood lock-nuts take advantage of elasticity in the nnt itself, which is 

obtained either by slitting the nut, or by reducing its thickness near the bolt hole. 

It is claimed that if tlie threads of an ordinary holt and nut are carefully cut, 
so as to he in contact with each other throughout, no lock-nut 

contrivance is necessary, because the friction between the two threads is distributed over a larger 
surf, and abrasion does not take place so readily as if the threads touched each other at only a few 
points. The nuts are therefore less apt to wear loose under repeated jarring. 



Owing to the difficulty of obtaining such perfect fitting bolts and nuts, due to the wear of the cut¬ 
ting tools used in their mfr, the Harvey Screw and Bolt, Co, 52 Wall St, 

New York, furnish bolts and nuts in which the thread on the bolt differs slightly in shape from that 
in the nut. They also furnish nuts in which the thread, instead of being of uniform shape through¬ 
out, gradually becomes deeper and thicker, by having its side angle a. Fig 1, p 406, made more 
acute, and its top truncated. These nuts are used with bolts having the usual uniform thread. 
The bolt enters the nut upon the side where the thread is of the same shape as its own; but Us 
thread encounters, and is forced into, the gradually narrowing and deepening path between the 
threads of the nut. In both devices, the enforced conformity between the two threads, is relied 
upon to give the desired completeness of contact between them. The greater force required in tcrew- 
iug on the nut, also increases the friction between the threads. 


Hill’s patent waslier nut-lock (J. F. Dill, Ridgway, Pa.), consists simply 

of two oblong interlocking 
washers of soft iron, about 
one twelfth of an inch thick, 
and shaped as in the Figure. 
In one end of each washer 
is a round hole through ! 
which the bolt passes. 
When the nuts are to be 
turned, the other ends of , 
the washers are bent up out I 
of the way by means of a pick or crow-bar, etc. When they are brought back to j 
the positions shown, by a blow of a hammer, they again lock the nuts. The aanm 
principle is also applied to tho locking of three or more nuts. For square nuts 
(and, indeed, generally for hexagon nuts also) the upturned ends of the two washers 
are left square. Actual tests in rail-joints under heavy traffic and extremes of 
weather have not produced a case of failure. Cost, 1888, about §10 per 1000. 



Table of diameters, weights, and approximate breaking 
strains, for round rolled iron bolts, ties, or bars; assuming the 

breaking strain per square inch of average quality of rolled iron to be as follows: Up to 1 inch 
square, or 1 inch diam. 20 tons, or 44800 lbs ; from 1 to 2 ins sq or diam, 19 tons ; 2 to 3 ins, 18 tons; , 
3 to 4 ins, IT tons; 4 to 5 ins, 16 tons; 5 to 6 ins, 15 tons. The first 4 columns of the table are to ' 
be used when the screw end of the bolt is enlarged or upset, so that the shank or body of the boll J 
shall not be weakened by the cutting of the screw threads. But when the shauk is so weakened the 
diam and weight of the bolt must be taken from the last 2 cols. * 


Rem. But it is very important to know that a long upset rod is no 
stronger than one not upset, against slowly applied loads or strains. Both will 
then break at about midlength, under equal pulls. Therefore in such cases tho 
col of greatest diams in the table should be used. 


Square bars. Strength or weight = 1.273 X strength or weight of round bar. 
Copper bars. -f = 8 X strength of similar iron bar. 

** \ Weight =1,14 X weight “ “ “ * 











WEIGHT OF METALS, 


409 


! 

i 

I 

s 

t. 

It 

it 

I 

I■ 
ti 

w 

4. 

t 

t, 

J 

it 

i 

v 

J 

i 

!t 

U 

A 

JI 

* 

i. 

IX 

of 

it 

to 

iD 

nil 

rs 


WEIGHT AXI> STRENGTH OF IRON BOLTS. (Original.) 
For square ones or for copper see preceding paragraph. 


Ends enlarged, or upset. 

Ends not 
enlarged. 

Ends enlarged, or upset. 

Ends not 
enlarged. 

Diana. 

Weight 

Break- 

Break- 

Diam. 

Weight 

Diam. 

Weight 

Break- 

Break- 

Diam. 

Weight 

of 

per foot 

ing 

ing 

of 

per foot 

of 

per foot 

ing 

ing 

of 

per foot 

shank 

run. 

strain. 

strain. 

shank 

run. 

shank 

run. 

strain. 

strain. 

shank 

run. 

Ins. 

Pds. 

Tous. 

Pds. 

Ins. 

Pds. 

Ins. 

Pds. 

Tons. 

Pds. 

Ins. 

Pds. 

* 

.0414 

.245 

549 



If 

8.10 

45.7 

10236S 

2.14 

12.0 

T 3 8 

.093 

.553 

1239 



m 

8.69 

49.0 

109760 

2.22 

12.9 

1 

.105 

.983 

2202 

.35 

.321 

n 

9.30 

52.5 

1170)0 

2.30 

13.8 

TB 

.258 

1.53 

3427 

.43 

.452 

lit 

9.93 

56.0 

125440 

2.38 

14.7 

$ 

.372 

2.21 

4950 

.50 

.654 

2 

10.6 

59.7 

133728 

2.45 

15.7 

T ? B 

.506 

3.00 

6720 

.58 

.897 

2| 

12.0 

63.8 

142912 

2.59 

17.5 


.661 

3.93 

8S03 

.66 

1.14 

2 i 

13.4 

71.6 

160384 

2 73 

19.5 

A 

.837 

4.97 

11133 

.73 

1.41 

2g 

14.9 

79.7 

178528 

2.88 

21.6 

5 

a 

1.03 

6.14 

13754 

.80 

1.67 

2 * 

16.5 

88.4 

198016 

3.02 

23.9 

u 

1.25 

7.42 

16021 

.88 

2.03 

2| 

18.2 

97.4 

218176 

3.16 

26.1 

$ 

1.49 

8.83 

19779 

.96 

2.41 

2| 

20.0 

106.9 

239456 

3.30 

28.5 

n 

1.75 

10.4 

23296 

1.04 

2.81 

2§ 

21.9 

116.8 

261632 

3.45 

31.1 

7 

15 

2.03 

12.0 

26880 

1.12 

3.26 

3. 

23.8 

127.2 

284928 

3.60 

33.9 

1 5 

1 8 

2.33 

13.8 

30912 

1.20 

3.77 

H 

27.9 

141.0 

315840 

3.86 

39.1 

1 in. 

2 65 

15.7 

3516S 

1.27 

4.27 

3* 

32.4 

163.6 

366464 

4.12 

44.4 

1A 

2.99 

16.8 

37632 

1.35 

4.77 

3 f 

37.2 

187.7 

420448 

4.41 

51.0 

’1 

3.35 

18.9 

42336 

1.42 

5.28 

4. 

42.3 

213.6 

478464 

4.70 

57.8 

3 A 

3.73 

21.1 

47261 

1.49 

5.81 

H 

47.8 

227.0 

508480 

4.98 

65.2 

li 

4.13 

23.3 

52192 

1.55 

6.39 


53 6 

254.5 

570080 

5.25 

72.9 


4.56 

25.7 

5756S 

1.64 

7.04 

4f 

59.7 

283.5 

635040 

5.53 

80.5 

lg 

5.00 

28.2 

63168 

1.72 

7.74 

57 

66.1 

314.2 

703808 

5.80 

88.1 

1 7 

5.47 

30.8 

68992 

1.80 

8.48 

5* 

72.9 

324.7 

72732S 

6.08 

97.0 

1 A 

5.95 

33.6 

75261 

1.87 

9.20 

5* 

80.0 

356.4 

798336 

6.36 

106. 

3 A 

6.46 

36.4 

81536 

1.94 

9.88 

5f 

87.5 

389.5 

8724S0 

6.63 

116. 

1 § 

6 99 

39.4 

88256 

2.00 

10.6 

67 

95.2 

424.1 

949984 

6.90 

126. 

m 

7.53 

42.5 

95200 

2.07 

11.3 


See Rem, p 408, 



BUCKLED PLATES 


of iron or steel are usually 3 or 4 feet square, from 3 to % inch thick, with 
a flat rim about 2 inches wide all around, with rivet or Dolt holes for holding the 

J date firmly down to its intended place. The rest of the plate is stamped into the 
orm of a kind of groined arch rising from 1 to 3 inches in the center. They are 
very strong, and are used for the floors of fire-proof buildings, and of city iron 
bridges, covered with asphalt or stone paving, &c. One of 3 feet square, .25 
inch thick, curved 1.75 inches, and with a 2-inch rim well bolted down on all 
sides, required a quiet, equally distributed load of 18 tons to crush it. 

Table of sale, quiet, uniformly distributed loads for buckled 
iron plates 3 feet square, arched 1.75 inches, and well bolted down on all sides. 
Keystone Bridge Co., Pittsburgh, Pa. Price, iron or steel, 3 cents per 8). 
at mill. 


| - ■ ■■ ... - - ■ 

Weight of one 

Safe load on one plate 

Thickness. 

plate. 

( = one-fourth of ultimate load.) 

Pounds. 

Pounds. 


t 3 «t incb 

68 

5600 

X “ 

90 

. 10080 

5 “ 

T5" 

113 

13888 

Vs “ 

133 

20160 





















































WIRE GAUGES, 


,410 

The Birmingham wire gauffe is the one in most general use for iron. The 
new British w g went into effect March 1st 1884. In the “ American ” w g of Dar¬ 
ling, Brown & Sharpe, Providence R. I., each diam or thick is = the next smaller 
one X 1.122932. We take the wt of wrot iron per cub ft at 485 lbs in the first two; 
and at 4Sf. in the last. For the wt ol' steel, mult that of iron by 1.01. For 
lead, mult iron by 1.46. For zinc, mult iron by .9. For brass (approx), mult 
iron by 1.06. For copper, mult iron by 1.134. See p 411 and Trenton Gauge p412. 


No. 


7-0 

6-0 

5-0 

4-0 

3-0 

2-0 

0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 
.16 

17 

18 
19 

• 20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 
- 46 

47 

48 

49 

50 


Birmingham W.Ga. 

New British W. Ga. 

American 1\ 

r . Ga. ] 

Diam of 
wire, or 
thickness 
of sheet, 
ius. 

Wt of 
iron wire, 
in tbs per 
lin ft. 

Wt of 
iron 
sheets, 
in tbs per 
sq ft. 

Diam of 
wire, or 
thickness 
of sheet, 
ins. 

Wt of 
iron wire, 
in lbs per 
lin ft. 

Wt of 
iron 
sheets, 
in tbs per 
sq ft. 

Diam of 
wire, or 
thickness 
of sheet, 
ins. 

Wt of 
iron wire, 
in lbs per 
iin ft. 

wt or 

iron 
sheets, 
in tbs 
persqft 




.500 

.661 

20.21 







.464 

.569 

18.75 







.432 

.494 

17.46 




.454 

.546 

18.35 

.400 

.423 

16.17 

.460000 

.561 

18.63 

.425 

.479 

17.18 

.372 

.366 

15.03 

.409642 

.445 

16.58 

.380 

.383 

15.36 

.348 

.320 

14.06 

.304796 

.353 

14.77 

.340 

.306 

13.74 

.324 

.278 

13.09 

.324861 

.280 

13.15 

.300 

.238 

12.13 

.300 

.238 

12.13 

.289297 

.222 

11.70 

.284 

.214 

11.48 

.276 

.202 

11.15 

.257627 

.176 

10.43 

.259 

.178 

10.47 

.252 

.168 

10.19 

.229423 

.139 

9.291 

.238 

.150 

9.619 

.232 

.142 

9.377 

.204307 

•111 

8.273 

.220 

.128 

8.892 

.212 

.119 

8.568 

.181940 

.0877 

7.366 

.203 

.109 

8.205 

.192 

.0976 

7.760 

.162023 

.0696 

6.561 

.180 

.0859 

7.275 

.176 

.0820 

7.113 

.144285 

.0552 

5.842 

.165 

.0721 

6.669 

.160 

.0677 

6.466 

.128490 

.0438 

5.203 

.148 

.0580 

5.981 

.144 

.0548 

5.820 

.114423 

.0347 

4.633 

.134 

.0476 

5.416 

.128 

.0434 

5.173 

.101897 

.0275 

4.125 

.120 

.03S2 

4.850 

.116 

.0357 

4.688 

.090742 

.0218 

3.674 

.109 

.0315 

4.405 

.104 

.0286 

4.203 

.080808 

.0173 

3.272 ( 

.095 

.0239 

3.840 

.092 

.0224 

3.713 

.071962 

.0137 

2.014 I 

.083 

.0183 

3.355 

.080 

.0169 

3.233 

.064084 

.0109 

2.595 [ 

.072 

.0137 

2.910 

.072 

.0137 

2.910 

.057068 

.00863 

2.310 

.065 

.0112 

2.627 

.064 

.0108 

2.587 

.050821 

.00684 

2.053 j 

.058 

.00891 

2.344 

.056 

.00832 

2.263 

.045257 

.00543 

1.832 

.049 

.00636 

1.9S0 

.048 

.00610 

1.940 

.040303 

.00430 

1.631 

.042 

.00467 

1.697 

.040 

.00423 

1.617 

.035890 

.00341 

1.452 

.03) 

.00325 

1.415 

.036 

.00344 

1.455 

.031961 

.00271 

1.293 

.032 

.00271 

1.293 

.032 

.00269 

1.293 

.028462 

.00215 

1.152 

.028 

.00208 

1.132 

.028 

.00207 

1.132 

.025346 

.00170 

1.026 

.025 

.00166 

1.010 

.024 

.00152 

.9700 

.022572 

.00135 

.913 . 

.022 

.00128 

.8892 

.022 

.00128 

.8S91 

.020101 

.00107 

.814 

.020 

.00106 

.8083 

.020 

.00106 

.8083 

.017900 

.000849 

.724 t, 

.018 

.000859 

.7225 

.018 

.000857 

.7275 

.015941 

.000673 

.644 ! 

.016 

.000678 

.6467 

.0164 

.000712 

.6628 

.014195 

.000534 

.574 ] 

.014 

.000519 

.5658 

.0148 

.000579 

.5982 

.012641 

.000423 

.511 i 

.013 

.000448 

.5254 

.0136 

.000489 

.5497 

.011257 

.000336 

.455 r 

.012 

.000382 

.4850 

.0124 

.000408 

.5012 

.010025 

.000266 

.405 } 

.010 

.000265 

.4042 

.0116 

.000357 

.4688 

.008928 

.000211 

.360 I 

.009 

.000215 

.3638 

.0108 

.000309 

.4365 

.007950 

.000167 

.3211 

.008 

.000170 

.3233 

.0100 

.000265 

.4042 

.007080 

.000133 

.286 k 

.007 

.000130 

.2829 

.0092 

.000224 

.3718 

.006305 

.000105 

.254 i 

.005 

.0000662 

.2021 

.0084 

.000187 

.3395 

.005615 

.0000837 

.226 a 

.004 

.0000424 

.1617 

.0076 

.000153 

.3072 

.0050(H) 

.0000662 

.202 f 




.0068 

.000122 

2748 

.004453 


ISO - 




.0060 

.0000952 

2425 

.003965 

00004.1 7 

1 50 

. 



.0052 

.0000714 

.2102 

.003531 

.0000330 

.142 jf 




.0048 

.0000608 

1940 

.003144 

.0000262 

127 L 




.0044 

0000513 

1778 






.0040 

0000423 

1617 







.0036 

0000344 

1455 







.0032 

0000271 

1293 







.0028 

0000207 

.1132 

HQ7A 







-0024- 

41000159 







.0020 

0000106 








0016 

0000068 

nnd7 







.0012 

.0000038 

.0485 







.0010 

.0000026 

.0404 



J 

























































WIRE GAUGES. 


411 


No trade stupidity is more thoroughly senseless than the adherence to 
the various Birmingham, Lancashire, &c, gauges; instead of at once denoting the 
thickness and diameter of sheets, wire, &c, by the parts of an inch; as has long 
been suggested. Thus, No. ]/g, or No. wire, or sheet-metal of any kind, should 
be understood to mean % or ^ of an inch diam, or thickness. To avoid mistakes, 
which are very apt to occur from the number of gauges in use; and from the absurd 
practice of applying the same No. to different thicknesses of different metals, in dif¬ 
ferent towns, it is best to ignore them all ; and in giving orders, to define the diam¬ 
eter of wire, and the thickness of sheet-metal, by parts of an inch. Or the weight 
per hundred ft for wire; or per sq ft for sheets, may be employed. We believe that 
the foregoing Birmingham gauge applies to zinc, copper, brass, and lead; although 
it is generally stated to be for iron and steel only. Another Birmingham gauge it 
used for sheet-brass, gold, silver, and some other metals; but we have never seen it 
stated what those others are. There are different gauges even for wire to be used 
for different purposes; and various firms have gauges of their own ; not even accord¬ 
ing among themselves. 

As Mr. Stubs makes various English gauges, the term “ Stubs gauge ” by 
itself means nothing. Generally, however, in our machine shops, it applies to the 
Birmingham gauge of the preceding table. 

Birmingham gauge for sheet Brass. Silver, Gold, and all metala 

except iron and steel ? 


No. 

Thickn’s. 

No. 

Tbickn’s. 

No. 

Thickn’s. 

No. 

Thickn’s. 

No. 

Thickri’s. 

No. 

Thickn’s. 

1 

Inch 

.004 

7 

Inch 

.015 

13 

Inch 

.036 

19 

Inch 

.064 

25 

Inch 

.095 

31 

Inch 

.133 

2 

.005 

8 

.016 

14 

.041 

20 

.067 

26 

.103 

32 

.143 

3 

.008 

9 

.019 

15 

.047 

‘21 

.072 

27 

.113 

33 

.145 

4 

.010 

10 

.024 

16 

.051 

22 

.074 

28 

.120 

34 

.148 

5 

.012 

11 

.029 

17 

.057 

23 

.077 

29 

.124 

35 

.158 

6 

.013 

12 

.034 

18 

.061 

24 

.082 

30 

.126 

36 

.167 


The mills rolling sheet iron in the United States generally 
use the following, which varies slightly from the Birmingham gauge: 


No. 

lbs per 
sq ft 

No. 

los per 
sq ft 

No. 

lbs per 
sq ft 

No. 

lbs per 
sq ft 

1 

12 50 

8 

6.86 

15 

2.81 

22 

1.25 

2 

12.00 

9 

6.24 

16 

2.50 

23 

1.12 

3 

11.00 

10 

5.62 

17 

2.18 

24 

1.00 

4 

10.00 

11 

5.00 

18 

1.86 

25 

.90 

5 

8.75 

12 

4.38 

19 

1.70 

26 

.80 

6 

8.12 

13 

3.75 

20 

1.54 

27 

.72 

7 

7.5)0 

14 

3.12 

21 

1.40 

28 

.64 


When wire, sheet-metal, &c., are ordered by gauge number, and it is 
not specified what gauge is intended; dealeis in the United States fill the order as 
follows: 

Brass, bronze or German Silver in sheets. German Silver wire, brazed brass, bronze, 
zinc or copper tubing, by Brown & Sharpe’s (or “American") gauge, last column, 

p. 410. 

Copper in sheets; brass and copper wire ; seamless brass, bronze or copper tubing; 
| and small brass rods; by Stubs’ (or Birmingham) gauge, first column, p. 410. 

Approximate prices per pound, of brass and copper wire, Nos. 
0 to 25, for 100 lbs. or more. Merchant & Co., 517 Arch St., Philadelphia. Copper, 
30 to 40 ets.; high brass, 22 to 32 cts.; low' brass, 26 to 36 cts. Discount, 1888, 10 to 
|20 per ceut. 

Unannealed or hard brass wire has about %tha the strengths of the table p. 412, 
and about l more weight. If annealed, only full half the strength. 

Hard copper wire may be taken at % of the tabular strengths, and full 
1 more weight. 






























412 


IRON WIRE. 




Table of Charcoal Iron Wire made by Trenton Iron Co., 

Trenton, N. J. The numbers in the first column are those of the Trenton Iron 
The corresponding diameters in th‘e second column will be seen to 
be somewhat less than those of the "Birmingham gauge, p 410. 


No. 

Diarn. 

ius. 

Lineal 
feet to the 
Pound. 

Tensile 

Str'gth 

Approx 

lbs. 

No. 

Diam. 

ins. 

Lineal 
feet to the 
Pound. 

Tensile 

Str'gth 

Approx 

lbs. 

No. 

Diam. 

ins. 

Lineal 
feet to the 
Pound. 

00000 

.450 

1.863 

12598 

11 

.1175 

27.340 

1010 

26 

.018 

1164.689 

0000 

.400 

2.358 

9955 

12 

.105 

34.219 

810 

27 

.017 

1305.670 

000 

.360 

2.911 

8124 

13 

.0925 

44.092 

631 

28 

.016 

1476.869 

00 

.330 

3.465 

6880 

14 

.080 

58.916 

474 

29 

.015 

1676.989 

0 

.305 

4.057 

5926 

15 

.070 

76.984 

372 

30 

.014 

1925.321 

1 

.285 

4.645 

5226 

16 

.061 

101.488 

292 

31 

.013 

2232.653 

2 

.265 

5.374 

4570 

17 

.0525 

137.174 

222 

32 

.012 

2620.607 

3 

.245 

6.286 

3948 

18 

.045 

186.335 

169 

33 

.011 

3119.092 

4 

.225 

7.454 

3374 

19 

.040 

235.084 

137 

34 

.010 

3773.584 

5 

.205 

8.976 

2839 

20 

.035 

308.079 

107 

35 

.0095 

4182.508 

6 

.190 

10.453 

2476 

21 

.031 

392.772 


36 

.009 

4657.728 

7! 

.175 

12.322 

2136 

22 

.028 

481.234 


37 

.0085 

5222.035 

8 

.160 

14.736 

1813 

23 

.025 

603.863 


38 

.008 

5896.147 

9 

.145 

17.950 

1507 

24 

.0225 

745.710 


39 

.0075 

6724.291 

10 

.130 

22.333 

1233 

25 

.020 

943.396 


40 

.007 

7698.253 


or unannealed, 
made with good 


The wire in this table is supposed to be hard, bright. 

The figures in the column of tensile strength are based upon tests 
charcoal iron wire from Trenton blooms. 

The tensile strength of wire made of is about 

Good refined iron.. 15 per cent, less 

Swedish charcoal iron. 10 “ “ 

Mild Bessemer steel. 10 

Ordinary crucible steel. 25 

Special crucible steel.30 to 120 

Annealing renders wire more pliable and ductile, but less elastic; and reduces the 
tensile strength by from 20 to 25 per cent. 


more 

44 


than that of 
bright charcoal 
wire, given in 
the above table. 


To find approximately the nnmber of straight wires that 
can be got into a cable of given diameter. 


Divide the diameter of the cable in inches, by the diameter of a wire in inches. 
Square the quotient. Multiply said square by the decimal .77. The result will be 
correct within about 4 or 5 per cent at most, in a cylindrical cable. 

The solidity, or metal area of all the wires in a cable, will be 

to the area of the cable itself, about as 1 to 1.3. In other words, the area of the 
voids is nearly % that of the cable; while that of the wires is fully % that of the 
cable. All approximate. 


Price-list of Charcoal Iron and Bessemer Steel Wire.* Bright 
and annealed. Trenton Iron Co. 


Nos. 


0 to 9, 

10 & 11 

, 12, 13 & 14, 

15 

& 16, 

17, 

18, 

19, 

20, 

21, 

22. 

Cts per 

lb. 

10 

11, 

n>i 12 K, 



15, 

16, 

19, 

20, 

21, 

22 

Nos. 


23, 24, 

25, 

26, 27, 28, 

29, 

30, 

31, 

32, 

33, 

34, 

35, 

36 

Cts per 

lb. 

23, 24, 

25, 

26, 28, 29, 

30, 

32, 

33, 

3o, 

37, 

40, 

45, 

55 


Price-list of Cast (crucible) Steel WTre.f Trenton Iron Co. 




i 


11 
i 

t 


r 

« 




Nos. 0 to 6, 7 to 9, 10 A 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 
Ctsperlb. 23, 24, 25, 26, 28, 30, 32, 33, 34, 36, 38, 40, 50, 60, 75 


The Co make other wire of specified quality to order; also W T ir« 
Ropes, and fittings for same. See p 413. 


* Discounts on iron and Bessemer steel wire (bright or annealed). Nos. 00000 t< 
36, about 60 to 75 per cent. Nos. 37 to 45, about 50 per cent. 1888. 

| Discount on crucible steel wire, about 50 to 60 per cent. 1888. 



























































WEIGHT AND STRENGTH OF WIRE ROPES, 


413 


a V w ,\ > !' '' * v nianufactured by John A. Roebling’s Sons 

rc nton, J m I rices, 1888, net, except on large orders. The prices 

HUM U’PKrhla trivun 4V».. 1. i . «... 0 r 


and weights given are lor ropes with hemp centers. ' When made with wire centers 
tiie prices aie one-tenth higher, and the weights one-tenth greater. 


ltope of 133 Wires (19 wires in a strand). 


Trade 

No. 

Diam. 

Ins. 

Cir- 

cumf. 

Ins. 

Pounds 
per foot 
run. 

Breaking load, lbs. 

Minimum diam of 
drum, in ft. 

Price per ft run, 
in cents. 



Iron.# 

Cast steel* 

Iron. 

Cast st'l* 

Iron.* 

Ca'tst* 

1 

2 

3 

4 

5 

&K 

6 

7 

8 

9 

10 

™K 

1"K 

10% 

2K 

2 

% 

ik 

1 

r{ 

6% 

6 

5U 

5 

i % 

f 

IK 

IK 

8.00 

6.30 

5.25 

4.10 

3.65 

3.00 

2.50 

2.00 

1.58 

1.20 

0.88 

0.60 

0.44 

0.35 

0.26 

148000 

130000 

108000 

88000 

78000 

66000 

54000 

40000 

32000 

23000 

17280 

10260 

8540 

6960 

5000 

310000 

250000 

212000 

172000 

154000 

126000 

104000 

84000 

66000 

50000 

36000 

28000 

18000 

15000 

8 

7 

6.5 

5 

4.75 

4.5 

4 

3.5 

3 

2.75 

2.5 

2 

1.75 

1.5 

1 

9 

8 

7.5 

6 

5.5 

5* 

4.5 

4 

3.75 

3.5 

3 

2.75 

2 

100 

78 

69 

58 

53 

43 

36 

29 

26 

20 

16 

14 

12 

10 

8 

152 

120 

100 

80 

71 

60 

50 

41 

34 

27 

21 

18 

17 

15 


Rope of 49 Wires (7 wires to the strand). 


Trade 

No. 

Diam. 

Ins. 

Circumf. 

Ins. 

Pounds per 
foot run. 

Breaking load, lbs. 

Price per foot run, 
in cents. 

Iron.* 

Cast steel.* 

Irou.* 

Cast st'l* 

11 

vx 

4K 

3.37 

72000 

124000 

48 

70 

12 

1 /8 

4K 

2.77 

60000 

104000 

39 

60 

13 

iK 

3K 

2.28 

50000 

88000 

34 

50 

14 

iK 

3K 

1.82 

40000 

72000 

27 

40 

15 

1 

3 

1.50 

32000 

60000 

23 

32 

16 

% 

2% 

1.12 

24600 

44000 

19 

25 

17 

% 

2% 

0.88 

17600 

34000 

14 

19 

18 

jb 

2K 

0.70 

15200 

28000 

12 

16 

19 

% 


0.57 

11600 

22000 

k>K 

14 

20 


1 % 

0.41 

8200 

16000 

8 

11 

21 

K 

IK 

0.31 

5660 

12000 

7 

8 

22 

TS 

IK 

0.23 

4260 


5K 


23 

% 

iK 

0.19 

3300 

8000 

5 

7 

24 

T 5 8 

1 

0.16 

2760 

6000 

4 

5 

25 

32 

% 

0.125 

2060 


3K 

... 


Notes on the Use of Wire Rope, by the Roeblings Co. 

The ropes with 19 wires per strand are the most pliable, and therefore best adapted for hoisting 
and running rope. Tbe others are stiffer and better adapted for guys, &c. Ropes of iron or steal, 
up to 8 ins diam, made to order. For the safe working load take one-fifth to one-seventh of 
1 the breaking load, according to speed. Hemp center rope is more pliable than wire center. Wire 
1 rope must not be coiled or uncoiled like hemp rope. When on a reel, tbe latter should be mounted 
on a spindle or Hat turn-table to pay off the rope. When forwarded in a small coil without a reel, 

1 roll it on the ground like a wheel, and thus run off the rope. Avoid untwisting and short bends. To 
, preserve wire rope, apply raw linseed oil (which may be mixed with an equal quantity of Span¬ 
ish brown or lamp-black) with a piece of sheepskin, keeping the wool against the rope. If for use 
in water or under ground, add 1 bushel of fresh slacked lime, and some sawdust, to t barrel of 
tar. Roil the mixture well, and saturate the rope with it while hot. Never use galvanized rope 
for running rope. The grooves of cast-iron pulleys should be filled with blocks of well-seasoned 
hard wood, set on end. Leather or india-rubber is better where the pulleys are large and run very- 
fast. Galvanized wire rope for rigging is cheaper and more durable than hemp rope; and does 
not stretch permanently under great strains. Its bulk is oue-sixth, and its weight one-half, that of 
hemp rope. Roehling’s wire rope has been made the standard by the U. States Navy Department. 
8hackles, sockets, swivel-hooks, and fastenings. <tc furnished and put on. and splices 
made. Pulley wheels furnished. Also, galvanized steel cables for suspension bridges. 
Crucible cast-steel wire ropes are much more durable than iron ones. They should be kept well lu¬ 
bricated. The foregoing is condensed from the ciroular of the Roeblings Co. 


* Ropes of Bessemer steel, and of Siemens-Martin steel, are sold at the same prices as iron ropes. 
They havp stood higher strains ; but. in view of the lack of uniformity in those steels, it is not advis¬ 
able to reduce their diam below that of iron rope for the same work. 






































































414 WEIGHT AND STRENGTH OF IRON CHAINS. 


On planes in Schuylkill Co. ko, a -wire rope generally lasts long enough to raise one mil- L 
lion of tons of coal up a plane half a mile long, and rising 1 in 10. The ordinary duration on inclined 
planes throughout the country is from l l 4 to 4 years, according to the amount of service; auu also 
greatly to the care taken of them, and of the sheaves and rollers upon which they move. « '' 

On the Mt Pisgah plane, 2500 ft long, rising 660 ft, for raising empty coal cars, aud lo wering loaded 
ones, thin iron bands, 7 inches wide, and about % inch thick, have been used instead of ropes, lhey j 
scarcely exhibit any sign of wear in several years. They should be riveted; being apt to break if , 
welded. Steel would probably be the best material in many cases. k 


Table of Manilla rope. 


Diam. 

Ins. 

Circ. 

Ins. 

Wt per 
foot, 
lbs. 

Breaki 

ng load. 

Diam. 

Ins. 

Circ. 

Ins. 

Wt per 
foot, 
lbs. 

Breaking load. 

Tons. 

lbs. 

Tons. 

lbs, 

.239 

X 

.019 

.25 

560 

1.91 

6 

1.19 

11.4 

25536 

.318 

1 

.033 

.35 

784 

2.07 

&A 

1.39 

13.0 

29120 

.477 

\'A 

.074 

.70 

1568 

2.23 

i 

1.62 

14.6 

32704 

.636 

2 

.132 

1.21 

2733 

2.39 

i'A 

1.86 

16.2 

36288 

.795 

•2\4 

.206 

1.91 

4278 

2.55 

8 

2.11 

17.8 

39872 

.955 

3 

.297 

2.73 

6115 

2.86 

9 

2.67 

21.0 

47040 

1.11 

Z'A 

.404 

3.81 

8534 

3.18 

10 

3.30 

24.2 

54208 

1.27 

4 

.528 

5.16 

11558 

3.50 

11 

3.99 

27.4 

61376 

1.43 


.668 

6.60 

14784 

3.82 

12 

4.75 

30.6 

68544 

1.59 

5 

.825 

8.20 

18368 

4.14 

13 

5.58 

33.8 

75712 

1.75 

5^ 

.998 

9.80 

21952 

4.45 

14 

6.47 

37.0 

82880 


I 


The strength of Manilla ropes, like that of bar iron, la very variable 
and so with hemp ones. The above table supposes an average quality. Ropes oi 
good Italian hemp are considerably stronger than Manilla; hut their cost exclude: 
them from general use. The tarring of ropes is said to lessen their strength 
and, when exposed to the weather, their durability also. We believe that the use oil 
it in standing rigging is partly to diminish contraction and expansion by alternat 
wet and dry weather. The common rules for finding the strength of rop 
by multiplying the square of the diam or circumf by a given coefficient are entirel; 
erroneous. Prices in Philada, 1888, Manilla, 18 to 14 cts per lb; Italian hemp, - 
20 cts; American hemp, 12 cts; Sisal hemp, 10 cts; jute, (E. Indies,) 7 cts. E. H 
Fitler & Co., 23 N. Water St, Phila. 

The strengths of pieces from the same coil may vary 25 per ct. 

A few months of exposed work, weakens ropes 20 to 50 per ct. 


WEIGHT AND STRENGTH OF IRON CHAINS. 


Table of strength of chains. 

Chains of superior iron will require ^ to % more to break them. (Original.) 


Diam of rod 
of which 
the links 
are made. 

Weight 
of chain 
per ft run. 

Breaking strain 
of the chain. 

j 

Diam of rod 
of which 
the links 
are made. 

Weight 
of chain, 
per ft run. 

Breaking strair 
of the chain. 

Ins. 

Pds. 

Pds. 

Tons. 

Ins. 

Pds. 

Pds. 

Tons. 

3-16 

.5 

1731 

.773 

1 

10.7 

49280 

22 00 

X 

.8 

3069 

1.37 


12.5 

59226 

26.41 

5-16 

1. 

4794 

2.11 

dJ 

16. 

73114 

32.61 

% 

1.7 

6922 

3.09 


18.3 

88301 

39.4'i 

7-16 

2. 

9408 

4.20 

i % 

21.7 

105280 

47 .Of 

X A 

2.5 

12320 

5.50 


26. 

123514 

55.14 

9-16 

3.2 

15590 

6.96 

m 

28. 

143293 

63 9; 

% 

4.3 

19219 

8.58 

v/ 8 

32. 

164505 

73.4- 

11-16 

5. 

23274 

10.39 

2 

38. 

187152 

83.51 

X 

5.S 

27687 

12.36 


54. 

224448 

100.2 

13-16 

6.7 

32301 

14.42 

2K 

71. 

277088 

123.7 

% 

8. 

37632 

16.80 

-k 

88. 

335328 

149.7 

15-16 

9. 

43277 

19.32 

3 

105. 

398944 

178.1 


































































LEAD, COPPER, ETC, 


415 


^***? ts ®**dinary iron chains are usually made as short as is 
onsistent with easy play, in order that they may not become bent when wound 
round drums, sheaves, Ac ; and that they may he more easily handled in slinging 
vrge blocks of stone, Ac. U. S. Govt, expts, 1878, prove that studs weaken the links. 

lundTrnn m> We ' Rh C Per f °° t 1 ru “ is I 1 " 1 ' 5 approximately 3^ times that of a single bar or the 

’ annno^ed tL nh» h i^ ilre composed. Since each link consists of two thicknesses of bar, it might 
' su PP° sed that a chain would possess about double the strength of a single bar; but the strength of 

ie bar becomes reduced about y’j, by being formed into links; so that the chain really has but about 
<1 of the strength of two bars. Asa thick bar of iron will not sustain as heavy a load in proportion as a 

bl^^’fl'tons^ner ™ U /? e h S •° U *’ cfla ' n ? are proportion ably weaker than slighter ones. In the foregoing 

w^r^led iron l ? ineh*i’n A* SSUmp<1 th u aVerage breakin « st,ah ‘ or a single straight bar of ordi- 
iry rolled iron, 1 inch in diam ; or 1 inch square; lit tons, from 1 to 2 ius ; and 18 tons, from 2 to 3 

Deducting from each of these, we have as the breaking strain of the two bars composing 

link, aa follows* 14. a _i i_ 1 J ^ 


18 


ms f f s • a- 14 tons / e *? up to 1 inch diam; 13.3 tons, from I to 2 ins; and 12.6 

ms, fiom 2 to 3 ins diam : and upon these assumptions the table is based. The wts are approxi- 
late : depending upon the exactness of diameter of the iron, and shape of link. 


Approximate prices of chains, in cents per pound, 
usquehanna Avenue and Beach Street, Philadelphia, 1888. 


Bradlee & Co., 


Diameter of rod from which the links are made; ins 

Ordinary proved or coil chain. 

Crane chain.' 

Chain of combined iron and steel. 




1 



ROLLED LEAD, COPPER, and BRASS: Sheets and Bars. 


lickness 

or 

tnieter, 
; side, 

‘ in 
kches. 


LEAD 

• 

COPPER. 

BRASS. 

Thickness 

or 

Diameter, 
or side, 
in 

Inches. 

Sheets, 

per 

Square 

Foot. 

Square 
liars ; 

1 Foot 
long. 

Round 
Bars; 

1 Foot 
long. 

Sheets, 

per 

Square 

Foot. 

Square 
Bars; 

1 Foot 
long. 

Round 
Bars; 

1 Foot 
long. 

Sheets, 

per 

Square 

Foot. 

Square 
Bars; 

1 Foot 
loiig. 

Round 
Bars; 
1 Foot 
long. 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 


1-32 

1.86 

.005 

.004 

1.44 

.004 

.003 

1.36 

.004 

.003 

1-32 

1-16 

3.72 

.019 

.015 

2.89 

.015 

.012 

2.71 

.014 

.011 

116 

3 32 

5.58 

.014 

.034 

4.33 

.034 

.027 

4.06 

.032 

.025 

3-32 

hi 

7.44 

.078 

.031 

5.77 

.000 

.017 

5.42 

.056 

.044 

y» 

5 32 

9.30 

.121 

.095 

7.20 

.094 

.074 

6.75 

.088 

.069 

5 32 

3 IS 

11.2 

.174 

.137 

8.66 

.135 

.106 

8.13 

.127 

.100 

3 16 

7 32 

13.0 

.237 

.187 

10.1 

.184 

.144 

9.50 

.173 

.136 

7-32 

Va 

14.9 

.310 

.214 

11.5 

.240 

.189 

10.8 

.226 

.177 


5 16 

18.6 

.485 

.381 

14.4 

.376 

.295 

13.5 

.353 

.277 

5 16 

% 

22.3 

.698 

.548 

17.3 

.541 

.425 

16.3 

.508 

.399 

% 

7-16 

26.0 

.950 

.746 

20.2 

.736 

.578 

19 0 

.091 

.543 

716 

34 

29.8 

1.24 

.974 

23.1 

.962 

.755 

21.7 

.903 

.709 


9-16 

33.5 

1.57 

1.23 

26.0 

1.22 

.955 

21.3 

1.14 

.900 

9-16 

% 

37.2 

1.94 

1.52 

• 28.9 

1.50 

1.18 

27.1 

1.41 

1.11 

% 

1116 

40.9 

2.34 

1.84 

31.7 

1.82 

1.43 

29.8 

1.70 

1.34 

11 16 

% 

41.6 

2.79 

2.19 

31.6 

2.16 

1.70 

32.5 

2.03 

1.60 

H 

13-16 

48.3 

3-«7 

2.57 

37.5 

2.55 

1.99 

35 2 

2.38 

1.87 

13-16 

% 

52.1 

3.80 

2.98 

40.4 

2.94 

2.31 

37.9 

2.76 

2.17 

% 

15-16 

56 0 

4.37 

3.42 

43.3 

3.38 

2.65 

40.6 

3.18 

2.49 

15 16 

1 . 

59.5 

4.96 

3.90 

46.2 

3.85 

3.02 

43.3 

3.61 

2.84 

1. 


66.9 

6 27 

4.92 

52.0 

4.87 

3.82 

48.7 

4.57 

3.60 

1 % 

Di 

74.4 

7.75 

6.09 

57.7 

6.01 

4.72 

54.2 

5.64 

4.43 



81.8 

9.37 

7.37 

63.5 

7.28 

5.72 

59.6 

6.82 

5 37 



89.3 

11.2 

8.77 

69.3 

8.65 

6.80 

65.0 

8.12 

6.38 


1?4 

96.7 

13.1 

10.3 

75. t 

10.2 

7.98 

70.4 

9.53 

7.49 

i H 

\% 

iu4. 

15.2 

11.9 

80.8 

11.8 

9.25 

75.9 

11.1 

8.68 

m 

1 % 

112 . 

17 5 

13.7 

86 6 

13.5 

10.6 

81.3 

12.7 

9.97 

1 % 

2 . 

119. 

19.8 

15.6 

92.3 

15.4 

12.1 

86.7 

14.4 

11.3 

2 . 


Approximate net prices. Merchant & Co., £17 Arch St., Philadelphia, 1888. 
pper sheets, Nos. 1 to 24, 25 cts per lb. Copper ingots, 18 els. per pound. Brass 
sets, No. 18 and heavier, 2 to 14 inches wide, 18 cts. per lb. 











































416 


WEIGHT OF METALS 


Roof copper is usually in sheets of 2^ ft X 5 ft; or 12*^ 

square feet, weighing 10 to 14 fts |>er sheet; and is laid ou boards.* No solder 
is used in the horizontal joints as it is in tin roofs; but both the horizontal and 
the sloping joints are formed by only overlapping and bending the sheets, much 
as shown by the figs on page 418; except that the horizontal joints are bent 
or locked together, as in this figure; and then flattened down close. 

Sheet lead. Price, Philada., 1888, about 6 % cts. per lb. 
Tatham Bros, 226 S Fifth St. List of standard wts in lbs per 
eq ft. Thicknesses in decimals of an inch. 





wt. 

Th. 

Wt. 

Th. 

Wt. 

Th. 

Wt. 

Th. 

Wt. 

Th. 

Wt. 

Th. 

2.5 

.042 

4 

.068 

6 

.102 

8 

.136 

10 

.170 

14 

.237 

3 

.051 

5 

.085 

7 

.119 

9 

.153 

12 

.203 

16 

.271 


WEIGHT OF BATES. 


Diameter 

in 

Inches. 

Cast 

Lead. 

Cast 

Copper. 

Cast 

Brass. 

Cast 

Ikon. 

Diameter 

in 

Inches. 

Cast 

Lead. 

Cast 

Copper. 

Cast 

Brass. 

Cast 

Iron. 


Lbs. 

Lbs. 

Lbs. 

Lbs. 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

34 

.026 

.021 

.019 

.017 

534 

30.1 

24.1 

21.5 

19.8 

H 

.088 

.070 

.063 

.058 

54 

34.7 

27.7 

24.7 

22.7 

1 . 

.209 

.167 

.148 

.136 

H 

39.6 

31.7 

28.3 

25.9 

34 

.408 

.325 

.290 

.266 

6. 

45.0 

36.0 

32.0 

29.4 

X 

.705 

.562 

.501 

.460 

X 

57.2 

45.8 

40.8 

37.4 

% 

1.12 

.893 

.795 

.731 

7. 

71.5 

57.2 

50.9 

46.8 

2. 

1.67 

1.33 

1.19 

1.07 

34 

88.0 

70.3 

62.6 

57.5 

34 

2.38 

1.90 

1.69 

1.55 

8. 

106. 

85.3 

76.0 

69.8 

X 

3.25 

2.60 

2.32 

2.13 

X 

127. 

102. 

91.2 

83.7 

H 

4.34 

3.47 

3.09 

2.83 

9. 

151. 

121. 

108. 

99.4 

3. 

5.63 

4.50 

4.01 

3.68 

X 

178. 

143. 

127. 

117. 

34 

7.15 

5.72 

5.10 

4.68 

10. 

208. 

167. 

148. 

136. 

X 

8.94 

7.14 

6 36 

5.85 

34 

241. 

193. 

172. 

158. 

H 

11.0 

8.79 

7.83 

7.19 

11 . 

277. 

222. 

198. 

182. 

4. 

13.4 

10.7 

9.50 

8.73 

X 

317. 

253. 

226. 

207. 

54 

16.0 

12.8 

11.4 

10.5 

12 . 

360. 

288. 

257. 

236. 

Yi 

18.9 

15.*2 

13.5 

1*2.4 






X 

22.7 

17.9 

15.9 

14.6 

The weights of balls are as the cubes of their 

5. 

26.0 

20.8 

18.6 

17.0 

dianis. 






Eead pipe. Price, Philadelphia, 1888, about 6J4 cts per lb. Tin-lined, 1 
cts. Tatham Bros, mfrs, 226 S. Fifth St. List of standard sizes. 


Inner 

diain. 

• 

-£ 

«i 

tu 

£ * 

H 

WtS per ft (F) 
and per rod (R) 
of \6X ft. 

Inner 

diain. 

Thick- 

ness. 

Wts per ft (F) 
and per rod (R) 
of IdX ft. 

Inner 

diain. 

Thick- 

ness. 

CC 

£ . 

0 > 

Inner 

diain. 

Thick. 

ness. 

CO 

43 

ai 

t* ! 

Ins. 

Ins. 



Ins. 

Ins. 



lus. 

Ins. 


Ins. 

Ins. 


% 

.06 

7 

lbs R 

% 

.08 

16 

fts R 

\x 

.14 

3.5 

3 

3-16 

9 

«• 

.08 

10 

oz F 

44 

.10 

134 

fts F 

44 

.17 

4.25 

44 

34 

12 . 

If 

.12 

1 

ft F 

44 

.12 

1*4 

fts F 

44 

.19 

5. 

4 4 

5-16 

16 

44 

.16 

134 

fts F 

44 

.16 

234 

fts F 

44 

.23 

6.5 

44 

% 

20 


.19 

'X 

fts F 

44 

.20 

3 

fts F 

44 

.27 

8 

334 

3 16 

9 

X 

.07 

9 

fts R 

44 

.23 

334 

fts F 

1?4 

.13 

4 


3* 

15 

« 4 

.09 

H 

ft F 

44 

.30 

4 H 

fts F 

• 4 

.17 

5 

44 

5-16 

18. 


.11 

1 

ft F 

1 

.10 

24 % 

fts R 

44 

.21 

6.5 

41 

% 

22 

44 

.13 

154 

fts F 

44 

.11 

2 

fts F 

»4 

.27 

8 5 

4 

3-16 

12 

14 

.16 

1*4 

fts F 

U 

.14 

234 

fts F 

2 

.15 

4.75 

<4 

34 

16 


.19 

2 

fts F 

44 

.17 

334 

fts F 

44 

.18 

6 

44 

5-16 

21 

44 

.25 

3 

fts F 

44 

.21 

4 

fts F 

44 

.22 

7 

44 

y* 

25 

H 

.08 

12 

fts R 

»4 

.24 

4*4 

fts F 

4 4 

.27 

9 

*x 

3-16 

14 


.09 

1 

ft F 

134 

.10 

2 

fts F 

234 

3-16 

8 

4 t 

34 

18 


.13 

IX 

fts F 

44 

.12 

234 

fts F 

( 4 

34 

11 

5 

34 

20 


.16 

2 

fts F 

44 

.14 

3 

fts F 

44 

5 16 

14 

44 

% 

31 

4 4 

.20 

'IX 

fts F 

44 

.16 

3*4 

fts F 

44 

% 

11 




44 

.22 


fts F 

<4 

.19 

4*4 

fts F 







44 

.25 

'AX 

fts F 

44 

.25 

6 

fts F 








Tend service pipes for single dwellings in Philadelphia are usually of fro 
V .1 inch bore ' wt 1 to ‘ 2l A ; to % inch bore, wt to 3 lbs per ft run, accordit 
to head. They rarely burst from sudden closing of stopcocks ; but sometim 
do so from the freezing of the contained water. 


* To w hich it is held by copper cleats; as at Fig y, page 4] 
Price of roofing copper in 1888, about 25 cts ; and copper nails, 35 cts per ft). 















































































BRASS AND COPPER TUBES. 


417 


Seamless drawn brass and copper tubes are made by American Tube 
Works, Boston, Mass.; Ansouia Brass ami Copper Co., Ansonia, Conn., office 19 and 
21 Cliff St., New York; Benedict & Burnham Mfg. Co., Waterbury, Conn., office 13 
Murray St., New York; Randolph & Clowes, Waterbury, Conn., and Bridgeport 
Brass Co., Bridgeport, Conn. The following sizes are kept in stock, iu 12 ft et lengths, 
jy Merchant & Co., 517 Arch St., Philadelphia. The five columns signify as follows: 

A = outside diameter of tube iu inches. 

B = thickness of side by Stubs’ (or Birmingham) gauge, (first column of table, 
•age 410). When seamless tubes are ordered to gauge number, it is understood that 
his gauge is intended unless otherwise specified. 

C = thickness of sides of tube in decimals of an inch. 

D = weight, in pounds per lineal foot, of brass tube for columus A, B aud C. 

For copper, add one-nineteenth). 

E = net price, in cents per pound, of brass tube, Philadelphia, February, 1888, 
for copper, add 3 cents per pound. 

Tubes will be furnished, bard, unless ordered annealed or soft. 


A 

B 

c 

D 

E 

A 

B 

c 

o 

E 

A 

B 

c 

I> 

j 

E 

j. 

18 

.049 

.11 

62 

if 

13 

.095 

1.68 

25 

24 

12 

.109 

3.02 

23 


18 

.049 

.15 

47 

n 

11 

.120 

2.10 

25 

24 

10 

.134 

3.68 

23 

ii 

H 

17 

.058 

.22 

41 

ii 

15 

.072 

1.40 

25 

2f 

14 

.083 

2.44 

24 

& 

17 

.058 

.25 

39 

u 

14 

.083 

1.61 

24 

2f 

12 

.109 

3.18 

23 

k 

17 

.058 

.29 

36 

12 

13 

.095 

1.82 

24 

2f 

10 

.134 

3.87 

23 

h 

17 

.058 

.34 

35 

ii 

11 

.120 

2.27 

24 

21 

14 

.083 

2.57 

24 

f 

16 

.065 

.42 

33 

1 u 

15 

.072 

1.50 

25 

22 

12 

.109 

3.37 

23 

2 

16 

.065 

.51 

32 

u 

14 

.083 

1.72 

24 

21 

10 

.134 

4.07 

23 

§ 

16 

.065 

.61 

31 

if 

13 

.095 

1.96 

24 

2f 

12 

.109 

3.50 

24 

1 

16 

.065 

.70 

30 

if 

11 

.120 

2.44 

24 

2s 

10 

.134 

4.26 

23 

H 

16 

.065 

.79 

301 

2 

14 

.083 

1.84 

24 

3 

10 

.134 

4.46 

23 

-11 

16 

.065 

.88 

27 1 

2 

13 

.095 

2.10 

24 

34 

10 

.134 

4.85 

23 

11 

14 

.083 

1.12 

26 

! 2 

10 

.134 

2.91 

23 

34 

10 

.134 

5.24 

23 

11 

11 

.120 

1.57 

26 

24 

14 

.083 

1.97 

24 

32 

10 

.134 

5.62 

23 

5 1 H 

15 

.072 

1.08 

27 

24 

13 

.095 

2.23 

24 

4 

10 

.134 

6.00 

24 

n 

14 

.083 

1.25 

26 [ 

24 

10 

.134 

3.10 

23 

44 

10 

.134 

6.39 

24 

m 

11 

.120 

1.76 

26 

21 

14 

.083 

2.08 

24 

44 

10 

.134 

6.78 

25 

n 

15 

.072 

1.19 

26 

21 

13 

.095 

2.38 

24 

44 

10 

.134 

7.17 

26 

m 

14 

.083 

1.36 

25 1 

21 

10 

.134 

3.29 

23 

5 

10 

.134 

7.56 

27 

-M 

13 

.095 

1.55 

25 

2t 

14 

.083 

2.20 

24 

54 

10 

.134 

7.94 

28 


11 

.120 

1.92 

25 § 

2f 

13 

.095 

2.51 

24 

54 

10 

.134 

8.33 

29 

-it 

15 

.072 

1.29 

261 

2f 

10 

.134 

3.49 

23 

54 

10 

.134 

8.72 

30 

if 

14 

.083 

1.48 

25 j 

24 

14 

.083 

2.33 

24 

6 

10 

.134 

9.11 

31 


Merchant & Co. supply sizes up to 7 inches outside or inside diameter, and up 
- 3 18 inches inside diameter, of other gauges as well as those given in the table; 

Iso tubes of special shapes, such as square, triangular, octagonal, etc.; and 
> ironze tubes. 

They also have in stock, in lengths of 12 feet, the following sizes of seamless 
rass and copper tubing, made of same outside diameter as standard 
izes of iron piping, so as to be used with the same fittings as the iron pipe. 

A = Nominal inside diameter of iron pipe, in inches. For actual inside diameters 
ee first table p. 405. 

B = Outside diameter of iron pipe, and of seamless tube, in inches. 

C — Inside diameter of seamless tube, iu inches. 

D = Weight per foot of brass pipe, cols. B and C. For copper, add one-nineteenth. 

E = Price in cents per pound of brass pips. For copper, add 3 cents per pound. 


A 

B 

c 

D 

E 

A 

B 

c 

D 

E 

A 

B 

c 

D 

E 

8 

if 

1 

4 

.28 

37 

2 

1 T*l) 

** 

1.15 

22 

2 

28 

2yV 

4.15 

22 

k 

if 

ii 

.43 

33 

1 

i/V 

ifV 

1.50 

22 

28 

2f 


4.50 

22 

8 

n 

if 

.58 

30 

U 

if 

m 

2.25 

22 

3 



8.00 

22 

i 

if 

£. 

8 

.80 

28 

H 

if 

m 

2.55 

22 4 

4£ 

48 

12.24 

24 
































































































418 


TIN AND ZING. 


TIN ANI> ZINC. 

The imre metal is called block, tin. When perfectly pure,(which it 

rarely is being purposely adulterated, frequently to a largo proportion, with the < 
cheaper metals lead or zinc,) its sp grav is 7.29; and its weight per cub ft is 4oo lbs. 

It is sufficiently malleable to l>c beaten into tin foil, only T oVt7 of an inch thick - 
Its tensile strength is but about 4600 lbs per sq inch; or about 7000 Ihs when made 
into wire. It melts at the moderate temperature of 442° Fah. Pure block tin is 
not used for common building purposes; but thin plates of sheet iron, covered with 
it on both sides, constitute the tinned plates, or, as they are called, the fin, used for i 
covering roofs, rain pipes, and many domestic utensils. For roofs it is laid on boards. | 

The sheets 
of tin are uni- j 
ted as shown in : 
this fig. First, sev- i 
eral sheets are 
joined together in 
the shop, end for ! 
end, as at tt; by 
being first bent j 
over, then ham- j 
mered flat,and then I 
___ soldered. These are | 
then formed into a j 
•roll to be carried I 
to the roof; a roll 

being long enongh to reach from the peak to the eaves. Different rolls being spread 
up and down the roof, are then united along their sides by simply being bent as at a 
and s, by a tool for that purpose. The roofers call the bending at s a double groove, 
or double, lock ; and the more simple ones at t, a single groove . or lock. 

To hold flic tin securely to the sheeting boards, pieces of the tin 3 or 4 ins long, i 
by 2 ins wide, called cleats, are nailed to the boards at about every IS ins along the i 
joints of the rolls that are to be united,and are bent over with the double groove s. 
This will be understood from y, where the middle piece is the cleat, before being 
bent over. The nails should be 4-penny slating nails, which have broader heads 
than common ones. As they are not exposed to the weather, they may be of plain iron. 

Much use is made of what is called learie*! tin, or tomes, for rooting. It is 
simply sheet-iron coated with lead, instead of the more costly metal tin. It is not« 
as durable as the tinned sheets, but is somewhat cheaper. 

The best plates, both for tinning and for ternes, are made of charcoal iron ; which, ; 
being tough, bears bending better. Coke is used for cheaper plates, but inferior as i 
regards bending. In giving orders, it is important to specify whether charcoal j 
plates or coke ones are required ;* also whether tinned plates, or ternes. 

Tinned and leaded sheets of Bessemer and other cheap steel, are now much used. ’ 
They are sold at about the juice of charcoal tin and terne plates. 

There are also in use for roofing, certain compound metals which resist tarnish 
better than either lead, tin, or zinc; but which are so fusible as to be liable to be 
melted by large burning cinders falling on the roof from a neighboring conflagration. 

A roof covered with tin or other metal should, if possible, slope not much less than 
five degrees, or about an inch to a foot; and at the eaves there should be a sudden 1 
fall into the rain-gutter, to prevent rain from backing up so as to overtop the double-; 
groove joint s, and thus cause leaks. Where coal is used for fuel, tin roofs should 
receive two coats of paint when first put up, and a coat at every 2 or 3 years after. 
Where wood only is used, this is not necessary ; and a tin roof, with a good pitch i 
will last 20 or 30 years.f F 

Two good workmen can put on, and paiut outside, from 250 to 300 sq ft of tin roof ' 
per day of 8 hours. 

Tinned iron plates are sold by the box. These boxes, unlike glass, have not cqua 
areas of contents. They may be designated or ordered either by their names oi 
sizes. Many makers, however, have their private brands in addition; and some ol 
these have a much higher reputation than others. See table of sizes, etc., p. 419. 

* l*rioes, Phila, 1888. Tinned plates, charcoal, IC —10 X 14 and 14 x 20, the staudardsize 3 (se« 
table, p 419) $5.50 to $10per boxof 112 lbs, according to grade. Coke, IC —14 x 20, $5 to $5.50 
Hooting ternes, 1C — 14 X 20, $4.50 to$7 50. IC, 20 X 28, $9 to $15 per box of 224 lbs. Win. F. Potts 
Son <fe Co, 1225 Market St, Pbila; Hall & Carpenter, 709 Market St ; Merchant & Co, 617 Arch St. 

tTlie cost, of till-roofing;, so-called, but actually terues, in Philadelphia, 1886, is about' 
or 8 cts per sq ft of roof, including terues, all labor, and one coat of paiut on each side. It is often laid oi 
old shingle roofs. Galvanized iron rain water-pipes, 3 ins diarn, about 20 cts per ft run, 
put up. 














TIN AND ZINC, 


410 


Table of Tinned and Terne Plates. 


tamfri!n^;r: B H Xe8 considerably less weight of tin plate than the 

table requires, the plates being rolled thin and plated thin, in order to enable 

*t liav TYvT* mnrn r.i .. 1 4-1... At /- . . ' 


mechanics to get pay for more material than they furnish. 
V 1 ie marks indicate the thicknesses, approximately as foil 


follows: 


Mark. 


Nam ber 
Birmingham 
wire gauge. 


Ins. 


Lbs 

per sq ft. 


Mark. 


Number 
Birmingham 
wire gauge. 


Ins. 


Lbs 

per sq ft. 


IC 
IX 
IXX 
IXXX 
IXXXX 


30 

28 

27 

26 

25 


.012 

.0U 

.016 

.018 

.020 


.48 

.56 

.64 

.72 

.80 


DC 

DX 

DXX 

DXXX 

DXXXX 


27 

25 

23 

22 

21 


.016 

.020 

.025 

.028 

.032 


.64 

.80 

1.00 

1.13 

1.29 


Size, 

inches. 

9 X 18 

i « 

10 x 10 
* < 

10 X 14 

t( 

a 

it 

a 

10 X 20 

4 ( 

11 X 11 

4. 

11 X 22 

it 

12 X 12 

44 

44 

12 X 24 

44 

44 

12% X 17 

* 4 
44 
44 
f 6 

13 X 13 

44 


Mark. 

No. of 
sheets 
in a box 

1 Weight 

1 per box, 

1 pounds. 

Size, 

inches. 

Mark. 

TO 

225 

130 

I 13 X 13 

IXX 

IX 

is 

162 

13 X 26 

IC 

IC 

(4 

80 

** 

IX 

IX 

44 

100 

4 4 

IXX 

1C 

44 

112 

11 X 11 

IC 

IX 

44 

140 

44 

IX 

IXX 

44 

161 

44 

IXX 

IXXX 

it 

182 

44 

IXXX 

IXXXX 

44 

203 

14 X 20 

IC 

IC 

44 

160 

•« 

IX 

IX 

• 4 

200 

44 

IXX 

IC 

44 

97 

4 4 

IXXX 

IX 

41 

121 

*4 

IXXXX 

1C 

112 

97 

U x 22 

IX 

IX 

4 ‘ 

121 

“ 

IXX 

IXX 

44 

139 

H X 21 

IX 

IC 

225 

112 

il 

IXX 

IX 

»4 

140 

11 X 25 

1C 

IXX 

4 4 

161 

4 4 

IX 

IC 

112 

115 

4 » 

IXX 

IX 

44 

144 

H X 26 

IXXX 

IXX 

1 i 

166 

14 X 28 

IC 

IXXX 

it 

187 

4 i 

IX 

DC 

100 

98 

44 

IXX 

DX 

44 

120 

14 X 30 

IXX 

DXX 

44 

147 

H X 31 

IX 

DXXX 

44 

168 

46 

IXX 

DXXXX 

44 

189 

15 X 15 

IX 

IC 

225 

135 

4 4 

IXX 

IX 

44 

169 

it 

IXXX 


1 No. of 
sheets 
in a box 

1 Weight 

1 per box, 
pounds. 

] Size, 

| inches. 

Mark. 

No. of 
sheets 
in a box 

Weight 

per box, 

pounds. 

225 

191 

16 X 16 

IC 

225 

205 

112 

135 

4 4 

IX 

44 

256 


169 

44 

IXX 

44 

294 


191 

16 X 19 

IX 

112 

152 

225 

156 

17 X 17 

IX 

44 

in 

4 4 

196 

4 4 

IXX 

44 

166 


225 

17 X 25 

DX 

100 

252 


251 

“ 

DXX 

44 

291 

112 

112 

18 X 18 

IX 

112 

162 

44 

110 

44 

IXX 

4 » 

180 

<4 

161 

44 

IXXXX 

44 

235 

41 

182 

20 X 20 

IX 

•« 

200 


203 

4 4 

IXX 

44 

230 

44 

151 

44 

IXXX 

4 4 

260 

44 

177 

4 4 

IXXXX 

44 

290 

44 

168 

20 X 28 

IC 

4 4 

224 

4 4 

193 

4 4 

IX 

44 

280 

4 4 

110 

4 4 

IXX 

44 

322 

44 

175 

<4 

IXXX 

44 

364 

44 

201 

44 

IXXXX 

4 4 

406 


2,-57 





4 4 

157 





(4 

196 

Terne Plates. 


225 





44 

241 

10 X 20 

IC 

112 

80 


217 

4 4 

IX 

(4 

100 


219 

14 X 20 

IC 

46 

112 

225 

225 

4 4 

IX 

44 

110 

il 

259 

20 X 28 

1C 

44 

224 

44 

326 

44 

IX 

44 

280 


Sheets of larger size may be made to special order; those of tinned iron, in England; but leaded 
ternes are made in Philada also, and elsewhere. 

A box of 225 sheets of 13% by 10. contains 214.84 sq ft, but, allowing for overlapping, it will cover 
but about 150 sq ft of roof; even without any allowance for the waste which occurs in cutting away 
portions in order to fit at angles, &c. J 

To find the area of roof covered by any sheet, first deduct 2 ins from its width, and 1 inch from its 
length. 

Zinc, in sheets, and laid in the same manner as slates, is much used in some 
parts of Europe for roofing. By exposure to the weather, it soon becomes covered bv a thin film of 
white oxide, which protects it from further injury, and renders the roor very durable'.* Corrugated 
sheet zinc is also used. See Galvanized Sheet Iron, page 403. 

Zinc sheets are usually about 3 ft by 7 or 8 ft. The gauge differs from that of iron ; thus No 13 is 
-032 of an inch thick, or 1.22 lbs per sq ft; No 14 = .035 inch, and 1.35 lbs; No 15 = .042 inch, and 
1 49 lbs; No 16 — 049 inch, and 1.62 lbs per sq ft. Any of these numbers may be used on roofs, for 
which purpose it should be very pure. 

Water kept in zinc vessels is said to become injurious to health; and 
recently an outcry has on that account arisen against galvanized-iron service-pipes in dwellings. 
Yet such have been in use for many years in New England, Philada, and elsewhere, without as yet 
any deleterious effects. This is possibly owing to the fact that service-pipes being short, the water 
is Usually all drawn through them several times a day; and hence does not remain in contact with 
the zinc or lead long enough to acquire a poisonous character. In taking possession of a house in 
which the water has remained stagnant in the service-pipes for some considerable time, such water 
should all be run to waste; otherwise sickness may ensue from its use. 


* The price of sheet zinc does not ordinarily differ much from that of sheet lead 
which in Philadelphia in 1888 is about 6 to 8 cts. per pound; or in pigs, from 4 to 6 cts. 

The price of block till, made into either pipes or sheets, about 60 cts. per pound • 
in bars, 40 to 42 cts. in 1888. Messrs. Tatliam &i Bros., 226 South Fifth St., make both. Merchant 
& Co., dealers, 517 Arch St., Philadelphia. 












































































420 


BOARD MEASURE, 


l 


BOARD MEASURE. 


Remark on following 1 tnl>lc. The table extends to 12 ins by 24 ins, but 

J,* 3 for greater sizes; thus, for example, the board measure in a piece of U) by 22, will 

' r L or 17 - 42 * 2 = ft board mcas ; or that of 19J* by 22, will bo 

, f ^ - 22 added to that of 9 by 22, or 18.79 -f- 16.50 — 35.29. A foot of board meas is equal to 

l foot square and 1 inch thick, or to 144 cub ins. Hence 1 cub ft = 12 ft board mcas. 


m 
& G 

S3 O 

S 0 


X 

Vi 

% 

1 . 

X 

X 

X 

2 . 

X 

X 

X 

8 . 

X 

X 

X 

4. 

X 

X 

X 

6. 

X 

X 

X 

6 . 

X 

X 

X 

7. 

X 

X 

X 

8 . 

X 

X 

X 

9. 

X 

X 

X 

10 . 

X 

X 

X 

11 . 

X 

X 

X 

12 . 

X 

n. 

X 

14. 

X 

15. 


X 

16. 

X 

17. 

X 

18. 

19. 

20 . 
21 . 
22 . 

23. 

24. 


Feet of Board Measure contained in one running foot of Scantlings 
of different dimensions. (Original.) 

__ 1000 ft board measure —83)^ cu b ft. 

THICKNESS IN INCHES. 


1 

1J4 

\x 

Ft. Bd.M 

Ft. Bd.M 

Ft. Bd.M. 

.0208 

.0260 

.0313 

.0417 

.0521 

.0625 

.0625 

.0781 

.0938 

.0833 

.1042 

.1250 

.1012 

.1302 

.1563 

.1250 

.1563 

.1875 

.1458 

.1823 

.2187 

.1667 

.2083 

.2500 

.1875 

.2344 

.2813 

.2083 

.2604 

.3125 

.2292 

.2865 

.3438 

.2500 

.3125 

.3750 

.2708 

.3385 

.4063 

.2917 

.3646 

.4375 

.3125 

.3906 

.4689 

.3333 

.4167 

.5000 

.3542 

.4427 

.5312 

.3750 

.4688 

.5625 

.3958 

.4948 

.5938 

.4167 

.5208 

.6250 

.4375 

.5169 

.6563 

.4583 

.5729 

.6875 

.4792 

.5990 

.7188 

.5000 

.6250 

.7500 

.5208 

.6510 

.7813 

.5417 

.6771 

.8125 

.5625 

.7031 

.8438 

.5833 

.7292 

.8750 

.6042 

.7552 

.9063 

.6250 

.7813 

.9375 

.6458 

.8073 

.9688 

.6667 

.8333 

1.000 

.6875 / 

.8594 

1.031 

.7083 

.8854 

1.063 

.7292 

.9114 

1.094 

.7500 

.9375 

1.125 

.7708 

.9635 

1.156 

.7917 

.9895 

1.188 

.8125 

1.016 

1.219 

.8333 

1.042 

1.250 

.8542 

1.068 

1.281 

.8750 

1.094 

1.313 

.8958 

1.120 

1.344 

.9167 

1.116 

1.375 

.9375 

1.172 

1.406 

.9583 

1.198 

1 438 

.9792 

1.224 

1 469 

1.000 

1.250 

1.500 

1.042 

1.302 

1.563 

1.083 

1.364 

1.625 

1.125 

1.406 

1.688 

1.167 

1.458 

1.750 

1.208 

1.510 

1.813 

1.250 

1.563 

1.875 

1.292 

1.615 

1.938 

1.333 

1.667 

2.000 

1.375 

1.719 

2.063 

1.417 

1.771 

2.125 

1.458 

1.823 

2.187 

1.500 

1.875 

2.250 

1.583 

1.979 

2.375 

1.667 

2.083 

2.500 

1.750 

2.188 

2.t)25 

1.833 

2.292 

2.750 

1.917 

2.396 

2.875 

2.000 

2.500 

3.000 


IX 

I 2 

| 2X 

Ft. Bd.M. 

Ft. Bd.M. Ft. Bd.M. 

.0365 

.0417 

.0469 

.0729 

.0833 

.0938 

.1094 

.1250 

.1406 

.1458 

.1667 

.1875 

.1823 

.2083 

.2344 

.2188 

.2500 

.2S13 

.2552 

.2917 

.3281 

.2917 

.3333 

.3750 

.3281 

.3750 

.4219 

.3646 

.4167 

.4688 

.4010 

.4583 

.5156 

.4375 

.5000 

.5625 

.4739 

.5116 

.6094 

.5104 

.5833 

.6563 

.5469 

.6250 

.7031 

.5833 

.6667 

.7500 

.6198 

.7083 

.7969 

.6ofi3 

.7500 

.8438 

.6927 

.7917 

.8906 

.7292 

.8333 

.9375 

.7656 

.8750 

.9844 

.8020 

.9167 

1.031 

.&385 

.9583 

1.078 

.8750 

1.000 

1.125 

.9115 

1.042 

1.172 

.9479 

1.083 

1.219 

.9844 

1.125 

1.266 

1.021 

1.167 

1.312 

1.057 

1.208 

1.359 

1.094 

1.250 

1.106 

1.130 

1.292 

1.453 

1.167 

1.333 

1.500 

1.203 

1.375 

1.547 

1.240 

1.417 

1.594 

1.276 

1.458 

1.641 

1.313 

1.500 

1.688 

1.349 

1.542 

1.734 

1.385 

1.583 

1.781 

1.422 

1.625 

1.828 

1.458 

1.667 

1.875 

1.495 

1.708 

1.922 

1.531 

1.750 

1.969 

1.568 

1.792 

2.016 

1.604 

1.833 

2.063 

1.641 

1.875 

2.109 

1.677 

1.917 

2.156 

1.714 

1 958 

2.203 

1.750 

2.000 

2.250 

1.823 

2.083 

2.344 

1.896 

2.167 

2.438 

1.969 

2.250 

2.531 

2.042 

2 333 

2.625 

2.115 

2.417 

2.719 

2.188 

2.500 

2.813 

2.260 

2.583 

2.906 

2.333 

2.667 

3.000 

2.406 

2.750 

3.094 

2.479 

2.833 

3.188 

2.552 

2.917 

3.281 

2.625 

3.000 

3.375 

2.771 

3.167 

3.563 

2.917 

3.333 

3.750 

3.063 

3.500 

3.938 

3.208 

3.667 

4.125 

3.354 

3.833 

4.313 

3.500 

4.000 

4.500 


2J4 

2X 

3 

Ft. Bd.M 

Ft. Bd.M 

Ft. Bd.M. 

.0521 

.0573 

.0625 

.1012 

.1146 

.1250 

.1563 

.1719 

.1875 

.2083 

.2292 

.2500 

.2604 

.2865 

.3125 

•SI 25 

.3438 

.3750 

.3646 

.4010 

.4375 

.4106 

.4583 

.5000 

.4688 

.5156 

.5625 

.5208 

.5729 

.6250 

.5729 

.6302 

.6875 

.6250 

.6£75 

.7500 

.6771 

.7448 

.8125 

.7292 

.8021 

.8750 

.7813 

.8594 

.9375 

.8333 

.9167 

1.000 

.8854 

.9740 

1.063 

.9375 

1.031 

1.125 

.9896 

1.086 

1.188 

1.042 

1.146 

1.250 

1.094 

1.203 

1.313 

1.146 

1.260 

1.375 

1.198 

1.318 

1.438 

1.250 

1.375 

1.500 

1.302 

1.432 

1.563 

1.354 

1.490 

1.625 

1.406 

1.547 

1.688 

1.458 

1.604 

1.750 

1.510 

1.661 

1.813 

1.563 

1.719 

1.875 

1.615 

1.776 

1.938 

1.667 

1.833 

2.000 

1.719 

1.891 

2.063 

1.771 

1.948 

2.125 

1.823 

2.005 

2.188 

1.875 

2.062 

2.250 

1.927 

2.120 

2.313 

1.979 

2.177 

2.375 

2.031 

2.234 

2.438 

2.083 

2.292 

2.500 

2.135 

2.349 

2.563 

2.188 

2.406 

2.625 

2.240 

2.463 

2.688 

2.292 

2.521 

2.750 

2.344 

2.578 

2.813 

2.396 

2 635 

2.875 

2.448 

2.693 

2.938 

2.500 

2.750 

3.000 

2.004 

2.865 

3.125 

2.708 

2.979 

3 250 

2.813 

3.094 

3.375 

2.917 

3.208 

3.500 

3.021 

3.322 

3.625 

3.125 

3.438 

3.750 

3.229 

3.552 

3.875 

3.333 

3.667 

4.000 

3.438 

3.781 

4.125 

3.542 

3.896 

4.250 

3.646 

4.010 

4.375 

3.750 

4.125 

4.500 

3.958 

4.354 

4.750 

4.167 

4.583 

5.000 

4.375 

4 812 

5.250 

4.583 

5.042 

5.500 

4.792 

5.270 

5.750 

5.000 

5.500 

6.000 


d . 
to 

T3 o 
£ £ 


x 

X 

X 

1. 

X 

X 

X 

2 . 

X 

X 

X 

3. 

X 

X 

X 

4. 

X 

X 

X 

5. 

X 

X 

X 

6 . 

X 

X 


X 

X 

X 


X 

X 

X 


9. 


X 

X 

X 


10 . 


X 

X 

11 . 

X 

X 

X 

12 . 

X 

13. 

X 

14. 

X 

15. 

X 

16. 

X 

17. 

X 

18. 

19. 

20 . 
21 . 
22 . 

23. 

24. 





































































V 


BOARD MEASURE, 


421 


Table of Board Measure — (Continued.) 


si 

T3 O 

•rS C3 

* 


Feet of Board Measure contained in one running foot of Scantlings 
of different dimensions. (Original.) 


THICKNESS IN INCHES. 


•« a 

23 



334 

, 3X 

3X 

1 4 

*x 

*x 

4X 

, 5 

, 5X 


X 

Ft. Bd M. 

.0677 

Ft.Bd.M 

.0729 

Ft.Bd.M 

.0781 

| Ft.Bd.M 
.0833 

FtJld.M 

.0885 

Ft.Bd.M 

.0938 

Ft Bd.M 

.0990 

Ft.Bd.M 

.1042 

[ Ft.Bd.M. 
.1094 

X 

X 

.1354 

.1457 

.1562 

.1667 

.1770 

.1875 

.1979 

.2083 

.2188 

X 

X 

% 

.2031 

.2187 

.2344 

.2500 


.2813 

.2969 

.3126 

.3281 

1. 

.2708 

.2917 

.3125 

.3333 

.3542 

.3750 

.3958 

.4167 

.4375 

1. 

X 

.3385 

.3646 

.3906 

.4167 

.4427 

.4688 

.4948 

.5208 

.5469 

X 

X 

.4063 

.4375 

.4688 

.5000 

.5113 

.5625 

.5938 

.6250 

.6563 

X 

X 

.4740 

.5104 

.5469 

.5833 

.6198 

•6563 

.6927 

.7292 

.7656 

X 

2. 

.5417 

.5833 

.6250 

.6667 

.7083 

.7500 

.7917 

•8333 

.8750 

2. 

X 

.6094 

.6563 

.7031 

.7500 

.7969 

.8438 

.8906 

.9375 

.9844 

X 

X 

.6771 

.7292 

.7813 

.8333 

.8854 

.9375 

.9896 

1.042 

1.094 

X 

X 

.7448 

.8021 

.8594 

.9167 

.9740 

1.031 

1.089 

1.146 

1.203 

X 

3. 

.8125 

.8750 

.9375 

1.000 

1.062 

1.125 

1.183 

1.250 

1.313 

3. 

X 

.8802 

.9479 

1.016 

1.083 

1.151 

1.219 

1.286 

1.354 

1.422 

X 

X 

.9479 

1.021 

1.094 

1.167 

1.240 

1.313 

1.385 

1.458 

1.531 

X 

X 

1.016 

1.094 

1.172 

1.250 

1.327 

1.406 

1.484 

1.563 

1.641 

X 

4. 

1 083 

1.167 

1.250 

1.333 

1.416 

1.500 

1.583 

1.667 

1.750 

4. 

X 

1.151 

1 240 

1.328 

1.417 

1.504 

1.594 

1.682 

1.771 

1.859 

X 

X 

1.219 

1.313 

1.406 

1.500 

1.593 

1.688 

1.781 

1 875 

1.969 

X 

X 

1.286 

1.384 

1.484 

1.583 

1.681 

1.781 

1.880 

1 979 

2.078 

X 

5. 

1.354 

1.457 

1.566 

1.666 

1.770 

1.875 

1.979 

2.083 

2.188 

5. 

X 

1.422 

1.530 

1.644 

1.750 

1.858 

1.969 

2.078 

2.188 

2 297 

X 

X 

1.490 

1.603 

1.722 

1.833 

1.917 

2 063 

2.177 

2.292 

2.406 

X 

■ X 

1.557 

1676 

1.800 

1.917 

2.035 

2.156 

2.276 

2.396 

2.516 

X 

6 . 

1.625 

1.750 

1.875 

2.000 

2.125 

2 250 

2.375 

2.500 

2.625 

6. 

X 

1.693 

1.823 

1.953 

2.083 

2.214 

2.344 

2.474 

2 604 

2.734 

X 

X 

1.761) 

1.896 

2.031 

2.167 

2.302 

2.438 

2.573 

2.708 

2.843 

X 

X 

1.828 

1.969 

2.109 

2.250 

2.391 

2.531 

2.672 

2.813 

2 953 

X 

7. 

1.896 

2.042 

2.188 

2.333 

2.479 

2.625 

2.771 

2.917 

3.063 

7. 

X 

1.96 4 

2.115 

2-266 

2.416 

2.568 

2 719 

2.870 

3 021 

3.172 

X 

X 

2.031 

2.187 

2.344 

2.500 

2 656 

2 813 

2.969 

3.125 

3.281 

X 

X 

2.099 

2.260 

2.422 

2.583 

2.745 

2.906 

3.068 

3.229 

3.391 

X 

8 . 

2.167 

2 333 

2500 

2.667 

2.833 

3.000 

3.167 

3.333 

3.500 

8. 

X 

2.234 

2.406 

2.578 

2.750 

2 922 

3.094 

3. ‘26f> 

3.438 

3.609 

X 

X 

2.302 

2.479 

2 656 

2.833 

3.010 

3.188 

3.365 

3.542 

3.718 

X 

X 

2.370 

2.552 

2 734 

2.916 

3.099 

3.281 

3.464 

3.646 

3.828 

X 

9. 

2.438 

2.625 

2.813 

3.000 

3.187 

3.375 

3.563 

3.750 

3.938 

9. 


2.505 

2.698 

2.891 

3.083 

3.276 

3.469 

3.661 

3.854 

4.047 

X 

X 

2.573 

2.771 

2.969 

3.167 

3.365 

3.563 

3.760 

3.958 

4.156 

X 

X 

2.641 

2.844 

3.047 

3.250 

3.453 

3.656 

3.859 

4.063 

4.266 

X 

10. 

2.708 

2.917 

3.125 

3.333 

3.542 

3.750 

3.958 

4.167 

4.375 

10. 

34 

2.776 

2.990 

3.203 

3.416 

3.630 

3.844 

4.057 

4.271 

4.484 

X 

2.844 

3.063 

3.281 

3.500 

3.719 

3.938 

4.156 

4.375 

4.594 

X 

X 

2.911 

3.135 

3.359 

3.583 

3.807 

4.031 

4.255 

4.479 

4.703 

X 

11. 

2.979 

8.208 

3.438 

3.666 

3.896 

4.125 

4.354 

4.583 

4.813 

11. 

34 

3.047 

3.281 

3.516 

3.750 

3.984 

4.219 

4.453 

4.688 

4.922 

X 

3 * 

3.115 

3.354 

3.594 

3.833 

4.073 

4.313 

4.552 

4.792 

5.031 

X 

3.182 

3.427 

3.672 

3.916 

4.161 

4.406 

4.651 

4.896 ■ 

5.141 

X 

12. 

3.250 

3.500 

3.750 

4.000 

4.250 

4.500 

4.750 

5.000 

5.250 

12. 

X 

3.385 

3.646 

3.906 

4.167 

4.427 

4.688 

4.948 

5.208 

5.469 

X 

13. 

3.521 

3.792 

4.063 

4.333 

4.604 

4.875 

5.146 

5.417 

5.688 

13. 


3.656 

3.938 

4.219 

4.500 

4.781 

5.063 

5.344 

5.625 

5.906 

X 

14. 

3.792 

4.083 

4.375 

4.667 

4.958 

5.250 

5.542 

5.833 

6.125 

14. 

X 

3.927 

4.229 

4.531 

4.833 

5.135 

5.438 

5.740 

6.042 

6.344 

X 

15. 

4.063 

4.375 

4.688 

5.000 

5.313 

5.625 

5.938 

6.250 

6.563 

15. 

X 

4.198 

4.521 

4.844 

5.166 

5.490 

5.813 

6.135 

6.458 

6.781 

X 

16. 

4.333 

4.667 

5.000 

5.333 

5.667 

6.000 

6.333 

6.667 

7.000 

16. 

34 

4.469 

4.813 

5.156 

5.500 

5.844 

6.188 

6.531 

6.875 

7.219 

X 

17. 

4.604 

4.958 

5.313 

5.667 

6.021 

6.375 

6.729 

7.083 

7.438 

17. 

34 

4.740 

5.104 

5.469 

5.833 

6.198 

6.563 

6.927 

7.292 

7.656 

X 

18. 

4.875 

5.250 

5.625 

6.000 

6.375 

6.750 

7.125 

7.500 

7.875 

18. 

19. 

5.146 

5.542 

5.938 

6.333 

6.729 

7.125 

7.521 

7.917 

8.313 

19. 

20. 

5.417 

5.833 

6.250 

6.667 

7.083 

7.500 

7.917 

8.333 

8.750 

20. 

21. 

5.688 

6.125 

6.563 

7.000 

7.438 

7.875 

8.313 

8.750 

9.188 

21. 

22. 

5.958 

6.417 

6.875 

7.333 

7.792 

8.250 

8.708 

9.167 

9.625 

22. 

23. 

6.229 

6.708 

7.188 

7.667 

8.145 

8.625 

9.104 

9.583 

10.06 

23. 

24. 

6.500 

7.000 

7.500 

8.000 

8.500 

9.000 

9.500 

10.00 

10.50 

24. 





































































422 


BOARD MEASURE 


Table of Board Measure — (Continued.) 


.2* 

XI.S 

Feet of Board Measure contained in one running foot of Scantlings 
of different dimensions. (Original.) 

S3 




THICKNESS IN 

INCHES. 



Sfl 


5X 

5X 

6 

ex 

ex 

ex 

7 

ix 

7X 

> 

X 

X 

X 

1 . 

x 

X 

% 

2 . 

x 

x 

X 

3. 

x 

x 

H 

4. 

y 

y 

% 

5. 

X 

y 

X 

6 . 

x 

y 

% 

7. 

X 

y 

X 

8 . 

X 

y 

X 

9. 

X 

y 

X 

10 . 

X 

X 

X 

11 . 

X 

X 

X 

12 . 

X 

13. 

X 

14. 

X 

15. 

X 

16. 

X 

17. 

X 

18. 

19. 

20 . 

21 . 

22 . 

23. 

24. 

Ft. Bd. M. 

.1146 

.2292 

.3438 

.4583 

.5729 

.6875 

.8021 

.9167 

1.031 

1 .H6 

1.260 

1.375 

1.490 

1.604 

1.719 

1.833 

1.948 

2.063 

2.177 

2.292 

2.406 

2.521 

2.635 

2.750 

2.865 

2.979 

3.094 

3.208 

3.323 

3.438 

3.552 

3.667 

3.781 

3.896 

4.010 

4.125 

4.240 

4.354 

4.469 

4.583 

4.698 

4.813 

4.927 

5.0(2 

5.156 

5.271 

5.385 

5.500 

5.729 

5.958 

6.188 

6.417 

6 .6(6 

6.875 

7.104 

7.333 

7.563 

7.792 

8.021 

8.250 

8.708 

9.167 

9.625 

10.08 

10.54 

11.00 

Ft.Bd.51. 

.1198 

.2396 

.3594 

.4792 

.5990 

.7188 

.8385 

.9583 

1.078 

1.198 

1.318 

1.438 

1.557 

1.677 

1.797 

1.917 
2.036 
2.156 
2.276 
2.396 
2.516 
2.635 
2.755 
2.875 
2.995 
3.115 
3.234 
3.354 
3.(74 
3.594 
3.714 
3.833 
3.953 
4.073 
4.193 
4.313 
4.432 
4.552 
4.672 
4.792 
4.911 
5.031 
5.151 
5.271 
5.391 
5.510 
5.630 
5.750 
5.990 
6.229 
6.469 
6.708 

6.918 
7.188 
7.427 
7.667 
7.906 
8.146 
8.385 
8.625 
9.104 
9.583 

10.06 

10.54 

11.02 

11.50 

Ft.Bd.M. 

.1250 

.2500 

.3750 

.5000 

.6250 

.7500 

.8750 

1.000 

1.125 

1.250 

1.375 

1.500 

1.625 

1.750 

1.875 
2.000 

2.125 

2.250 

2.375 

2.500 

2.625 

2.750 

2.875 
3.000 

3.125 

3.250 

3.375 

3.500 

3.625 

3.750 

3.875 
4.000 

4.125 

4.250 

4.375 

4.500 

4.625 

4.750 

4.875 
5.000 

5.125 

5.250 

5.375 

5 500 

5.625 

5.750 

5.875 
6.000 

6.250 

6.500 

6.750 
7.000 

7.250 

7.500 

7.750 
8.000 

8.250 

8.500 

8.750 
9.000 

9.500 
10.00 

10.50 
11.00 

11.50 
12.00 

Ft.Bd.M. 

.1302 

.2604 

.3906 

.5208 

.6510 

.7812 

.9115 

1.042 

1.172 

1.302 

1.432 

1.562 

1.693 

1.823 

1.953 

2.083 

2.214 

2.344 

2.474 

2.604 

2.734 

2.865 

2.995 

3.125 

3.255 

8.385 

3.516 

3.646 

3.776 ' 

3.906 

4.036 

4.167 

4.297 

4.427 

4.557 

4.687 

4.818 

4.948 

5.078 

5.208 

5.339 

5.469 

5.599 

5.729 

5.859 

5 990 
6.120 
6.250 
6.510 
6.771 
7.031 
7.292 
7.552 
7.812 
8.073 
8.333 
8.594 
8.854 
9.115 
9.375 
9.896 
10.42 
10.94 
11.46 

11.98 
12.50 

Ft. Bd.M 

.1354 

.2708 

.4063 

.5417 

.6771 

.8125 

.9479 

1.083 

1.219 

1.354 

1.490 

1.625 
1.760 
1.896 
2.031 
2.167 
2.302 
2.438 
2.573 
2.708 
z844 
2.979 
3.115 
3.250 
3.385 
3.521 
3.656 
3.792 
3.927 
4.063 
4.198 
4.333 
4.469 
4.604 
4.740 
4.875 
5.010 
5.146 
5.281 
5.417 
6.552 
5.688 
5.823 
5.958 
6.094 
6.229 
6.365 
6.500 
6.771 
7.0(2 
7.313 
7.583 
7.854 
8.125 
8.396 
8.667 

8.938 

9.208 

9.479 

9.750 

10.29 

10.83 

11.38 

11.92 

12.46 

13.00 

Ft.Bd.M. 

.1406 

.2813 

.4219 

.5625 

.7031 

.8438 

.9844 

1.125 

1.266 

1.406 

1.547 

1.688 
1.828 
1.969 
2.109 
2.250 
2.391 
2.531 
2.672 
2.813 
2.953 
3.094 
3.234 
3.375 
3.516 
3.656 
3.797 
3.938 
4.078 
4.219 
4.359 
4.500 
4.641 
4.781 
4.922 
5.063 
5.203 
5.344 
5.484 
5.625 
5.766 
5.906 
6.047 
6.188 
6.328 
6.469 
6.609 
6.750 
7.031 
7.313 
7.594 
7.875 
8.156 
8.438 
8.719 
9.000 
9.281 
9.563 
9.844 
10.13 
10.69 
11.25 

11 81 
12.38 
12.94 
13.50 

Ft. Bd.M. 

.1458 

.2917 

.4375 

.5833 

.7292 

.8750 

1.020 

1.167 
1.313 

1 458 
1.604 

1.750 
1.896 
2.042 
2.188 

2.333 
2.479 

2.625 
2.771 

2.917 
3.063 
3.208 
3.354 
3.500 
3.646 
3.792 
3.938 
4.083 
4.229 
4.375 
4.521 
4.667 
4.813 
4.957 
5.103 
5.249 
5.395 
5.541 
5-687 
5.833 
5.979 
6.125 
6.271 
6.417 
6.563 
6.708 
6.854 
7.000 
7.292 
7.583 
7.875 

8.167 

8.158 

8.750 
9.042 

9.333 

9.625 

9.917 
10.21 
10.50 
11.08 
11.67 
12.25 
12.83 
13.42 
14.00 

Ft. Bd.M. 

.1510 

.3021 

.4531 

.6042 

.7552 

.9062 

1.057 
1.208 
1.359 
1.510 

1.661 
1.813 

1.964 
2.115 

2 266 
2.417 
2.568 
2.719 
2.870 
3.021 
3.172 
3.323 
3.474 
3.625 
3.776 
3.927 
4.078 
4.229 
4.380 
4.531 
4.682 
4.833 
4.984 
5.135 
5.286 
5.438 
5.589 

5 740 
5.891 
6.042 
6.193 
6.344 
6.495 
6.646 
6.797 
6.948 
7.099 
7.250 
7.552 
7.854 
8.156 
8.458 
8.760 
9.063 
9.365 
9.667 
9.969 
10.27 
10.57 
10.88 
11.48 
12.08 
12.69 
13.29 
13.90 
14.50 

1 

Ft. Bd.M. 

.1563 

.3125 

.4688 

.6250 

.7813 

.9375 

1.094 

1.250 

1.406 

1.563 

1.719 

1.875 
2.031 
2.188 

2.344 

2 500 
2.656 

2.813 
2.969 

3.125 
3.281 
3.438 
3.594 

3.750 
3.906 
4.063 
4.219 

4.375 
4.531 

4.688 
4.844 
5.000 
5.156 
5.313 
5.469 
5.625 
5.781 
5.938 
6.004 

6.250 

6.406 

6.563 

6.719 

6.876 
7.031 
7.188 

7.344 
7.500 

7.813 

8.125 

8 . 138 

8.750 
9.063 

9.375 

9.688 
10.00 
10.31 
10.63 
10.94 
11.25 
11.88 
12.50 
13.13 
13.75 
14.38 
15.00 

X 

X 

X 

1 . 

X 

X 

X * 

2 . 

X 

X 

X 

3. 

X 

X 

4* 

X 

X 

X 

5. 

X 

X 

X 

6 . 

X 

X 

X 

7. 

X 

X 

X 

j 

X 

10 . 

X 

X 

X 

11 . 

X 

X 1 
X 

12 . 

X 

13. 

X 

14. 

X 

15. 

X 

16. 

X 

17. 

X 

18. 

19. 

20 . 

21 . 

22 . 

23. ; 

24. 

































BOARD MEASURE. 


m 


Table of Board Measu re — (Continued.) 


a ■ 

-1 to 

•o ° 

E2 a 


Feet of Board Measure contained in one running foot of Scaatliuga 
of different diineusious. (Origiuai.) 


THICKNESS IN INCHES. 



IX | 

8 

8X 

8X 


Ft.Bd.M. 

Ft.Bd.M. 

FtBd.M. 

Ft.Bd.M. 

X 

X 

.1615 

.1667 

.1719 

.1771 

.3229 

.3333 

.3438 

.3542 

X 

.4844 

.5000 

.5156 

.5313 

1. 

.6456 

.W8i7 

,687o 

.74183 

X 

.8073 

.8333 

.8594 

.8854 

X 

.9688 

1.04*0 

1.031 

1.063 

X 

1.130 

1.167 

1.203 

I 240 

2. 

1.292 

1.333 

1.375 

1.417 

X 

1.453 

1.500 

1.547 

1.594 

X 

1.615 

1.667 

1.719 

1.771 

X 

1.776 

1.833 

1.891 

1.948 

3. 

1.938 

2.090 

2.063 

2.125 

x 

2.099 

2.167 

2.234 

2.302 

X 

2.260 

2.333 

2.406 

2.479 

X 

2.422 

2.500 

2.578 

2.4156 

4. 

2.583 

2.667 

2.750 

2.833 

X 

2.745 

2.833 

2.922 

3.010 

X 

2.906 

3.04)0 

3.094 

3.188 

X 

3.068 

3.167 

3.24!6 

3.365 

5. 

3.229 

3.333 

3.438 

3.512 

X 

X 

3.391 

3.500 

3.609 

3.719 

3.552 

3.667 

3.781 

3.896 

X 

3.714 

3.833 

3.953 

4.073 

6. 

3.875 

4.000 

4.125 

4.250 

X 

4.0.36 

4.167 

4.297 

4.427 

X 

4.198 

4.333 

4.469 

4.4104 

X 

4.359 

4.500 

4.4)41 

4.781 

7. 

4.521 

4.667 

4.813 

4.958 

X 

4.682 

4.833 

4.984 

5.135 

X 

4.844 

5.0041 

5.156 

5.313 

X 

5.005 

5.167 

5.328 

5.490 

8. 

5.167 

5.333 

5.500 

5.4J67 

X 

5.328 

5.500 

5.672 

5.844 

X 

5.490 

5.667 

5.844 

6.021 

X 

5.651 

5.833 

6.4)16 

6.198 

9. 

5.813 

6.4X10 

6.188 

6.375 

X 

5.974 

41.167 

6.359 

6.552 

X 

6.135 

6.333 

6.531 

6.729 

X 

6.297 

41.500 

6.703 

6.906 

10. 

6.458 

6.f><>7 

6.875 

7.083 

X 

6.620 

6.833 

7.047 

7.260 

X 

5.781 

7.000 

7.219 

7.438 

X 

..943 

7.167 

7.391 

7.615 

11. 

7.104 

7.333 

7.563 

7.792 

X 

7.266 

7.500 

7.735 

7.969 

Vi 

7.427 

7.6417 

7.906 

8.146 

X 

7.589 

7.833 

8.078 

8.323 

12. 

7.750 

8.4X10 

8.250 

8.500 

X 

8.073 

8.333 

8.594 

8.854 

13. 

8.396 

8.666 

8.938 

9.208 

u 

8.719 

9.0410 

9.281 

9.563 

14. 

9.042 

9.333 

9.625 

9.917 

X 

9.365 

9.6416 

9.969 

10.27 

15. 

9.688 

10.000 

10.31 

10.63 

X 

10.01 

10.33 

10.66 

10.98 

16. 

10.33 

10.67 

11.00 

11.33 

X 

10.66 

11.00 

11.34 

11.69 

17. 

10.98 

11.33 

11.69 

12.04 

X 

11.30 

11.66 

12.03 

12.40 

18. 

11.63 

12.00 

12.38 

12.75 

19. 

12.27 

12.67 

13.06 

13.46 

20. 

12.92 

13.33 

13.75 

14.17 

21. 

2*2. 

13.56 

14.21 

14.4)0 

14.641 

14.44 

15.13 

14.88 

15.58 

2.4. 

14.85 

15.33 

15.81 

16.29 

24. 

15.50 

16.4)0 

16.50 

17.00 


8X 

9 

9X 

9X 

9% 

FCBd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft-Bd.M. 

Ft.Bd.M. 

.1823 

.1875 

.1927 

.1979 

.2031 

.3646 

.3750 

.3854 

.3958 

.4063 

.54457 

.54)25 

.5781 

.5938 

.6094 

.7292 

.7500 

.7708 

.7917 

.8125 

.9115 

.9375 

.9635 

.9896 

1.016 

1.094 

1.125 

1.156 

1.188 

1.219 

1.276 

1.313 

1.349 

1.385 

1.422 

1.458 

1.500 

1.542 

1.583 

1.4725 

1.641 

1.688 

1.734 

1.781 

1.828 

1.822 

1.875 

1.927 

1.979 

2.031 

2.005 

2.063 

2.120 

2.177 

2.234 

2.188 

2.250 

2.313 

2.375 

2.438 

2.370 

2.438 

2.505 

2.573 

2.641 

2.552 

2.625 

2.698 

2.771 

2.844 

2.734 

2.813 

2.891 

2.969 

3.047 

2.917 

3.004) 

3.083 

3.167 

3.250 

3.099 

3.188 

3.2^6 

3.365 

3.453 

3.281 

3.375 

3.469 

3.563 

3.656 

3.44)4 

3.563 

3.661 

3.760 

3.859 

3.646 

3.750 

3.854 

3.958 

4.063 

3.828 

3.938 

4.047 

4.156 

4.266 

4.010 

4.125 

4.240 

4.354 

■4.469 

4.193 

4.313 

4.432 

4.552 

4.672 

4.375 

4.500 

4.625 

4.750 

4.875 

4.557 

4.688 

4.818 

4.948 

5.078 

4.740 

4.875 

5.010 

5.146 

5.281 

4.922 

5.(Hr, 

5.203 

5.344 

5.484 

5.104 

5.250 

5.360 

5.542 

5.088 

5.286 

5.438 

5.590 

5.740 

5.891 

5.44)9 

5.625 

5.782 

5.938 

6.094 

5.4>51 

5.813 

5.975 

6.135 

6-297 

5.833 

6.000 

6.167 

6.333 

6.500 

6.016 

6.188 

6.359 

6.531 

6.703 

6.198 

6.375 

6.552 

6.729 

6.906 

6.380 

6.563 

6.745 

6.927 

7.109 

6.54i3 

6.750 

6.938 

7.125 

7.313 

6.745 

6.938 

7.130 

7.323 

7.516 

6.927 

7.125 

7.323 

7.521 

7.719 

7.109 

7.313 

7.516 

7.719 

7.922 

7.292 

7.500 

7.708 

7.917 

8.125 

7.474 

7.688 

7.901 

8.115 

8.328 

7.656 

7.875 

8.094 

8.3 J 3 

8.531 

7.839 

8.063 

8.286 

8.510 

8.734 

8.021 

8.250 

8.479 

8.708 

8.938 

8 203 

8.4:18 

8.672 

8.90*; 

9.141 

8.386 

8.625 

8.865 

9.104 

9.344 

8 568 

8.813 

9.057 

9.302 

9.547 

8.750 

9.000 

9.250 

9.500 

9.750 

9.115 

9.375 

9.635 

9.896 

10.16 

9.479 

9.750 

10.02 

10.29 

10.56 

9.844 

10.13 

10.41 

10.69 

10.97 

10.21 

10.50 

10.79 

11.08 

11.38 

10.57 

10.88 

11.18 

11.48 

11.78 

10.94 

11.25 

11.56 

11.88 

12.19 

11.30 

11.63 

11.95 

12.27 

12.59 

11.67 

12.00 

12.33 

12.67 

13.00 

12.03 

12.38 

12.72 

13.06 

13.41 

12.40 

12.75 

13.10 

13.46 

13.81 

12.76 

13.13 

13.49 

13.85 

14. Tl 

13.13 

13.50 

13.88 

14.25 

i4.y 

13.85 

14.25 

14.65 

15.04 

15.44 

14.58 

15.00 

15.42 

15.83 

16.25 

15.31 

15.75 

16.19 

16.63 

17.06 

16.04 

16.50 

16.96 

17.42 

17.88 

16.77 

17.25 

17.73 

18.21 

18.419 

17.50 

18.00 

18.50 

19.00 

19.50 


a 

*5® 
T3 O 


ME 

X 

H 

]. 

X 

X 

X 

2 . 

X 

X 

X 

3. 

X 

X 

% 

4. 

X 

X 

X 

5. 

X 

X 

X 

«. 

X 

X 

% 

7. 

X 

X 

X 

8 . 

X 

X 

X 

9. 

X 

X 

X 

10 . 

X 

X 

X 

11 . 

X 
X 
X 
1 % 

X 

13. 

X 

14. 

X 

15. 

X 

16. 

n 

17. * 

X 

18 . 

19. 

20 . 
21 . 
22 . 

23. 

24. 



























































424 


BOARD MEASURE, 


Table of Board Measnre— (Continued.) 


1 

j Width in 
Inches. 

Feet of l>oard Measure contained in one running foot of Scantlings 
of diHereut dimensions. (Original.) 

a 

43 S 

10 

J0X 

TI 

| H>X 

IICKNESS II 
10% | 11 

7 INCHES. 

| >1% 1 nx 

11% | 12 

•c O 

ii 3 

X 

X 

X 

1. 

x 

x 

X 

2. 

x 

X 

x 

3. 

X 

X 

X 

4. 

X 

X 

X 

5. 

x 

X 

X 

s. 

X 

X 

X 

7. 

X 

X 

X 

a. 

X 

X 

X 

9. 

X 

X 

X 

30. 

X 

X 

»* 

X 

13. 

X 

u. 

X 

15. 

x ; 

16 

X 

IT. 

X 

TK 

19. 

20. 

21. 

22. 

23. 

24. 

Ft. Bd.ar. 

.208:1 

.4167 

-625C 

.8333 

1.012 

1.250 
1.158 

user 

1.875 
2.063 
2.202 
2.600 

2.708 
2.017 

3.125 
&33S 
3.542 

3.750 
3.058 

4.167 

4.375 
4.585 
4.702 
5.000 
5.206 
5.417 
6J>25 
5.833 
3.012 

6.250 
S.456 
6.667 

6.875 
7.063 
7.292 
7.500 

7.708 
7.017 

8.125 
aaxi 
8542 

8.750 
8958 

9.167 

9.375 
9.583 
9.792 

10.00 

10.42 

10.83 
11.25 

11.67 
12.08 

12.50 
12.92 

13.33 
13.75 

14.17 
14.58 
15.00 

15.83 

16.67 

17.50 

18.33 

19.17 
20.00 

Ft. Bd.M 

.2135 

.4271 

.6406 

.8542 

1.068 

1.821 

1.495 

1.708 

1.922 

2.135 

2.349 

2.563 

2.776 

2.000 

3.208 

3.417 

3.630 

3.844 

4.057 

4.271 

4.484 

4.698 

4.911 

5.125 

5.339 

5.552 

5.766 

5.079 

6.193 

6.406 

6 620 
6.83:3 
f.OI7 
7.260 
TAU 
7.S88 
7.901 
8.115 
8.323 
a542 
8.755 
8.069 
9.182 
9.396 
9.609 
9.823 
10.01 
10.25 
10.68 
11.10 
11.53 
11.06 
12.39 
12.81 

13 24 
13.67 
14.09 

14 52 
14.95 
15.38 
16.23 
17.08 

IT.04 
1879 
10.65 
20.50 

Ft. M.M. 

.2188 

.4375 

.6563 

.8750 

1.094 

1.313 
1.531 

1.750 

1.969 
2.188 

2.406 

2.625 

2.844 
3.06:5 
3.281 
3.500 
3.719 
3.038 
4.156 
4.375 
4.594 
4.813 
5.031 
5.250 
5.469 
5.6S8 
5.906 
6.125 
6-3 44 
6.563 
6.781 
7.000 
7.210 
7.438 
7.656 
7.875 
8.094 

8.313 
8.53L 

5.750 

8.969 
9.188 

9.406 

9.625 

9.844 
10.06 
10.2& 

10.50 
10.94 

11.38 
11.81 

12.25 
12.69 

13.13 
13.56 
14.00 
14.44 
14.88 
15.31 
15.75 
16.63 

17.50 

18.38 

19.25 

20.13 
21.00 

Ft. Bd.M 

.224C 
.4479 
.6719 
.8958 
1.120 
1.344 
1.568 
1.792 
2.016 
2.2 k) 
2.464 
2.688 
2.911 
3.135 
3.359 
3.583 
3.807 
4.031 
4.255 
4.479 
4.703 
4.927 
5.151 
5.375 
5.599 
5.823 
6.047 
6.2T1 
6.495 
6.719 
6.943 
7.167 
7.391 
7.615 
7.839 
8.0S3 
3.286 
8.510 
8.734 
8.958 
9.182 
9.406 
9.630 
9.854 
10 08 
10.30 

10.53 
10.75 
11.20 
11.65 
12.09 

12.54 
12.99 
13.44 
13.89 
14.33 
14.78 
15.23 
15.77 
16.13 
17.02 
17.92 
18.81 
19.71 
20.60 
21.50 

Ft BdM 

.2292 
.4583 
.6875 
.9167 
, 1.146 
1.375 
1.604 

1.813 
2.063 
2.292 
2.521 
2.750 
2.979 
3.208 
3.438 
3.667 
3.896 
4.125 
4.354 
4.583 

4.813 
5.042 
5.271 
5.500 
5.729 
5.958 
6.188 
6.417 
6.616 
6.875 
7.104 
7.333 
7.5S1 
7.792 
8.021 
8.250 
8.479 
a 709 
8.939 
9.167 
9.396 
9.625 
9-854 

10.08 

10.31 

10-54 

10.77 

11.00 

11.46 

11.92 

12.38 

12.83 

13.29 
13.75 
14.21 
14.67 
15.13 
15.58 
16.04 
16.50 
17.42 
18.33 
19.25 
20.17 
21.08 
22.00 

Ft Bd M 

.2344 

.4688 

.7031 

.9375 

1.172 

1.406 

1.641 

1.875 

2.109 

2.344 

2.578 

2.813 

3.047 

3.281 

3.516 

3.750 

3.984 

4.219 

4.453 

4.688 

4.922 

5.156 

5.:$9i 

5.625 

5.859 

6.094 

6.328 

6.563 

6.797 

7.031 

7.206 

7 500 

7 734 
7.969 
8.203 
8.438 
8.672 

8.90(5 
9.141 
9.375 
9.609 
9.844 
10.08 

10 31 

10.55 
10.78 
11.02 
11.25 
11.72 
12.19 
12.66 
13.13 
13.59 
14.06 
14.53 
15.00 
15.47 
15.94 
16.41 
16.88 
17.81 
18.75 
19.69 
20.63 

21.56 
22.50 

Ft Bd M 

.2396 
.4792 
.7188 
.9583 
1.198 
1.438 
1.677 
1.917 
2.156 
2.396 
2.635 
2.875 
3.115 
3.354 
3.594 
3.833 
4.073 
4.313 
4.552 
4.791 
5.031 
5.270 
5.510 
5.750 
5.990 
6.229 
6.469 
6.70S 
6.948 
7.188 
7.427 
7.667 
7.906 
8.146 
8.385 

a62> 

9.104 
9.344 
9.583 
9.823 
1C. 96 
10.30 
10.54 
10.78 
11.03 
11.26 
11.50 
11.98 
12.46 
12.94 
13.42 
73.90 
14.38 
14.85 
15.33 
15.81 
16.29 
16.77 
17.25 
18.21 
19.17 
20.13 
21.08 
22.04 
23.00 

Ft Bd.M 

.2448 

.4894 

.7344 

.9792 

1.224 

1.469 

1.714 

1.958 

2.203 

2.448 

2.683 

2.938 

3.182 

3.427 

3-672 

3.917 

4.161 

4.406 

4.651 

4.896 

5.141 

5.385 

5 630 
5.875 
6.120 
6.365 
6.609 
6.854 
7.099 
7.344 
7.589 
7.833 
8.078 
8-323 
8.568 
8.813 
9.057 
9.302 
9.54T 

9 792 
10.01 
10.28 

10.53 
10.77 
11.02 
11.26 

11.51 
11.75 
12.24 
12.73 
13.22 
13.71 
14.20 
14.69 
15.18 
15.67 
16.16 

16.65 
17.14 

17.65 
18.60 
19.58 
20.56 

21.54 

22.52 
23.50 

Ft. Bd M 

.2500 

.5000 

.7500 

1.000 

1.250 

1.500 

1.750 
2.000 

2.250 

2.500 

2.750 
3.000 

3.250 

3.500 

3.750 
4.000 

4.250 

4.500 

4.750 
5.000 

5.250 

5.500 

5.750 
6.000 

6.250 

6.500 

6.750 
7.000 

7.250 

7.500 

7.750 

8 000 

8.250 

8.500 

8.750 
9.000 

9.250 

9.500 

9.750 
10.00 

10.25 

10.50 

10.75 
11.00 

11.25 

11.50 

11.75 
12.00 

12.50 
13.00 

13.50 
14.00 
14-50 
15.00 

15.50 
16.00 

16.50 
17.00 

17.50 
18.00 
19.00 
20.00 
21.00 
22.00 
23.00 
24.00 

X 

X 

X 

1. 

X 

X 

X 

2. 

X 

X 

X 

3. 

X 

X 

X 

4. 

X 

X 

X 

5. 

X 

X 

X 

6. 

X 

X 

X 

7. 

X 

X 

X 

8. 

X 

X 

X 

9. 

X 

10* 

X 

X 

X 

11. 

X 

X 

X 

12. 

X 

13. 

X 

14. 

X 

15. 

X 

16. 

X 

17. 

X 

18. 

19. 

20. 

21. 

22 

23. 

24. 














































TIMBER. 


425 


PRESERVATION OF TIMBER. 


Art. 1. (a) Tliedoeayof timber is caused by the fermentation of its 
sap. If dry air circulates freely about the sides and ends of the sticks, tlie sap 
evaporates. If air is excluded, as when timber is kept constantly and entirely 
immersed in salt or fresh water, the sap cannot ferment. In either case timber 
may resist decay for centuries. Sap, contined in timber with air, ferments, pro¬ 
ducing dry rot; as where beams are enclosed air-tight in brickwork etc; and 
where green timber is painted or varnished, or treated with creosote etc. The 
sap then not only prevents the thorough penetration of the oil etc, but may 
cause the greater part of the wood to rot although its firm outer shell gives it a 
deceptive appearance of strength. (t>) Sap should therefore be first removed by 
Mpsisoiiiiis is, either by drying the wood in air at natural or higher tem¬ 
peratures, or by first steaming the wood under pres so as to vaporize the sap, 
land then removing the latter by means of a vacuum. Thorough seasoning ot 
large timbers in dry air at ordinary temperatures may require years; and too 
ramd kiln-drying cracks and weakens the wood. But it is questionable whether 
steaming and" vacuum remove sap as thoroughly as do the slower dry processes, 
(c) Alternate exposure to water and air is very destructive. It causes wet rot. 

Sea-worms. The limnorici terebrans works from near high-water 
mark 10 a little below the surface of mud bottom ; the teredo navalis within some¬ 
what less limits. The teredo is said to be rendered less active by the presence 
of sewage in water. 

I Art. S. (a) The best timber-preserving processes are practically useless 
luuless thoroughly well done. If the gain in durability will not war¬ 
rant the expenditure of time and money reqd for this, it is more economical to 
lose the wood in its natural state, (b) TBae woods best adapted to 
treatment are those of an open or porous texture, as hemlock etc. I hey ab- 
Isorb the oil etc better than the denser woods; and their cheapness renders the 
use of the treatment more economical, (c) Most of the processes in common 
use seem to render wood less combustible, (d) Alter treatment by any piocess, 
the wood should be well dried before using. 


Art. 4. (a) Creosote oil, or dead oil, is the best known preservative. 
Against sea-worms it is effective for at least 25 years, and is the only known pro¬ 
tection. (b) As temporary expedients, piles are sometimes covered with sheet 
metal or with broad-headed nails driven close together. These rust or wear 
away in a few years. Oak piles, cut in January, and driven with the bark on, 
have resisted the teredo for 4 or 5 years; and cypress piles, well charred, for 9 
vears. (c) For ordinary exposures on land, 8 to 10 lbs of creosote oil. per cub 
ft are reqd = say 670 to 830 lbs per 1000 ft board measure = 30 to 40 lbs per cross 
tie of 4 cub ft. For protection agninst sea-worms 10 to 12 lbs per cub ft suffice in 
climates like those of Great Britain and the Northern U. S.; but in warmer 
waters where the teredo is very active, from 14 to 20 lbs per cub ft are used. 
Lar^e timbers mav not require saturation throughout, and thus may take less 
per'cubft. But se'e (i) and end of Art. 1 (a), (d) Creosote oil weighs about 
8.8 lbs per U. S. gallon. Its cost (1888) is about 1 ct per lb: that of the process, 
applied to pine and similar woods, including oil, from 15 cts per cub ft of tim¬ 
ber for 10 lbs of oil per cub ft, to 30 cts per cub ft for 18 lbs per cub ft. Special 
prices for hard wood. See (j). It is cheaper in England, (e) The sticks should 
be reduced to their intended final dimensions and framed (if framing is reqd) 
before treatment: especially if for exposure to teredo, which is sure to attack 
anv spots which (as by subsequent cutting) are left unprotected, (f) Creosoted 
ties have remained sound after 22 years’ exposure. The creosote protects the 
spikes from rusting. (*?) Spruce, owing to its irregular density, is unsuitable 
lor creosoting. (h) Creosote renders wood stiller and slightly more brittle. In 
hot weather it exudes to some extent and discolors the wood. Its smell excludes 
it from dwellings, (i) It does not wash out from the wood, but often fails to 
penetrate the heart-wood. Then, if any sap remains, decay begins at the cen¬ 
ter See end of Art 1 (a). Burnettizing the cen of the stick (see Art 7) and us¬ 
ing a coating of creosote outside, has long been suggested as the best, possible 
method This is cheaper than thorough creosoting. For cost, etc, oi a “ ztne- 
creosote” process, address J. P. Card, Chicago Ill. ( j) Freosoting^is done 
l,v the Eppinger & Russell Creosoting Works, office 160 Water St, New \ ork. 
Their arrangements are such that the process can, if desired be confined to a 
portion of the length of the stick. For their prices see (d). 1 he Carolina Oil A 

Creosote Co, Wilmington N. C. treat timber with creosote oil obtained from 
wood. 





425 a 


TIMBER. 


Art. 5. (a) Mineral solutions are inferior to creosote, even on land; 
and useless in running water or against sea-worms; but they approximately 
double the life of inferior timber under ordinary land exposures; and their 
cheapness permits their use where that of creosote is too expensive. (I>) They 
render wood harder; and brittle if the solution is too strong. They are liable 
to be washed out by rain etc. Hence the outer wood decays first. See Art 4 (i)‘ 
Art 8 (1>) (c) (d). (c) A committee of the American Soe of Civ Engrs,* after col¬ 
lating a large number of experiments, recommended Kurnettizing (Art 7) 
for (lamp exposure, as that of eross ties, damp floors etc; and liyaniz- 
iiiK- (An lor comparatively dry situations with exposure to air 
and sun-light, as in bridge timbers, for which it is better suited than Burnettiz- 
ing because it seems to weaken wood less. In such exposures it preserves wood 
for 20 to 80 years. 

Art, 6. (a) Kyanizing consists in steeping the wood in a warm solu¬ 
tion of 1 lb of l>i-clilori<(eot mercury (corrosive sublimate) in 100 

lbs of water, (b) It is usual to allow the wood to soak a day for each inch of the 
thickness, or least dimension, of the piece, and one dav in addition, whatever 
the size, (c) With the sublimate at 70 cts per lb (1888) and 4 to 5 lbs per 1000 ft 
bm, it costs about $7 per 1000 ft bm, or 8J cts per cub ft, or 84 cts per tie of 4 
cub ft. (d) Gen’l Cram found the process very unhealthy, “ salivating all 
the men”; but Mr. J. B. Francis, at Lowell, and Mr. H. Bissell of the Eastern 
R. It. ot Mass, had little or no trouble in this respect. The sublimate, however 
which is very poisonous, is apt to effloresce, and the use of the timber is thus 
rendered dangerous, (e) The wood decays sooner under than above ground- 
but spruce ties, kyanized in 1840, were perfectly sound in 1855. The sublimate 
readily washes out, and the process is therefore unsuitable for damp situations. 
It is carried on by the Proprietors of the Locks and Canals on Merrimac 
River, at Lowell, Mass. 

Art. 7. (a) Ituriicttizing consists in immersing the wood for several 
hours in a solution of 2 lbs chloride of zinc in 100 lbs of water, under a 
pres ot troiu 100 to 800 lbs per sq inch, (b) It seems to render wood more brit¬ 
tle than kyanizing, and is therefore less adapted for timbers bearing tensile or 
transverse strain. Spikes rust away rapidly in Burnettized ties, (c) It costs 
(1888) about $5 per 1000 ft bm = 6 cts per cub ft = 20 cts per tie of 4 cub ft. 

Art. 8. Other preventives, (a) Steeping in a solution of sulphate 

ot eopper (bine vitriol) has been extensively used, but does not seem to 
have been permanently successful. The blue vitriol washes out readily. (I>) 
In the 1 hi I many process, as practised by the Wisconsin Wood Preserv- 1 
ing ( o., Milwaukee, VV is., the timber is tirst steamed. The steam and air are 
then exhausted by an air-pump, after which are injected tirst a solution of sul¬ 
phate ol copper (blue vitriol) or ot sulphate ot zinc (white vitriol) and then one 
of chloride ot barium, both under pressure. It is claimed that this tills the pores 
with insoluble sulphate of baryta. Railroad ties require about 12 hours. Cost, 
S™. 1 .. 0 ceuts tie > or frotu to $5 per 1000 feet, board measure, (c) The 
Wcllhouse process^ as employed by the Chicago Tie Preserving Co. 
injects first a solution of chloride of zinc with glue, and then one of tannin 
(both under pres), in order to diminish the subsequent washing out of the 
chloride. The process costs (1888) about $7 per 1000 ft bm = say 8 cts per cub 
ft. It is not recommended for sub-aqueous use. (<1) The “Gypsum pro¬ 
cess” of the American Wood Preserving Co. of St Louis Mo uses gvpsum and 
chloride of zinc in order to retain the latter more perfectly. The wood is then 
kiln-dried, (e) Fence-posts etc seem to be preserved to some extent by having 
only their lower ends dipped in tar well boiled to remove the ammonia which 
last is destructive to wood. The upper end must, be left uutarred to let the sap 
evaporate, (f) Attempts at wood preservation by means of vapor of creo¬ 
sote etc have proved failures, (g) While wood is thoroughly saturated with 
petroleum it does not decay. But unless the supply is kept up the oil 
evaporates and leaves the wood unprotected, (h) Cottonwood ties laid upon a 
soil containing about 2 per cent carbonate of lime, 1 per cent salt and A per 
cent, each of potash and oxide of iron, on the Union Pacific R. R in 1868 were 
found in 1882 “as sound and a good deal harder than when first laid,”although 
sue b ties in other soils lasted but from 2 to 5 years, (i) The use of solutions of 
lime and of salt; and charring the surface; are sometimes found useful 
in damp situations. 


* See Transactions, July, Aug and Sept 18S5, to which we are indebted for many of the above sue. 

I SLlOilM, ® 









TIMBER. 


425 b 


Price of lumber, Philadelphia 1888; Spruce joists, $20 to $24 per 1000 feet 
board measure. Hemlock joists, $13 to $16. Yellow pine floor boards, $20 to $35. 
White pine boards, $18 to $50, according to quality, degree of seasoning, &c. Sawed 
white pine timbers, $28 to $35. Heart yt llow pine, $20 to $35. Hemlock, $16 to $20. 
Gillingham, Garrison & Co., 943 Richmond Street. 

Beards of oak or pine, nailed together by from 4 to 16 tenpenny 
common cut nails, and then pulled apart in a direction lengthwise of the boards, 
and across the nails, tending to break the latter in two by a shearing action, required 
about 300 to 400 lbs. per nail to separate them, as the average of many trials.. 


TABLE OF CUT NAILS. 

Base price, Philadelphia, 1888, about $2.10 per keg containing 100 lbs,* 
*< Extras ” adopted by Atlantic States Nail Association, February 9th, 1888. 


Name. 

Length. 

Inches. 

N umber 
of Nails 
per lb. 

Extra over 
base price, 
cts. per keg. 

Name. 

Length. 

Inches. 

Number 
of Nails 
per lb. 

Extra over 
base price, 
cts. per keg. 

“Common ” Nails. 

2 ppnny 

3 *• fine 

3 “ 

4 “ 

5 “ 

6 “ 

7 “ 

8 “ 

h 

m 

2 

2K 

2K 

716 

626 

440 

300 

210 

163 

123 

93 

225 

175 

150 

75 

75 

50 

50 

25 

10 penny 
12 “ 

20 “ 

30 “ 

40 « 

50 “ 

60 “ 

3 

3% 

4 

5 

6 

66 

50 

32 

19 

16 

13 

10 

0 

0 

0 

0 

25 

25 

25 

Finishing Nails. 

4 penny 

5 “ 

6 “ 

8 “ 

$ 

2 

2 Xt 

470 

330 

196 

116 

175 

175 

150 

125 

10 penny 
12 “ 

20 “ 

3 

3^4 

4 

84 

65 

50 

100 

100 

100 

Slating Nails. 

3 penny 

4 “ 

1 1 4 
VA 

280 

200 

200 

125 

5 penny 

6 “ 

1 % 

2 

160 

128 

125 

100 

Fence Nalls. 

Extras same as for corresponding sizes of common nails. 

2 inch, 80 per ft). 

ol / 4 C.R “ 

2\4 inch, GO per ft). 3 inch, 40 per lb. 

234 “ 48 “ 

UVA. ^- 



Cut Spikes, 25 cts. per keg extra. 

Similar, in shape, to the common noils (above) but some what heavier. 



6i nc h> 6 P er 

7 “5 “ 

8 “ 3 ^ “ 


3 inch, 29 per lb. 

2\4 “ 21 “ 

4 “ 15 

4V£ “ 13 “ _ 

Tile at.es a».l 

ihove are machine JJ*. ’ , |,y* hlacksiniih, or by machinery, from 

Si.rLnTrXft. Naif, “cut S Boomer and similar .tee.s .cost about 

10 cents per keg more than the abo ve. - --- 

"^MorritTwiieTlerTtCm7Pott8towiTlroirC<h)7uit)77ind Market Sts., Philadeiphia 

manufacturers. 































































































426 


PLASTERING. 


PLASTERING. 


The plastering of the inside walls of buildings, whether done on laths bricks or 
stone, generally consists of three separate coats ot mortar. The first of these is called 
by workmen the rough or scratch coat; and consists of about 1 measure of quicklime 
to 4 ot sand ; (which latter need not be of the purest kind;) and K4 measure of bul¬ 
lock or horse hair; the last of which is for making the mortar more cohesive, and 
less hable to sj.ht oft m spots This coat is about % to ^ inch thick; is put on 
roughly , and should be pressed by the trowel with sufficient force to enter perfectly 
between and behind the laths; which for facilitating this should not he nailed 
nearei together than 14 an inch. In rude buildings, or in cellars, &c, this is often 
the only coat used. \\ hen this first coat has been left for one or more days accord- 
ng o the dryness of the air, to dry slightly, it is roughly scored, or scratched, (hence 
ni„!i m ’ W1 , t , h a p0I " ted stick, or a lath, nearly through its thickness, by lines run- 
,i l tf mi y across each other, and about 2 to 4 ins apart. This gives a better 
hold to the second coat, which might otherwise peel off. If the first coat has be¬ 
come too dry it is well also to dampen it slightly as the second one is put on. 

comZ 7/n/r' 1 R°f 18 >"i OU abou l t K to . % iuch thick > of the same hair mortar, or 
cnaise stuff. Before it becomes hard, it is roughed over by a liickorv broom or 
some substitute, to make the third coat adhere to it better. ’ 

whiter t imi d nT a f t,abOUt ^ i,lch tllick » contains no hair; and forgiving it a still 
wlntei and neater appearance, more lime is used, say 1 of lime to 2 of sand - ™d 

he purest sand is used This mortar is by plasterers calledsinecVTa nam, 

( li nf l'i 0 'TZ r H hen used V ,r l ,lasteri,| g tLe outsides of buildings Or in- 

which consists Of t e , ? Ry l,e » , a,K | usnalI . v is- of hard finish, or gauge staff; 

i u 1 , n ' ea ? l ! re of ground plaster of Paris, to about 2 of quicklime 
Without sand. Hard finish works easier; hut is not as good as stucco for walls in’ 
tended to be painted m oil. The plaster of Paris is for hastening the hardening 

te^adhere Wa if : ’iJ'S.^on^mo^th wahs’ wlthouT^r^l" w Sh0U '1 b ? ,Cft VCr - v rou ^- “> ™ the plas- 
an inch, it is very apt to fall off: especially from outside .'Jm.? 1 1le D1 £ rtar ,0 the depth of nearly 

con t^of'Ji as h,' *81 igh tly^in ted' by » receives a 

thi torecUf/rn’ls perfectTv^T^vVa!, 6 maThr^ ° r ? Ut ,° f Hne; and U is not Possible for 

ssr ■**—•'Ss xrs-zz&xss i’Xasr., sr«s 

made perfectly straight, and out of wind w ith eooh -ccond one; and while soft are carefully 

&c. When they become dry, the second coaUs nu^ on fil'.^ on Z°vf the , ?‘ Un,b - liu f- straight-edge^ 
them; and is readilv brought to a nerfeetlv flu » lnfi U P thebroad horizontal spaces between 

- •». «»4iW«ss5S!i;!s,rsa s r sss i * i,h ,hu ° r **• •«*. >» 

A clay’s work at plastering. 

*veragTfrom'lO(fto & square ya°rds'adl?, of S SoltTaboui S".KS 













SLATING, 


427 


much of third, which requires more care. The amount will depend upon the number of angles, sine 
of rooms, whether on ceilings or on walls, &c, (Sic. 


Gcu Gillmore's estimate of cost of plastering?* 100 square yardo 
with 2 or with 3 coats. Common labor $1 per day. 



Three Coats. 

Two Coats. 

Ill ALL FI dlb« 

Hard finished work. 

Slipped coat finish. 

Quicklime. 

4 casks. 

$4.00 

334 casks. 

$3.33 

** for fine stuff. 

% “ 

.85 



Plaster of Paris. 

34 “ 

.70 



Laths. 

2000 

4.00 

2000. 

4.00 

Hair. 

4 bushels. 

.80 

3 bushels. 

.GO 

Common Sand. 

7 loads, t 

2.00 

6 loads. 

1.80 

White Sand. 

234 bushels. 

.25 



Nails. 

13 lbs. 

.90 

13 lbs. 

.90 

Mason's labor... 

4 days. 

7.00 

3J4 days. 

6.12 

Laborer. 

3 days. 

3.00 

2 days. 

2.00 

Cartage. 


2.00 


1.20 

Cost of 100 square yards. 


$25.50 


$19.95 





This amounts to 2534 cts per sq yd for 3 coats; and say 20 cts for 2 coats. See Art 5, P 674. 


Plastering: laths are usually of split white or yellow pine, in lengths of 
about 3 to 4 feet; and hence called 3 or 4 ft laths. They are about 1)4 ins wide, by 54 inch thick. 
They are nailed up horizontally, about J4 inch apart. The upright studs of partitions are spaced at 
such distances apart, (generally about 15 ins from center to center,) that the ends of the laths may 
be nailed to them. Laths are sold by the bundle of 1000 each. A square foot of surface requires 134 
four feet laths; or 1000 such laths will cover 606 sq ft. Sawed laths may be had to order, of any re¬ 
quired length. A carpenter can nail up the laths for from 40 to 60 sq yds of plastering in f day of 
10 hours ; depending on the number of angles in the rooms, &c. 

-- —— ---- 

SLATING. For prices, see note* p 429. 


Roofing slates are usually from 14, to % inch thick; about -A. being a common 
average. They may be nailed either to a sheeting of rough boards (c, g, in the fig) 
from % to 1]4 inch thick, (which should be, but rarely are, tongued and grooved,) 



* Averase prices of plastering: in Philada, 1888, in cts per sq yard. 

Three coats, including laths, scaffold, &c, 50 to 55 cts. Two coats, 35 to 40. Three coats on brick or 
stone (no laths reqd), 30 to 40. Outside plastering, 60; or if to imitate marble, 75. Simple plaster 
cornices, 1 to 2 cts per inch of girth, per ft run. Plaster center flowers for parlors, fc to $15yach, 
put up. The plastering of a 20-ft front, 3-story dwelling, with large 3-story back buildings, SoOO to 
.1700. Stipulate expressly to pay only for surfaces actually plastered; and thus avoid extras, even 

if you have to pay a few cts more per yard. _ ,, , , 

t A load (one-horse), both in the U. S. and in England, usually means a cub yd; 

but many dealers adopt 20 struck bushels = 25 cub ft — fully a ton. 
























































428 


SLATING. 


laid 
may 
to 


id horizontally from rafter to rafter; or sloping, from purlin to purlin, as the case 
ay he; or to stout laths ttt about 2 to 3 ins wide, and from 1 to thick, nailed 
the rafters at distances apart to suit the gauge of the slates. Two nails arc used to 
each slate; one near each upper corner. They may be either of copper, (which is the 
most durable, but most expensive,) of zinc, or of either galvanized or tinned iron. 
The last two are generally used; or in inferior work, merely plain iron ones, pre- 

VlOllsl V lioilidl in liiK<wwl stil i)U *» Tiui'tiol t\rncniriroti Vo fiwrw 4 1) ....i 1. .... 


^ uouu, hi 11 j iiiiui wum, merely juain iron ones, pre- 

viously boiled in linseed oil, as a partial preservative from rust. Rust, however, 
sometimes weakens them so much that they break; and the slates are blown off in 
high winds, to the danger of passers by. Since good slate endures for a long series 
of years, it is true economy to use nails that are equally durable. In iron roofs, the 
slates, instead of being nailed to boards, are sometimes tied directly to the iron 
purlins, by wire. A square of slating, shingling, &c, is 100 sq ft. 

In laboratories, chemical factories, &c, subject to acid fumes, it is difficult to 

provide a metal fastening that will not be eaten away. In such cases it is best to depend chiefly upon 
! iil el 'if.i ll ' 0r l tar between the slates. • This will harden before the inetal fastenings give way ; and 
will hold the slates in place, while new fastenings are being inserted. 

The least pitch considered advisable for a roof, to prevent rain or snow from being driven 
through the interstices between the slates, is about 26^°; or 1 vert to 2 hor; which corresponds to a 
rise or the span in a common double pitched roof. Rut even at steeper pitches, rain, and more 
particularly snow, wtll be forced through the roof by violent winds ; especially if laths alone be used 
to even boarding alone. To avoid this, a layer of mortar about % inch thick, may be spread over 
the touching surfaces of the slates if on laths. If on boards, the same process may be adopted • or 
the more common one of first covering the boards with a layer of what is called slat inn felt: hut 
which in reality is merely thick brown paper, soaked in tar. This is sold in long continuous rolls 
28 ins wide, and weighing from 40 to 60 lbs. A 50 !b roll will cover about 500 sq ft of roof With 
3 Tiny beadopted 3 a8alUSt the 1(11115881011 of rain and snow, a pitch as flat as 1 in 2%, or even 1 in 

The thickness of slate on a roof is double; except at the taps is, is, kc., where it is triple. The 
Jap is measured from the nail hole (under t) of the lower slate, to the lower edge or tail s of tho 
upper one ; and is usually about 3 ins. In order that the showing lower edges of the slates shall 
when laid, form regular straight lines along the roof, the nail holes are made at equal distances from 
said lower edges ; so that any irregularity of length is concealed from view at the hidden heads et 
the slates. The slater estimates the length or his slate from the nail hole to the tail: discarding the 
narrow strip between the nail hole and the head. If from this reduced length the lap be deducted 
t^ 0n .K hal K° r . the ren ' a,n,ic r wil1 be the gauge, weathering, or margin, of the slating; or, in other 
T Width of the courses of slates. The gauge in ins multiplied bv the 

th of a slate iu ms, gives the area in sq ins of finished roof covered by a single slate ; and if 14 4 
(the sq ins in a sq foot) be divided by this area, the quotient will be the number of slates required per 
sq ft of roof The upper side of a slate is called its hack ; the lower one, its bed. q l )er 

i U J' 8, «• ., e 8l ) ,n gling, must evidently he commenced at the eaves, and extended upward. Since 
the beds of the slates are not exactly parallel to the boarding, and consequently do not rest flat upon 
’.h lio l 'f lower edge w would easily be broken. To prevent this, a tilting strip (a stout wide 
Uth, with its Upper side planed a little bevelling, to suit the slope of the slates) is first nailed around 
near the eaves, for the tails of the lowest course of slates to rest on. This is shown on a larger scale 

or S nanpr°rMhh] , r t iTv ia il! t - 7 iI 1 ? S * liste " ,n * semi-metallic appearance, somewhat like that of a surface 
o paper rubbed with black lead pencil. That of a dull earthy aspect, is softer, more absorbent and 
consequently more liable to yield to atmospheric influences, rain, frost. <fec. Iron pyrites freatientlv 

W two nniuM ’ a 'f d alW ??Vdecomposes and lea ves holes, should never he admitted on a roof 

< r *"° fl ,la| >ties of slate, that which absorbs the least weight of water, when pieces of equal size are 

isst.'ts. *-« k *“ »*»-» * Arfflte 

may be cut to order, of almost any prescribed dimensions or shane T1 h»<?p in pnmmrm ncr» , „ ♦ ^ 

about 7 by 14, to 12 by 18. The first forms about 5 to fi iich courses •£? toeabout ? T 

wlS , Uhe i sfaZg n be h ° Wfar fr ° lU ^ ^ ,1Hil hoIes are P 5ei0 ^’ The farther this is! the firmed 

neath. They are also liable to break when walked on; less so when beddeTin LortaT P ' MtCr ** 

Weight of slate roofs. Slate weighs about 175 lbs per cub foot; therefore, 


* Sq ft ’ * inch thick ’ W0i8hs abo,,t 18 lbs - A- 2 * ; and x thick, 3.6 lbs. But owing to the 

1 J 4 inch boards 2.8 J> S . Lulu RSr - * ta “ «* * * « 


“ 316 


Approx 'Weight 
of one sq ft of 

Slate X inch thick on laths. Slating, in lbs. 

on 1 inch boards. . p '»? 

on laths. . 

“ “ " on 1 inch boards. . 

11 44 44 on 44 “ .l l,2 ° 

1 A . 11 80 

it slating felt is used, add X B>; or if tu e slates are bedded in X inch of mortar, add 3 9*. 




















SHINGLES, 


Tor the total weight borne by the roof trusses, that of the purlins also must be added. This 
ot vary much from the limits of 1^ to 3 lbs per sq ft in roofs or moderate span. Add tor wind auu 
now, say 20 lbs per sq ft;* aud finally add the weight of the truss itself. 

I'or the ioints between slates (or shingles, &c) anrl chimneys, 

ormer window*, Ac, a mixture of stiif white lead paint, as sold by the keg, wit .k s *" d *“[ )U S h t0 
ent it from running, is very good; especially if protected by a covering of st ^ .. li n io s which 
in. Ac. nailed to the mortar-joints of the chimneys, after being bent so as to enter• said jo nts, which 
lu hi Id be scraped out for an inch in depth, and afterward refilled. Mortar protected in the sam* 
■av, or even unprotected, is often used for the purpose; but is not equal to the paint -indl staid.M»r 
xr a few days old, (to allow refractory particles of lime to slack,) mixed with blacksmith s unde s, 
nd molasses, is much used for this purpose . and becomes very hard, aud ctlectivc. 

I'or prices of slating, see foot note* below. 




42'J 


SHINGLES, 


White cedar shingles are the best in use; and when of good quality will last 10 or 
,0 v Is ?n oir Northern States. They are usually 27 ins long; by from 0 to 7 in 

V | || 0 js 

They are usually laid in three \ ^fhs^r^ak^yeUo’w phieTahouTie' ft long; 2>$ ins 

ire four. They are nailed to sawed shingling-laths or oas oi ) 1 These are nailed to the rat't- 

vide, and 1 inch thick; placed in horizontal rows abou /a t ‘|^ m o re than 2 ft apart from center 
rs. or purlins; which, for laths of the foregoing size, i q'i, ev should not be of less size 

n center. Two nails are used to each 3 ^i^^the strongeTare the cut ones are apt to break 

ban 400 to a tb. Wrought nails being f, w;H su ffice for 100 sq ft of roof, including 

>y the warping of the shingles. Two poutids 0 r 3 b exposes C>3 3 4 sq ins; making 2'A shingles 

rrt, svsriSrR»s » «.&* »*»■« >» pemm t« » 

•%$S&£ must plainly ho begun .. tb. ««,: C '“'" g “* 

joints between the shingles, and chimneys, dornur win ows, > mU( , h Reaper, but scarcely half as 

Cypress and white pine are also much used for slung . 8 warm damp climates they 

lurabled All shingles wear quite thin in time by rain and exposure, in wa.ui * 

til decay within C to 12 years. _.__ 


PAINTING. 


_ • • , mutorisl nsiod in house-painting, is cither white lead, or oxide of 

The principal matoH, by a mill, to the consistency of a thick 

-me, gtout" condition it is sold by the manufacturers m kegs of 25, 50, and 100 
uistc. In this actual use merely requires the addition of more linseed oil, 

3 or t$nU to id*.of theTog paint!forVinniog it sufficiently to flow readily 

“£!i 5 KS! 

lues. Kach coat must be allow ei y I i «■ n r st coat • 3 vds of secoud: and 4 yds of each 

saint will, after being of 4 coats, 1* lbs; of 5 coats 1.58 

^ bS Tbe n Iea°son why th/flrst coats require so much more than the subsequent ones, is that the bare 

mrfi.ee of the wood absorbs u more thinning, dryers must be added to it; otherwise 

When, as is usual, rau or unboiled ml is a ea with drvers, from 1 to 3 days, according 

;he paint might require severs.^ weekstc^bar , h u h t ’ 0 reoe ive the next one. The dryers most 
io the weather suffice for each coa.to bwome, I , * of one heaped teaspoonful; or Japan var- 

jommonly used, are powdered b thar £ e ■>“ [ iu , er sugar of lead, or sulphate of zinc, may also 

aish. 1 table-spoonful, to 10 tts othe keg^a • * on . A f th0UB h both litharge and Japan varnish 
be used instead of litharge, and in f ! t l0 appreciably affect the whiteness of the paint, 

are dark-colored, yet the quantity is so small ^ not to appr > work finished, it produces 

If the varnish is used in excess ^ ° ^ ‘ e ^painters' billed oil be used for thinning. Mere 

sracks all over the surface. No> dryer Is; nc ^-. ( ? that intended for painters, has litharge added 

boiling will not cause oil to b *r de “„ 1 , j l0 eae h 10 gallons of raw oil. In some works 

to it previously to boiling , m the pr p „ sserted t i iat boiling renders the oil too thick for any but 
written for the use of house painters, . • • , f if h boiling be properly done, the oil 

w moreover be Nearer than wh ile raw^and 

* Price Of slate, felt, ami slating inPJ^ada, }Jnidte 

according to quality of slate, kind ofnails Ax , bute clu^veof ^ ^ <? Q the g tate . [ t commands 
ct per sq ft. The slate from Peach' BottoT?/„f leaded tin'will cost about the same as one of slate; 
] or 2 cts per sq ft more than the others. A r ahinirles (in Philada.) Felt about 2 cts per lb. 

and not much more than half as much as ‘ _ . £ fi 'j ns y ;{(»i n8 , $25; 6 ins X 24 ins, $20. 

t Price of shingles per lb 0 °, in Philada, in la - - hn , an(J shingling complete, 20 Cts pel 

JSSfffl SulSuiSMk exclusive ef CuurUs. 

















PAINTING 


430 


Will impart to the paiuted surface a more shiuing appearance. The heat should be barely^sufficient 
to produce bcdliug ; or about S00° Fab. The boiling should continue about lt$ hours, the oil being 
thoroughly stirred at short intervals, to preveui the litharge from settling at the bottom. The hie 
may then be allowed to subside; when the operation will be completed. A sednueul will theu foiiu 
at the bottom ; which must be left behiud when the oil is poured off. Although uo dryer is necessary 
with this oil, still a little litharge may be added when great expedition demands it. .Painters lately 
use this oil. on account of its tritiiug increase of cost. 


Another substance much used with the thinning oil, (except for the first coat,; is spirits of turpen¬ 
tine ; called “turp" by the workmen. The quantity of oil may be diminished, to the extent ot the 
idded turp This being more fluid than oil, causes the paint to work more pleasantly under the brush. 


added turp. This being ....... *--- - - .. .. . ._. 

It moreover diminishes the tendency of the paint to become yellow; especially in rooms kept closed 
for some time. It is also much cheaper than oil. It should not be used, or but spanugl^ . foi exposed 
outdoor work ; inasmuch as its teudeucy is to impair the hruiness of the paint; and although its 
effects are scarcely appreciable indoors, they are quite apparent when the work has to lesist tlie 
weather. As the fashious change in hoiise-paintiug, the surface is at times required to presents 
akining or glossy finish ; at other times a dead one is in vogue. The glossy one is that which tht 
paint will uaturally have, provided that no more turp than oil be used in the thiuuing. The dead 
finish is obtained by using uo oil, but turp alone, for the last coat; which in that case is called c 
flatting coat. Although turp is not properly a dryer, still, as it evaporates quickly, it facilitates tht 
hardeuing of the paint. 

In outdoor work it is usually advisable to use more dryer than inside, so that the paint may soonei 
become hard enough not to be injured by dust or rain. Otherwise less would be belter. 

When, instead of a white finish, one of some other color is required, the coloring ingredient ii 
mixed with the w hite paint to be used iu the last coat only ; although two coloring coats are some 
times found to be necessary before a satisfactory effect is produced. The coloring iugredieuts may b< 
indigo, lampblack, terra sienna, umber, ochre, chrome yellow, Venetian red, red lead, &c, &c; whict 
are ground in oil, ready for sale, by the manufacturers of the white lead aud zinc paiuts. They an 
simply well stirred into the white paiut. 

All surfaces to be painted, should first be thoroughly dry, and free from dust. If on wood, al 


plane-marks, aud other slight irregularities, should first be smoothed off by sand-paper, when tin 
neatest finish is required. Also, all heads of nails must be punched to about % inch below the sur 
face. To prevent knots from showing through the fiuished work, (as those in w hite or yellow pin' 
would do, on account of the contained turpentine,) they must first be killed, as it is termed. A usua 1 


and effective way of doing this, is by coveriug them with two coats of shellac varnish ; which, whei 
dry, should be smoothed by sand paper. Another mode, not quite so certain, is by one or two coat 
of white lead mixed with thiu glue-water, or size, as it is called. 

After these preparations, the first, or priming coat, is put on ; in which there should be no turp 
because it would siuk at ouce into the bare wood, leaving the w hite lead behiud it, iu a nearly dr 


friable condition. After this the nail holes, cracks, <Stc, must be filled with common glaziers' putty 

i) and raw linseed oil; boiled oil will not answer; the putt 


made of whiting (fine clean w'ashed chalk) 
would be friable. The putty would be apt to fall out, if put iu before priming; because the woo 
would absorb the oil, and the putty would theu shrink. After the first coat is perfectly dry, tl 
second one is put on; and for it about 1 measure of turp may be mixed with 3 measures of the thii 
liing oil. In the third, aud any subsequent coats, equal measures of turp and oil, may he used f> 
thinning, if the work is required to dry with a gloss; but if it is to finish dead, the last coat mu; 
be a flatting one; or one in which the thinning oil is entirely omitted, and turp alone substitute 

for it-* . 

I’ainters generally clean their brushes by merely pressing out most of the paint with a knife; an 
then keep them in water until farther use. If to be put away for some time, they may he thoroughl 1 
cleaned by turp; or by soap and water. To prevent a hard skin from forming on the top of the 
paint when not, used for some days, they pour on a little oil, 


The best paints for preserving iron exposed to the weatlici 

appear to be pulverized oxides of iron, sucli as yellow and red iron ochres; or brown hematite iro 
ores finely ground ; and simply mixed with linseed oil, and a dryer. White lead applied directly i |, 
the iron, requires incessant renewal: and indeed probably exerts a corrosive efiect. It may, ho\ 
ever, be applied over the more durable colors, when appearance requires it. Red lead is said to 1 
very durable, when pure. An instance is recorded of pump-rods, in a well 200 It deep, near Loudoi 
which, having first been thus painted, were in use for 45 years; and at the expiration of that tim 
their weight was found to be precisely the same as when new; thus showiug that rust had u 
affected them. See p 403. 

When the size of the exposed iron admits of it, its freedom from rust may be very much promote 
by first healing it thoroughly ; and then dipping it into, or washing it well with, hot linseed oi 
which will then penetrate into the interior of the iron. For tinned iron exposed to the weather, < 
roofs, rain pipes, <fee, Spanish brown is a very durable color. The tiu is frequently found perfect, 
bright and protected, when this color has been used, after an exposure of 40 or 50 years. M hi 
paiut washes off in a few years bv rain. 

Plastered walls should if possible be allowed to dry for at least a year, before being painted in oi 
otherwise the paint will be liable to blister. They may, if preferred, be frescoed (w T ater-coioi 
mixed with size) to the desired tint during the interval. 

The painting of unseasoned wood hastens its decay. If the surface to be painted is greasy, t 
grease must first be removed by water iu which is dissolved some lime. 

Washes for outside work. Downing, in liis work on country houst 

recommends the following: For wood-work ; in a tight bushel, slack half a bushel of fresh lime, 
pouring over it boiling water sufficient to cover it 4 or 5 ins deep; stirring it until slacked. Ad< 
lbs of sulphate of zinc (white vitriol) dissolved in water. Add water enough to bring all to the cc 
sistence of thick whitewash. Apply with a whitewash brush. This wash is white; but it may- 
colored by adding powdered ochre, Indian rod, umber, &c. If lampblack is added to water-colors, i 


* Average cost of Painting 1 in Philada, 1888, including scaffold, &c, ] | 
square yard. Four coats iu plain colors, 30 cts; 3 coats, 25. Graining in imitation of oak, wain 
&c, 50.' White lead ground in oil, in kegs, 8 cts per fl>. The cost of painting and glazing a 20-ft fre 
3-story dwelling, with large 3-story back buildings, $300 to $100. A church of 00 by 80 ft, with bw 
nient story, and galleries, $900 to $1000. Avoid extras; or stipulate for them iu advance. 











GLASS, AND GLAZING. 431 


in°a few weeksdissolved ia alcohol. The sulphate of zinc causes the wash to become hard 


For brick, masonry, or rougrh-cast. Slack }A a bushel of lime as 

before; then fill the barrel % full of water, and add a bushel of hydraulic cement. Add 3 fts of sul. 
v pnate of zinc, previously dissolved in water. The whole should be of the thickness of paint; and 
X °® Wl th a whitewash brush. The wash is improved by stirring in a peck of white sand, 

rf ust before using it. It may be colored, if desired, like the preceding. 

He also gives the following cheap oil-paiut for outside work on wood, brick, stone, &c ; and savs it 
becomes far harder and more durable than common paint: Cue measure of ground fresh quicklime; 
add the same quantity of fine white sand, or fine coal ashes ; and twice as much fresh wood ashes; 
all the foregoing to be passed through a fine sieve. Mix well together dry. Mix with as much raw 
linseed oil as will make the mixture as thiu as paint. Applv with a painter's brush. It may be col¬ 
ored like the foregoing, taking care to mix the colors well with oil before adding them. It is best to 
put on two coats; the first thin, and the second thick. 


Also, another, said to stand 15 to 30 years : 50 lbs best white lead ; 10 quarts raw liuseed oil • ^ lb 
dryer; 50 lbs finely sifted sharp clean sand ; 2 tbs raw umber. Add very little, sav U pint of tur- 
I pentine. Apply with a large brush. 

Cement for stopping: .joints, such as around chimneys, &c, &c. White 

lead ground in oil, as sold by the keg; mixed with enough pure saud to make a stiff paste that will 
, not run. It grows hard by exposure, and resists heat, eold. and water. Pieces of stone may ba 
strongly cemeuted together by it, allow ing a few' months for proper hardening. 


Whitewash for inside work, according to Mr. Downing, “is made more 
fixed and permanent, by adding 2 quarts of thiu size to a pailful of the wash, just before usiug. 
The best size for this purpose is made of shreds or glove leather; but any clean size of good quality- 
will answer,” as thin glue-water. We will add, that the common practice of mixing salt with white¬ 
wash, should not be permitted. Paper pasted on a wall whieh has previously been covered with salt 
whitewash, is very apt to become wet, and loose, and to fall off during damp weather. The white¬ 
wash should be scraped off, and the wait or partition covered with a coat or two of thin size, to pro¬ 
tect t he paper from the effect of the salt that may still adhere to the plaster. 


GLASS, AND GLAZING. 

Window glass is sold by the box. Whatever may be the size of the panes, a box 
contains as nearly 50 sq ft of glass as the dimensions of the panes will admit of. 

Paues of any size may be made to order by the manufacturers. The sizes given in the following 
table, as well as many others, are generally to be had ready made. Ordinary window glass of all the 
sizes in the table, is about one-sixteenth ot' an inch thick ; and this is the thickness supposed to be 
intended when a greater one is not specified. Double-thick glass is nearly % iuch; and its price is 
50 per ct more than the single thick. It is of course much stronger than the single. 

The panes are confined to the sash by glaziers’ putty, made of whiting (powdered chalk) and raw 
linseed oil ; and by small triangular pieces of thiu tin, about % inch on a side, which uphold the 
glass while the putty is being put on; and are allowed to remain afterward, as a protection while the 
putty continues soft. 


TABLE OF NUMBERS OF PANES IN A BOX. 


Size in 
ins. 

Panes 

to 

a box. 

Size in 

ids. 

Panes 

to 

a box. 

Size in 
ins. 

Panes 

to 

a box. 

Size in 
ins. 

Panes 

to 

a box. 

Size in 
ins. 

Panes 

to 

a box. 

6 X 8 

150 

12 X 36 

17 

16 X 42 

11 

24 X 24 

12 

30 X 66 

4 

7X0 

115 

13 X H 

40 

48 

9 

26 

12 

70 

3 

8 X 10 

90 

16 

35 

54 

8 

30 

10 

32 X 34 

7 

12 

75 

18 

31 

60 

8 

36 

9 

36 

H 

0 X 12 

67 

20 

28 

18 X 20 

20 

42 

7 

42 

6 

14 

57 

24 

23 

22 

18 

48 

6 

48 

5 

16 

50 

32 

17 

24 

17 

54 

6 

60 

4 

18 

45 

14 X 16 

32 

30 

14 

60 

5 

66 

3 

10 X 12 

60 

18 

29 

36 

11 

66 

5 

34X36 

6 

14 

52 

20 

26 

42 

10 

28X28 

10 

44 

5 

16 

45 

24 

22 

50 

8 

32 

9 

48 

i 

18 

40 

30 

17 

60 

7 

36 

8 

54 

4 

20 

36 

36 

14 

20 X 22 

17 

42 

7 

60 

4 

24 

30 

42 

12 

24 

15 

48 

6 

66 

3 

30 

24 

46 

11 

30 

12 

54 

5 

36X 40 

5 

11 X 12 

55 

15 X 16 

30 

38 

10 

60 

5 

44 

5 

14 

47 

18 

27 

42 

9 

28 X 30 

9 

48 

4 

16 

41 

20 

24 

48 

3 

36 

7 

54 

4 

18 

37 

24 

20 

54 

7 

42 

6 

60 

3 

20 

33 

30 

16 

64 

6 

56 

5 

70 

3 

24 

27 

36 

13 

22 X 24 

14 

66 

4 

38 X <4 

4 

12 X 14 

43 

40 

12 

30 

11 

30 X 34 

7 

52 

4 

16 

38 

16 X 18 

25 

36 

9 

36 

7 

40 X 4« 

4 

IS 

34 

20 

23 

42 

8 

42 

6 

54 

3 

20 

30 

24 

19 

48 

7 

48 

5 

72 

3 

24 

25 

30 

15 

56 

6 

54 

4 

44X50 

3 

28 

22 

36 

13 

60 

5 

60 

4 

56 

3 

30 

20 













































GLASS. 


432 


The best qualitips of American glass made in the vicinity of Philadelphia, 
Boston Pittsburg, Ac, are for most purely useful purposes, as good as those frm 
for! X countries’ but when the highest degree of beauty is required as in the 

fowiffront wTdow, of first-clas, flwelliofc ftnejSriwfoJ 
class of England, France, or Germany, must be used, although the price >r 
moderate sized panes is from 5 to 8 times as great as that of the hes qua ity 
single-thick American, as given in the following table* Its peilectiy smoom 

surface, free from distorted reflections, also makes itthe^nurobse few 
tures- still, if carefully selected American panes be used for this purpose, lew 

except critics in glass will detect the difference. 

A tfliick crlass is made expressly for flooring, up to 1 inch thick, 

and up to 50 inches by 9 feet dimensions. Also, for ft 

tl.ick 1 This can be furnished to order of any size up to 40 inches by 8 or 10 lect. 
The smaller sizes can also be had ground. Grinding prevents the entrance o 
SS Ml «lSe TlhesJ; and, moreover, diffuses the light over a much greater 
width of space below. 


Strength of class. Tensile 2500 to 9000 lbs per square inch. Boston rod: 
bv author a?00 to 5200. Crushing strength, 6000 to 10000 bs per square inch 
Transversely (bv the writer’s trials,) flooring glass 1 inch square, and 1 foo 
between the’end supports, breaks under a center load of about 1/0 lbs, con 
sequently, it is considerably stronger than granite, except as regards crushing 

'"iTkmakk. 16 Window and other glass which contains an excess of potash or o 
soda is very liable to become dull in time, owing to the decomposition of thos 
ingredients by atmospheric influences. 


* Price list of American single-thickplass, ndoptdL bj DoJIhh 
can Window Glass Manufacturers’ National Association, Jan \S, TJ8_ 1 out b 
thick about 50 per cent. more. For ground glass add atrout $2.50 per box. Ois 
count. 1888, about 75 per cent. Kenj. II. Shoemaker, dealer in trench a 
American window glass, 205 N. 4th St, Philadelphia; also Malaga Glass an 
Manufacturing Go., office 400 Chestnut St, 1 hiladelphia. 


Size in Inches. 


From 

fix 8 

11 X 14 

18 X 22 
15 X ••16 
26 X 28 
26 X 36 
26 X 46 
30 X 52 
30 X 56 
34 X 58 
36 X 60 


to 


10 X 15 
16 X 24 
20 X 30 
24 X 30 
24 X 36 
26 X 44 
30 X 50 
30 X 54 
34 X 56 
34 X 60 
40 X 60 


1st Quality. 

2d Quality. 

3d Quality. 

A 

4th Quality. 

Per box. 

Per box. 

Per box. 

Per box. 

$10.50 

$ 9.00 

$ 8.50 

$8.00 

11.50 

10.75 

10.25 

9.75 

15.50 

14.00 

13.00 

12.50 

16.50 

15.00 

13.50 


17.75 

1625 

14.75 


19.00 

17.50 

15.25 


21.00 

19.50 

17.00 


22.00 

20.25 

18.00 


23 00 

21.25 

19.00 


24.00 

22.75 

21.00 


26.50 

24.50 

23.00 



In small quantities, the following are also approximate prices for A merit 
glass: Large plates of U inch thick, rough, 40 cents per square loot- One n 
thick 75 cents to Si; if either is ground, 10 to 15 cents additional per squ 
foot. ’Riblied glass, % inch thick, 20 cents; % inch, 25 cents per square f< 
Stained glass, single thickness, ( T V inch,) or figured white enameled glass, (sir 
thickness > 20 to 25 cents per square foot. Superior thicker strong figured gl 
first ground, and the transparent figures then formed by polishing away ] 
lions of the ground surface, $1.00 per square foot. .. . . 

Mufted glass is an inferior article of fanciful colored patterns, attached 

n v .. 11 __ _ ik/w,-. 4-^v a ff oftnr o vnar Ar i w A Ai 


some imperfect process which allows them to peel off after a j ear or two of 

posure to the weather. . , . 

The charge by glaziers for putting the glass into new windows, indue g 
putty tins, and two coats of paint to the sash, (one of which is a priming c< 
is ’ equal to the cost of the glass at the above prices. , 

For reglazing old sash and removing the broken panes, the charge is at t 
twice as great. 


t i 


- 






































PAPER. 


433 


PAPER.* 


24 sheets 1 quire. 20 quires 1 ream. 


Sizes of drawing papers. 


Ins. Ins. 


Antiquarian. 31 X 52 

Double Elephant. 26 X 40 

Atlas. 26 X 34 

Imperial. 21 X 30 


Ins. Ins. 


Super Royal. 19 X 27 

Royal. 19 X 24 

Medium. 17 X 22 

Demy. 15 X 20 

Cap .'. 13 X 17 


The English drawing-papers are stronger and superior to the American. Those 
by Whatman have a high reputation; they are, however, of different qualities. When 
paper is pasted on muslin, the difference in quality is not so important. Of paper 
in rolls, the German makes are the best. There is but little of other makes imported. 

Rotli white and tinted papers, for the use of engineers, are made in 
continuous rolls, without seams. Widths 36, 42, 54, 58, and 62 ins; usual lengths 40 
yds; but can be had to order to 400 yds or more. These may also be purchased ready 
pasted on muslin, in rolls 10 to 40 yds long. This last, on account of its strength, 
should be used for all drawings which undergo frequent rolling and unrolling; or 
: other hard usage; particularly working-drawings. For the last purpose, strong 
! cartridge or pattern paper answers very well. It is for sale in long rolls, of same 
lengths as white paper, mounted or not; widths up to 54 ins. Color, a light buff. 


j Tracing paper. Most of that sold, whether domestic or foreign, tears so 
j readily as to be of comparatively little service, except for tracings to be enclosed in 
i letters for mailing. Some of what is called French vegetable tracing-paper , is, liow- 
] ever, quite stout and strong,and good for line drawings; but it shrivels badly under 
] broad washes of color, even w hen stretched, forming little puddles, which make it 
difficult to produce a uniform tint. Sizes 19 X 25, 21 X 26, 28 X 40 ins; also in 
rolls of 11 and 22 yds. Parchment paper, 37 and 38 ins wide, rolls of 20 and 33 yds, 
j is better, but does not take ink perfectly. 

Tracing cloth, usually called tracing muslin, and sometimes vellum cloth , is 
altogether preferable to tracing paper, pn account of its great strength. Widths 18, 
30, 36, and 42 ins ; lengths to 24 yds. 

Common inks dry pale on either tracing muslin or tracing paper; therefore use 
India ink. Neither the muslin nor the paper takes colors as kindly as drawing 
paper. 

Profile paper is made in widths of 9 ins and 20 ins, and in single sheets or 
in long, continuous rolls. 

Ruled squares, or cross-section paper. Paper carefully ruled in 
small squares, so that the divisions answer for a scale for the drawing, is exceedingly 
useful for sketching out plans, &c. It is sometimes ruled on both sides of the sheet. 

Colors. A good draughtsman needs but few colors; say India ink, Prussian 
blue, lake, or carmine, light red, burnt umber, burnt sienna, raw sienna, gamboge, 
Roman ochre, sap green. Winsor & Newton’s colors are among the best in use. 
Purchase none but the very best India ink. Cakes of colors should alw ays be wiped 
dry on paper, after being rubbed iu water; and but little water should be used while 
rubbing; more being added afterward. 

Lead pencils. Genuine A. W. Faber’s Nos. 2, 3, and 4, are very good. The 
hardness increases with the number. Nos. 3 and 4 are good for field-book use: which 
to prefer, will depend on the character of the paper; No. 3 for smooth, and No.4 for 
the coarser or more granular papers. His lettered pencils are ot a higher grade and 
better suited for draughting. “ II ” stands for “ hard,” “ B ” for “soft.” The degree 
of hardness or of softness is indicated by the number of II’s or of B’s. One II cor¬ 
responds with No. 3. Dixon’s American pencils are good. The office draughtsman 
diould have a flat file, or a piece of fine emery paper glued to a strip of wood, upon 
which to rub his lead to a fine point readily, after using the knife. 


# James W. Queen & Co, No 924 Chestnut St, Philadelphia. 
















434 


STRENGTH OF MATERIALS. 


STRENGTH OF MATERIALS. 


GENERAL. PRINCIPLES. 

Art. 1 (a) Stress or Strain.* As explained in Art. 27 a, p. 318 h , stress or 
strain takes place when force acts upon a body in such a way that its particles 
tend to move (at the same time) with different velocities or in different direc¬ 
tions; to do which they must either separate from each other or come closer 
together. This occurs, for instance, when a body is so placed as to oppose the 
relative motion of two other bodies; as when a block is placed between a weight 
and a horizontal table. In this case, each of the two bodies (the weight and the 
table i imparts a force (Art. 5 c, p. 308 and Art. 25 a, p. 318 ), to the opposing body 
(the block); and the stress is the opposition of these equal forces. The tendency 
of the particles of the block to separate or to come closer together calls into 
action the Inherent forces of its material, as explained in Art. 27 (a), p. 318 k, 
and these act between the particles and tend to keep them in their original rela¬ 
tive positions. 

(h) If two opposite forces are simultaneously imparted to a 
body in the same straight line, the stress is either compressive 

(when the forces act toward each other) or tensile (when they act from each 
other.) 

Compressive stress tends to push the particles closer together. Tensile stress 
tends to pull them farther apart, f 

(c) If two imparted forces, as c a, b a, Fig. 9%, p. 320, meet at an 

angle, as at a ; then two equal and opposite components, c i and b o, will cause 
compressive or tensile stress in the body, w r hile the other two, i a and o a, unite 
to form the resultant, n a, which moves the body in its own direction, and, in 
doing so, produces another stress among the particles of the body, as explained 
in Art. 27 a, p. 318 i. 

(d) If the two forces are parallel, forming a “couple” (Art. 56 g, p. 

317 d), as in a punch and die, the stress is a shear (tending to slide some of the 
particles over the others), and is accompanied also by a transverse stress 
(causing a tensile stress in some of the particles and a compressive stress in 
others) as in the case of a beam. The transverse stress is proportional to the 
distance between the two forces (i. e., to the arm or leverage of the couple), so 
that, when they are very close together, as in a pair of shears, the transverse 
stress is very small and is neglected, and the shearing stress alone is considered. 

(e) If two contrary couples, in different planes, act upon a body, 
the stress is called torsion or twisting. Thus, torsion takes place in a brake 
axle when we try to turn it while its lower end is held fast by the brake chain. 

(f) But the ultimate tendency of any of these forms of stress is either to 
separate certain particles or to drive them closer together , as in cases (tensile 
or compressive stress) where the two forces are in one line.f We shall, 
therefore, in these introductory articles, consider only this simple and 
fundamental form of stress, assuming that it is caused by the action of two 
opposite imparted forces, acting in the same straight line so that they are 
entirely employed in causing the stress. 

(§>) A stress may be stated in any unit of weight, as in pounds, and is 
equal to one of the two opposite forces. See Art. 27 (/) p. 318,/. 

* For another use of the word “ strain,” see Art. 2 (d), p. 434 a. 

f Indeed, even in cases of compressive stress, it is only by the separation of th< 
particles that the structure of the body and its inherent forces can bo destroyed 















STRENGTH OF MATERIALS. 


434 a 


Art. 3 (a) Stretch and Rupture. It appears from experiment that the 
inherent cohesive forces called into action by the first application of any stress 
are always less than that stress, however small it may be. In other words ; any 
stress, however slight, is believed to produce some derangement of the particles. 
But the inherent forces increase with this derangement (up to a certain point); 
and thus, in many cases, they become equal to the stress and so prevent further 
derangement. When the stress exceeds the greatest inherent force which the 
body can exert, the particles separate to such an extent that the inherent forces 
cease to act. The body is then said to be broken, or ruptured. 

(t>) Different materials behave very differently when under stress. 
Brittle ones seem to resist almost perfectly up to a certain point, allowing no 
perceptible derangement of the particles; and then yield suddenly and entirely. 
In ductile materials, on the contrary, considerable derangement takes place 
before the inherent resisting forces finally yield. 

(c) The ultimate stress of a body is that which is just sufficient to break 
it or crush it, or, in short, to destroy its structure so that it can no longer resist. 
In other w ords, a stress just less than the ultimate is the greatest stress to which 
the body can be subjected. 

Caution. In brittle materials, such as brick, stone, cement, glass, cast-iron, 
etc., especially w r hen subjected to tension, the point of rupture is clearly marked, 
and hence the ultimate strength may in such cases be stated with precision. 
But w T ith ductile or malleable materials, such as copper, lead and wrought iron, 
especially when under compression, it is often difficult or impossible to state the 
ultimate strength definitely. For instance, a cube of lead may be gradually 
crushed into a thin flat sheet without rupture. In other words, there is 
practically no load which can break it by crushing. In such cases, we may 
arbitrarily assume some given amount of distortion as marking the point of 
ultimate stress. Thus, by the “ultimate” load of a rolled iron beam (p. 521) 
we mean “that one which so cripples the beam that it continues to yield 
indefinitely without increase of load.” Such assumptions, how r ever, necessarily 
give rise to some ambiguity, and care should therefore always be taken to define 
or to ascertain clearly in what sense the term “ultimate stress” is employed. 

The ultimate strength of a material (or, more briefly, its strength) is 
the greatest inherent force which its particles can exert in opposition to a 
stress. In other words, it is that inherent, resistance which is just equal to the 
ultimate stress. Hence strength, like stress, is stated in units of weight, and 
we may use the terms “ultimate strength” and “ultimate stress” indifferently, 
as denoting practically the same thing. 

(<l) For want of a convenient and appropriate name for the change of 
shape caused by stress; modern writers have, rather unfortunately, given to it 
the name of strain,* which, in ordinary language, (as in our articles on Force 
in Rigid Bodies, pp. 306, etc.) is used to signify stress, as above defined. We prefer 
to use the word “stretch” for change of shape, in inches, etc. (regarding 
compression as negative “Stretch”), and “strain” or “stress” for the action 
of the two opposing forces, in pounds, etc. 

(e) By the ‘‘length” of a body, we mean its dimension measured in the 
line of the stress; and, by “area,” the area of the resisting cross section at 
right angles to that line. Thus, if a slab of iron, 2 inches thick and 10 inches 
square, be laid flat upon a smooth and horizontal surface, and if a load be placed 
upon it so as to be uniformlydistributed over its upper flat surface,the “length” 
is 2 inches, and the “ area,” 10 x 10 = 100 square inches.f 

(f) Units adopted. Unless otherwise stated, we shall understand the 
stress in any case to be given in pounds, the stretch and the length in inches, 
and the area in square inches. 

♦The word “strain” is not thus defined, even as a scientific term, in either 
Webster’s or Worcester’s dictionary. 

fUnder stresses approaching the ultimate stress, the area of the cross section 
generally increases under compression, and diminishes under tension, to differ¬ 
ent extents m different materials; but we are here concerned only with cases 
within the limit of elasticity, (Art. 4a, p. 434 d) and in such cases the chango of 
area is generally very slight and may be neglected. 







4346 


STRENGTH OF MATERIALS. 


Art. 3 (a). If the total stress (in lbs., etc.) upon a body be divided by the 

area of the resisting surface (in square inches, etc.) the quotient, stre33 , is the 

area 

mean stress per unit of area, or (as it is sometimes called) the intensity of 
the stress. Or, * 

Stress per unit of area = total stress . 

area 

Thus, if a bar of iron, 2 inches wide by 1 inch thick, (having therefore 2 square 
inches of area of cross section) and 10 feet long, be subjected to a total tensile 
stress of 20,000 lbs. in the direction of its (10 ft.) length, we have 

mean stress ) 
per unit of area j 


total stress _ 20,000 lbs. 


area 


2 square in. 


10,000 lbs per square inch. 


strictly speaking, the stress on a surface is seldom distributed 
uniformly over it. thus, in the case of the bar just referred to, if the stress is 
applied by means of grips, clamping the sides and edges of the bar, the stress 
per square inch near those sides and edges is probably greater than that near 
the center of the bar, because the stress is not perfectly and uniformly trans¬ 
mitted from the outer to the inner fibres. And, in cases of compression, the 
load instead of being uniformly distributed over the surface, as it appears to be. 
is often in fact supported by a few projecting portions of it. In practice, these 
considerations are often of the greatest importance, but in studying the 
principles of resistance, we may, for convenience, temporarily neglect them, and 
assume the stresses to be uniformly distributed over their respective areas. 

I 1 , the total stretch of a body (in inches, etc.), under any given stress, be 
dnided by the original length of the body (in the same measure), the quotient 
is the stre tell per umt of length. Or, H 

Stretch per unit of length = total stretch 

original length * 

Thus : if the foregoing bar 10 feet (or 120 inches) long, is found to stretch 
.04 inch, under its load ot 20,000 lbs. total, or 10,000 lbs. per square inch, we have 


stretch per unit of length 
under said load 


total stretch 


.04 


—. —n——r — -= .00033 inch per inch. 

original length 120 

(c) The Modulus of Elasticity. In materials which undergo a per¬ 
ceptible stretch before rupture, it has been found bv experiment that up to a 
certain degree of stress, called the limit of elasticity (Art. 4 a, p. 434 d), the ratio 
total stress . ’ 9 

total Stretch ’ in aDy giv<m body ’ remaius vei T nearly constant. In other 

words, within the limit mentioned, equal additions of stress cause practically 
equal additional stretches. * * 

In order to compare bodies of different dimensions, we state the same fact by 
saying that, within the elastic limit the quotient, stress per unit of area 
remains practically constant. 


stretch per unit of length 


This quotient as found by experiment with any given material, is called the 
Mo .‘ l « of that material, and is usually denoted by the 

capital letter E. It is of course expressed in the same unit'as the stress per 
unit of area* as, for instance, in pounds per square inch. 

■ S I* wil L b ? n °ticed that the greater the stress required to produce a given 
stretch in a body the greater is its modulus of elasticity. Hence the modulus 
13a measure of the resistance which the body can make against a change in 
shape. This resistance we call the ** elasticity ” of the body although in 
,r n -? U » ge (and, indeed, often in a scientific sense also) we apply the 
J?™, elasticity rather to the ability of a body to sustain considerable distor- 
tion without losing its power of returning to its original shape. 











STRENGTH OF MATERIALS. 


434 c 


(e) Since 


and since 


Stress per unit of area 


stretch per unit of length = 


total stress 
area 

total stretch 
original length 


we may find the modulus of elasticity of any material, from experiment upon 
any specimen of it, thus: 

total stress X original length 


Modulus of elasticity 

total stretch X 

From this we have the following equations: 


area 


Total stress, in lbs. 
required for a given 
total stretch, in inches 


modulus of 
elasticity in lbs X 
per square inch 


total stretch y, 
in inches *• 


area 

in square ins 


Stress per unit of area, 

in lbs. per square inch, required 
for a given stretch, in inches 

Total stretcli, in 

inches, under any stress in lbs. 


original length in inches 

modulus of stretch per 

per 3 square inch X hoit ofleogth 

total stress ^ original length, in inches 


modulus of elasticity, y area, in 
in lbs. per square inch A sq. ins 


original length v stress, in lbs. per square inch 
in inches * ~, . ..... ,v-_ 


modulus of elasticity in lbs. per sq. inch 


Stretch per unit of length 


stress, in lbs. per square inch 


modulus of elasticity, in lbs per sq. in. 


( 1 ) 


( 2 ) 


( 3 ) 


..( 4 ) 


.(5) 


( 6 ) 


The modulus of elasticity is used chiefly in connection with the stiffness of 
beams (see pp. 505 b, etc.), and is most readily found by means of their deflections. 
Thus, in any beam, supported at both ends and loaded at the center: 


Modulus of 
elasticity, in 

lbs. per sq. inch 


/ load 
\in lbs. 


+ 


% weight of clear \ y cube of span 
span of beam, in lbs./ * in inches 


.q v deflection, v moment of inertia 
40 * in inches * (p. 486) in inches 


(7) 


or, 


E 


(W + % W) 1 3 

48 A I 


( 8 ) 


If the beam is rectangular , this becomes 

Modulus of ( load - 4 - % weight of clear \ v 

elasticity, in = lbs. + span of beam, in lbs/ X _ 

lbs per square inch 4 y deflection, y breadth, y cube of depth 

* in innVina ^ inoliua A in innKoa 


cube of span 
in inches 


(9) 


or, 


in inches ^ inches 
E _ (W+ %w)l* 

4 A & d 3 


in inches 

( 10 ) 


Corresponding formulae for modulus of elasticity in beams otherwise sup¬ 
ported and loaded, may be readily deduced from those for deflection on p. 505 a. 

(f) If equal additions of stress could produce equal additional stretches in a 
body to an indefinite extent, both within and beyond the elastic limit, then a 
stress equal to the Modulus of Elasticity would double the length of a bar when 
applied to it in tension, or would shorten it to zero when applied in compression. 
In other words, if equation (5), 


total stretch, 


original length 
in inches 


y, stress per square inch 
modulus of elasticity 


held good beyond the elastic limit, as it does (approximately) within that limit, 
and if we could make the stress per square inch equal to the modulus of 
elasticity, we should have total stretch = original length. 




















434 d 


STRENGTH OF MATERIALS. 


For example, a one-inch square bar of wrought iron will, within the limit ol 

elasticity, stretch or shorten, on an average, about fjljTTTT of its length under 
each additional load of 2240 lbs. If it could continue to stretch or shorten 
indefinitely at this rate, it is evident that 12000 times 2240 lbs., or 26 880 000 lbs., 
(which is about the average modulus of elasticity for such bars) could either 
stretch the bar to double its length or reduce it to zero. 

If equal additional stresses applied to a bar could indefinitely produce 
stretches, each bearing a constant proportion to the increased length of the bar, if 
in tension; or to the diminished length, if in compression; then the same load 
which would double the length of the bar if applied in tension, would reduce 
it to half its length, if applied in compression. 

(R) We give below a table of average Moduli of Elasticity, in round 

numbers, for a few materials; remarking, by way of caution, that, even in the 
case of ductile materials, the stretches produced by stresses within the elastic 
limit are so small, and (owing to differences in the character of the material) so 
irregular, that a satisfactory average can be arrived at only by comparing many 
experiments; while, in the case of materials, such as stone, brick, etc., where 
almost no perceptible stretch takes place before rupture, it is scarcely worth 
while to give any values as representing the actual moduli. Thus, eighteen 
experiments upon a single brand of neat cement for the St. Louis bridge, indi¬ 
cated a Modulus varying from 800 000 to 6 930 000 (!) pounds per square inch 
in tension, and from 500 000 to 1 500 000 in compression. 

. (h) Owing to the fact that the stretches within the elastic limit are seldom, 
if ever, exactly proportional to the stresses, but only approximately so, the 
modulus of elasticity, as found by experiment for a given material, will 
generally vary somewhat with the stress at which the stretch is taken. 


^ ( a ) d ^e stress beyond which the stretches in any body increase per¬ 
ceptibly faster than the stresses, is called the limit of elasticity of that 
body. Owing to the irregularity in the behavior of different specimens of the 
same material, awl to the extreme smallness of the distortions caused in most 1 
materials by moderate loads, and because we often cannot decide just when the 
stretch begins to increase faster than the load, the elastic limit is seldom, if 
evei, determinable with exactness and certainty.* Rut by means of a large ] 
number of experiments upon a given material we may obtain useful average ! 
or minimum \ allies for it, and should in all cases of practice keep the stresses 1 
well within such values; since, if the elastic limit be exceeded (through mis- ! 
calculation, or through subsequent increase in the stress.or decrease in the 
strength ol the material) the structure rapidly fails. The table, below gives 
approximate average elastic limits for a few materials i 

. l 

(to) Brittle materials, such as stones, cements, bricks, etc., can scarcelv be said 
to have an elastic limit; or, il they have, it is almost impossible to determine 
it, since rupture, in such bodies, takes place before anv stretch can be satis¬ 
factorily measured 1 bus, in the 18 specimens of one brand of cement, 
referred to in Art 3 g above, the experiments indicated an elastic limit varying 
between 16 and 104 (!) pounds per square inch in tension , and from 424 to 1502 
in compression. 

(c) Experiments show that a small permanent, “set” (stretch) probably 
takes place in all cases of stress even under very moderate loads; but ordinarily 
it first becomes noticeable at about the time when the elastic limit is exceeded 
Many writers define the elastic limit as that stress at which the first marked 
permanent set appears. 

(<1) The elastic ratio of a material is the quotient, clastic limit 


is usually expressed as a decimal fraction. 


ultimate strength 


. * ■ U - S - Board appointed to test Iron, Steel, Ac., found a variation of nearly 
4000 lbs. per square inch in the elastic limit of bars of one make of rolled iron 
prepared with great care and having very uniform tensile strength- and in 
another very carefully made iron, a difference of over 30 per cent, between two 
bars of the same size. Report, 1881, Vol. 1, p 31 








STRENGTH OF MATERIALS. 434 e 

(©) Table of Moduli of Elasticity and of Elastic Limits for 

different materials. 

The values here given are approximate averages compiled from many sources. 
Authorities differ considerably in their data on this subject. See Art. 3 (q) and 
(h), and Art. 4 (a) and (6). JJ 


MATERIAL. 

Modulus 
or Coeff 
of 

Elasticity. 

8tretoh or 
in a lengt 
under 

1000 lbs per 
sq in. 

Compression 
h of 10 ft, 
load of 

1 ton per 
sq in. 

Approx clas 
limit. 


lbs per sq in. 

Ins. 

Ins. 

Ibs per sqin. 

Ash. 

1 600 000 

.075 

.168 

4500 

Beech. 

1 300 000 

.092 

.207 

4000 

Birch. 

1 400 000 

.086 

.192 

5000 

Brass, cast. 

9 200 000 

.013 

.029 

6000 

“ wire. 

U 200 000 

.009 

.019 

16000 

Chestnut. 

1 000 000 

.120 

.269 


Copper, cast. 

18 000 000 

.007 

.015 

6300 

“ wire. 

18 000 000 

.007 

.015 

10000 

Elm. 

1 000 000 

.120 

.269 

2000 

Glass. 

8 000 000 

.015 

.034 

3200 

Iron, cast . 

12 000 000 

.010 

.022 

4500 


to 

to 

to 

to 


23 000 000 

.005 

.012 

8000 

“ “ average. 

17 500 000 

.007 

.015 

6250 

“ wrought, in either < 

18 000 000 

.006 

.015 

20000 

bars, sheets or plates.V 

to 

to 

to 


t 

40 000 000 

.003 

.007 

40000 

“ “ average. 

29 000 000 

.004 

.009 

30000 

“ wire, hard. 

26 000 000 

.005 

.010 

27000 

“ wire ropes. 

15 000 000 

.008 

.018 

13000 

Larch. 

1 100 000 

.109 

.244 

2300 

Lead, sheet. 

720 000 

.1fi7 



“ wire. 

1 000 000 

.120 



Mahogany. 

1 400 000 

.086 

.192 

2700 

Oak . 

1 000 000 

.120 

269 



to 

to 

to 


; 

2 000 000 

.060 

.134 


“ average. 

1 500 000 

.080 

.179 

3300 

Pine, white or yellow. 

1 600 000 

.075 

.168 

3300 

Slate . 

14 500 000 

.008 

.018 

3700 

Spruce. 

1 600 000 

.075 

.168 

3300 

Steel bars. 

29 000 000 

.004 

.009 

34000 


to 

to 

to 

to 


42 000 000 

.003 

.006 

44000 

'* “ average. 

35 500 000 

.003 

.007 

39000 

Sycamore. 

1 000 000 

.120 

.269 

4000 

Teak. 

2 000 000 

.060 

.134 

5000 

Tin, cast.1 

4 600 000 

.026 


1500 




























































434/ 


STRENGTH OF MATERIALS. 


Art. 5 (a) ReHilience is a name given to the work (as in inch-pounds) 
which must be done in order to produce a certain stretch in a given body. 
This work is equal to 

resilience _ said stretch ^ mean stress in pounds employed in producing 
in inch-pounds in inches the stretch. 

The total resilience is the work done in causing rupture. The elastic resilience 
(frequently called, simply, the resilience) is that done in causing the greatest 
stretch possible within the elastic limit. 

(to) Suddenly applied loads. Place a weight of 4 lbs. in a spring 

balance, but let it be upheld by a string fastened to a firm support in such a 
way that the scale of the balance shall show only 1 lb. By now cutting this 
string with a pair of scissors, we suddenly apply 4 — 1=3 lbs.; and the weight 
will descend rapidly, until, for an instant, the scale shows about 1 + twice 
3 = 7 lbs. In other words, the load of 3 lbs. applied suddenly (but without jar 
or shock) has produced nearly twice ihe stretch that it could produce if added 
grain by grain, as in the shape of sand. 

For, when the load is first applied, the inherent forces, as noticed in Art. 2 (a), 
are insufficient to counteract its stress. Hence the load begins to stretch the 
spring. The work thus done is equal to the product, suddenly applied weight 1 
of 3 lbs. X the stretch of the spring; and it has been expended (except a small 1 
portion required to counteract friction) in bringing the resisting forces into 
action, thus storing in the spring potential energy (Art. 22, p. 318 d), nearly 1 
sufficient to do the same work; i. e., to lift the weight (3 lbs.) to the point (1 lb. 1 
on the scale) from which it started. But a portion of this energy has to work 
against friction and the resistance of the air. Therefore the weight does not 1 
rise quite to its original height. 1 


The shortening of the spring nearly to its original length has now reduced 
its inherent forces almost to zero ; and the weight again falls, but not so far as 
before. It thus vibrates through a less and less distance each time, and finally 
comes to rest at a point (4 lbs. on the scale) midway between its highest and 
lowest positions (1 lb. and 7 lbs.) Thus, within the limit of elasticity, a load 
applied suddenly (though without shock) produces temporarily a 
stretch nearly equal to twice ttoat which. it could produce it 
applied gradually ; i. e., twice that which it can maintain after it comet 
to rest. 


Remark. If the load is added in small instalments, each applied suddenly, 
then each instalment produces a small temporary stretch and afterward main¬ 
tains a stretch half as great. Thus, under the last small instalment, the bar 
stretches temporarily to a length greater than that which the total load can 
maintain, by an amount equal to half the small temporary stretch produced by 
the sudden application of the last small instalment. 


(c) The ^lodiilus of Klastlc Resilience (often called, simply, the 

Modulus of Resilience) ot a material, is the work done upon one cubic inch of it 
by a gradually applied load equal to the elastic limit. Or, 


Modulus stretch in inches mean stress 

nfrcdlipnoA = P er incfl °f length X in lbs. per square inch 
at the elastic limit causing that stretch 


If, as is usually done, we assume this mean stress to be % the elastic limit 
then, by formula (6) p. 434 c, 


Modulus 
of resilience 


elastic limit 


modulus of elasticity 


X elastic limit 


==K 


square of elastic limit 
modulus of elasticity 


y 

Sl 

t 

11 

tl 

li: 


The elastic resilience of any piece is then 


« 

ft 


Resilience = modulus of resilience X volume of piece in cubic inches. 














STRENGTH OP MATERIALS 


435 


Fatig-ne of Materials. In the following articles on Strength of Mate- 
■ rials, the ultimate or breaking load is that which will, during its first application, 
rupture the given piece within a short time. But Wohler’s and Spaugenberg’s 
r experiments show that a piece may he ruptured hy repeated applica* 

’ t ions of a load much less than this; and that the oftener the load is applied the 
less it needs to be in order to produce rupture. Thus, wrought iron which re- 
i quired a tension of 58000 lbs per sq inch to break it in 800 applications, broke 
l with 35000 lbs per sq inch applied about 10 million times ; the stress, after each 
application, returning to zero in both cases. 

The diff between the maximum and minimum tension in a piece subjected to 
! tension only, or between the max and min compression in a piece subjected to 
1 ifomp only; or the sum of the max tension and max comp in a piece subjected 
* nlternately to tension and comp; is called the range of stress in the piece. 
1 When this is less than the elastic limit, the application may be repeated an 
6 “enormous ” number (say about 40 million) of times without rupture.* 

| For a given number of applications, the load required for rupture is least when 
1 the range of stress is greatest. If the stress is alternately comp and tension, 
rupture takes place more readily than if it is always comp or always tension. 
That is, it takes place with a less range of stress applied a given number of 
5 times, or with a less number of applications of a given range of stress. For a 
j given range of stress and given number of applications, the most unfavorable 
I condition is where the tension and comp are equal. 

The above facts are now generally taken into consideration in designing 
members of important structures subject to moving loads. For instance, Mr. 
Jos. M. Wilson, C. E., Mem. Inst. C. E. (London Eng.), Mem. Am. Soc. C. E., uses 
j the following formulae for determining the “ permissible stress” in iron bridges, 
j in lbs per sq inch; in order to provide the proper area of cross section for each 
member. 


For pieces subject to Oiie kind of stress only (all comp or all tension) 


i 

i 

1 

[ 

i 

( 



min stress in the piece 
max stress in the piece 


For a piece subject alternately to comp and tension , find the max comp and the 
max tension in the piece. Call the lesser of these two maxima “max lesser”, 
and the other or greater one, “ max greater ”. Then 





max lesser \ 
2 max greater/ 


For a piece whose max comp and max tension are equal , this becomes 



u 

2 


The above a is the permissible tensile stress in lbs per sq inch on any mem¬ 
ber; but the permissible compressive stress is found by “ Gordon’s formula” for 
pillars, p 439, using a (found as above) as the numerator, instead of/. For« in 
the divisor or denominator of Gordon’s formula (which must not be confounded 
with the a of the foregoing formulae) Mr. Wilson uses for wrought iron: 


when both ends are fixed. 36000 

when one end is fixed and one hinged. 24000 

when both ends are hinged. 18000 


Experiments show that materials may fail under a long* continued 
stress of much less intensity than that produced by the ult or bkg load. 


* This does not always hold in cases where the elastic limit has been artificially raised 
by process of manufacture, etc. Oft-repeated alternations between tension and compres¬ 
sion below such a limit reduce it to the natural one. A slight flaw may cause rupture 
under comparatively few applications of a range of stress but little greater, or even less, 
than the elastic limit. Best between stresses increases the resisting power of a piece. 
In many cases, stresses a little beyond the elastic limit, even if oft-repeated, raise that 
limit and the strength, but render the piece brittle and thus more liable to rupture from 
shocks ; and a little further increase of stress rapidly lessens, or may entirely destroy, 
the elasticity. A tensile stress above the elastic limit greatly lowers, or may even destroy, 
the compressive elasticity, and vice versa. If a tensile stress/by stretching a piece, reduces 
its resisting area, it may thus reduce its total strength, even though the strength per 
sq in has increased. Mr. B. Baker finds that hard steel fatigues much faster under re¬ 
peated loads than soft steel or iron, 
t u = 6500 tbs per sq inch for rolled iron in compression 

= 7000 tbs “ “ “ tension (plates or shapes). 

= 7500 jbs “ for double rolled iron in tension (links or rods). 













436 


STRENGTH OF MATERIALS. 


Art. 1. Compressive strengths of American woods, when 

slowly awl carefully seasoned. Approximate averages deduced from many experi¬ 
ments made with the U S Govt testing machine at Watertown, Mass, l>y Mr. S. P. 
Sharpies, for the census of 1880. Seasoned woods resist crushing much better 
than green ones; in many cases, twice as well. This must be allowed for when 
building bridges, <fcc, of timber recently cut. Different specimens of the same wood 
Vary greatly ; frequently as 5 to S, 9, or more. See Rem and foot-note p 611. 


The strengths in all these 
tables may readily vary as 
much as one-third part more 
er less than our average. 

End¬ 

wise.* 

lbs per 
sq in. 

Side- 

wise.IT 

lbs per 
sq in. 

The strengths in all these 
tables may readily vary as 
much as one-third part more 
or less than our average. 


.01 

.1 

Ash , red and white. 

6800 

1300 

3000 

Maple, broad - leafed. 

Aspen . 

4400 

800 

1400 


Beech . 

7000 

1100 

1900 

“ sugar and black.. 

Birch . 

8000 

1300 

2600 


Buckeye . 

4400 

600 

1400 

Oak, white, post for iron) 

Butternut . 

5400 

700 

1600 

swamp white, red 

Buttonwood (sycamore).. 

6000 

1300 

2600 

and black. 

Cedar , red. 

6000 

700 

1000 

“ scrub and basket... 

“ white (arbor vitae) 

4400 

500 

900 

“ chestnut and live... 

Catalpu (Indian bean)... 

5000 

700 

1300 

“ pin. 

Cherry , wild. 

8000 

1700 

2600 

Piwe^ whit -a 

Chestnut . 

5300 

900 

1600 

“ Red or Norway..... 

Coffee tree, Kentucky. 

5200 

1300 

2600 

“ pitch and Jersey 

Cypress, bald. 

6000 

500 

1200 


Elm, A in’ll or white. 

6800 

1300 

2600 

“ Georgia. 

“ red. 

7700 

1300 

2600 

Poplar t1 

Jlemlock . 

5300 

600 

1100 

Sas&afra .<? 

Hickory . 

8000 

2000 

4000 


Lignum vitae . 

10000 

1600 

13000 

ii vvhifft 

Linden, American. 

5000 

500 

900 

Sycamore (buttonwood).. 

Locust , black and yellow 

9800 

1900 

4400 

Walnut, black. 

“ boney. 

7000 

1600 

2600 

“ white (butter- 

Mahogaiiy . 

9000 

1700 

5300 

mi*'* 





Willow . 


End- 

Side- 

wise.* 

wise.11 

lbs per 

lbs per 

sq in. 

sq in. 


.01 

.1 

5300 

1400 

2600 

8000 

1900 

4300 

6800 

1300 

2900 

7000 

1600 

4000 

6000 

1700 

4200 

7500 

1600 

4500 

6500 

1300 

3000 

5400 

600 

1200 

6300 

600 

1400 

5000 

1000 

2000 

8500 

1300 

2600 

5000 

600 

1100 

5000 

1300 

2100 

5700 

700 

1300 

4500 

600 

1200 

6000 

1300 

2600 

8000 

1300 

2600 

5400 

700 

1600 

4400 

700 

1400 


Hence it appears that seasoned white and yellow pines, spruce, and ordinary oaks 
which are the woods most employed in the United States for bridges, roofs, etc., crush 
endwise with from 5000 to7000fbsper sqinch, in short blocks; average, 6000. 

But it is well to bear in mind that in practice perfectly equable pressure is rarely 
secured. In a few trials on sidewise compression, with fairly seasoned white pine 
blocks, 6 ins high, 5 ins long, and 2 ins wide, we found that under an equally dis¬ 
tributed pressure of 5000 lbs total or 500 lbs per sq inch, they compressed about from 
V* to *4 » n cb ; which is equal to from % to inch per foot of height; or from A 
‘?*° t ,i h « 1 L e, « ht; , the mea, L i> e 'ng about, % inch to a foot, or J* of the height. 
Under 10000 lbs total, or 1000 lbs per sq inch, they split badly; and in some cases 
large pieces flew off. See Rem, p 500. 

The tensile or cohesive strengths of pine and oak average about 10000 lbs per sq 
inch, or /, as much as average cast-iron, or nearly double their resistance to crush¬ 
ing. Hie tensile strength does not change with the length of the piece; so that in 
practice we may take its safe strain at from 1000 to 2000 lbs per sq inch, depending 
upon the character of the structure, &c., without regard to the length, except when 
this-is so great that two or more pieces have to be spliced together to make it: thus 
weakening the piece very much. 

* Specimens 4 centimetres (1.57 inch) square, 32 centimetres (12.6 ins) long 
When the length exceeds 10 times the least side, see Wooden Pillars p 458 

IT Specimens 4 centimetres (1.57 inch) square, 16 centimetres (6.3 ins) long - laid 
upon platform of testing machine. Pressure applied at their mid-length, by means 
ot an iron punch 4 centimetres square, or just covering the entire width of the 
specimen, and one-fourth of its length. The first column (headed “.01 ”) gives the 
loads producing an indentation of .01 inch. The second column (headed “ 1”) gives 
those producing an indentation of .1 inch. B 













































































STRENGTH OF MATERIALS. 


437 


Art. 2. Ultimate average crushing* loads in tons, per square 

?]Vtn°ho ® ton f *•;*®\ The stones are supposed to be on bed, and the heights 
o all to be from1.5 to 2 times the least side. Stones generally begin to crack or 
split under about one-half of their crushing loads. In practice, neither stone nor 
brickwork should be trusted with more than % to Ath of the crushing load, ac- 

f Ci n CUr i nS H a !l Ce ?- , Wh *‘« thoroughly wet some absorbent sand¬ 
stones lose fully half their strength. See head of next page. 


- 


0 

0 

J 


0 

J 

:■« 

A 

0 

J 

0 

hO 

10 

0 

■JO 

00 

DOS 

JO 

JO 

10 


5, 

■k 

3; 

1 

t 

j> 

a 

h 

t. 

•j 


1 


Granites and Syenites. 

Basalt. 

Limestones and Mar¬ 
bles *. 

Oolites, good. 

Sandstones fitfor build¬ 
ing* .. 

Sandstone, red, of Con¬ 
necticut and N. Jer¬ 
sey, to crack. 

Brick*.. 

Brickwork, ordinary, 

cracks with*. 

Brickwork, good, in ce¬ 
ment* . 

Brickwork, first-rate, 

in cement. 

Slate . 

Caen Stone. 


Tons per 
sq. ft. 

300 to 1200 

250 to 1000 
100 to 250 

150 to 550 


40 to 300 

20 to 30 

30 to 40 

50 to 70 
400 to 800 
70 to 200 


“ “ to crack. 

Chalk, hard,. 

Plaster of Paris, 1 day 
old. 


20 to 30 


Mean. 

Tons. 


750 

700 

625 

175 

350 


200 

170 

25 

35 

60 

600 

135 

70 

25 

40 



Tons per 
sq. ft. 

Mean. 

Tons. 

Cement, Portland, 
neat.,U. S. or foreign, 
7 days in water. 

75 to 150 

112.5 

Common U.S.cements, 
neat, 7 days in water 

Concreteof Port. 

cement, sand, and 
gravel or brok stone 
in the proper pro por¬ 
tions,rammed 1 m old 

15 to 30 

22.5 

12 to 18 

15 

6 months old. 

48 to 72 

60 

12 months old. 

74 to 120 

97 

With good com ition 
hy<l cements, 

abt .2 to .25 as much 

Coignet heton, 3 

months old. 

100 to 150 

125 

Bubble masonry, 
mortar, rough. 

15 to 35 

25 

Glass, green,crown and 
flint. 

1300to2300 

1800 

or 3 times that of granite. 


12 to 18 

15 


Crushing height of Brick and Stone. 


If we assume the wt of ordinary brickwork at 112 lbs per cub ft, and that it would 
crush under 30 tons per sq ft, then a vert uniform column of it 600 ft high, would 
crush at its base, under its own wt. Caen stone, weighing 130 lbs per cub ft, would 
require a column 1376 ft high to crush it. Average sandstones at 145 lbs per cub ft, 
would require one 4158 ft high ; and average granites, at 165 lbs per cub ft, one 
of 8145 feet. But stones begin to crack and splinter at about half their ultimate 
crushing load; and in practice it is not considered expedient to trust them with more 
than 3^th to j^th part of it, especially in important works; inasmuch as settlements, 
and imperfect workmanship, often cause undue strains to be thrown on certain 
parts. 

The Merchants’ shot-tower at Baltimore is 246 ft high; and its base sustains 6% 
tons per sq ft. The base of the granite pier of Saltash bridge, (by Brunei.) of solid 
masonry to the height of 96 ft, and supporting the ends of two iron spans of 455 ft 
each, sustains 9% tons per sq ft. The base of a brick chimney at Glasgow, Scotland, 
468 ft high, bears 9 tons per sq ft; and Professor Rankine considers that in a high 
gale of wind, its leeward side may have to bear 15 tons. The highest pier of Rocque- 
favour stone aqueduct, Marseilles, is 305 ft, and sustains a pressure at base of 13]^ 
tons per sq ft. For greater pressures on arch-stones, see p 694. 


* Trials at St. Bonis bridge, by order of Capt James B. Eads, C. E., 
I showed that some magnesian limestone did not yield under less than 1100 tons per sq ft. A column 
j 8 ins high and 2 ins diam shortened % inch under pressure; and recovered when relieved. 

8 Experiments made with the Govt testing machine at Water- 
town. Mass, 1882-3, gave 1400 tons per sq ft ultimate crushg load for white 
and blue marble from Lee, Mass, 700 for blue marble from Montgomery Co, Pa, 960 for limestone from 

[, Consholiocken, Pa, 500 for limestone from Indiana, 840 for red sandstone from Hummelstowu, Pa, 
260 to 1000 for yellow Ohio sandstone ; Phila bricks, flatwise; hard, machine-made, 650 to 700 tons ; 
j hand-made, 700 to 1300; pressed, machine-made, 450 to 580; Brickwork columns, 13 ins sq and 13 ins 
high ; in lime, 100 tons ; iu cement, 150. 

3 t Experiments by Col. Wm. Ludlow, U. S. A., with Govt testing machines, in 1*881, gave from 21 

9 to 64 tons per sq ft for pure, hard ice, and 16 to 59 tons for inferior grades. The specimens (6 and 
S 12-inch cubes) compressed to 1 inch before crushing. 


1 














































438 


STRENGTH OF MATERIALS, 


Sheet lead is sometimes placed at the Joints of stone col- 

limns, with a view to equalize the pressure, and thus increase the strength of the column. But 
experiments have proved that the effect is directly the reverse, and that the column is materially 
weakened, thereby. Does this singular fact apply to cast iron aud other materials ? 

Art. 3. Average crushing' load for Metals. 

It must be remembered that these are the loads for pieces but two or three times their least side in 
height. As the height increases, the crushing load diminishes. See “ Strength of Pillars,” p 439. 


The crushing load per sq inch, of any material, is frequently 
called its constant, coefficient, or modulus, of crushing or of com¬ 
pression. 


Cast Iron, usually.. 

It is usually assumed at 100000 lbs, or say 45 tons per sq inch. Its 
crushing strength is usually from 6 to 7 times as great as its tensile. 
Within its average elastic limit of about 15 tons per sq inch, average 
cast iron shortens about I part in 5555; or % inch in 58 ft under each 
ton per sq inch of load; or about twice as much as average wrought 
iron. Hence at 15 tons per sq inch it will shorten about 1 part in 370; 
or full % inch in 4 feet. Different cast irons may however vary 10 to 
15 per ct either way from this. 

II. S. Ordnance, or gun metal : Some. 

Wrought iron, within elastic limit. 

Its elastic limit under pressure averages about 13 tons per sq inch. 
It begins to shorten perceptibly under 8 to 10 tons, but recovers when 
the load is removed. With from 18 to 20 tons, it shortens permanently, 

about -jpg-th part of its length; and with from 27 to 30 tons, about -j-Lr-tli 
part, as averages. The crushing weights therefore in the table are 
not those which absolutely mash wrought iron entirely out of shape, 
but merely those at which it yields too much for most practical build¬ 
ing purposes. About 4 tons per sq inch is considered its average safe 
load, in pieces not more than lOdiams long; and will shorten it % inch 
in 30 ft. average. 

Brass, reduced y^th part in length, by 51000; and % by 

Copper, (cast,) crumbles. 

(wrought) reduced ^th part in length, by. 

Tin, (cast,) reduced yLth in length, by 8800; and x / s by 
Load, (cast,) reduced ]4 of its length, by 7000 to 7700.... 

“ By writer. A piece 1 inch sq, 2 ins high, at 1200 lbs the com¬ 
pression was 1-200 of the ht, at 2000, 1-29; at 3000, 1-8; at 
5000, 1-3; at 7000, 1-2 of the ht. 

Spelter or Zinc, (cast.) By writer. A piece 1 inch 

square, 4 ins high, at 2000 lbs was compressed 1-400 of its ht; at 4000, 
1-200; at 6000, 1-100; at 10000, 1-38 ; at 20000, 1-15; at 40000 yielded 
rapidly, and broke into pieces. 

Steel, 224000 lbs or 100 tons shorten it from .2 to .4 part. 

“ American. Black Diamond steel-works, Pittsburg, Penn, 
experiments by Lieut W. H. Shock, U. S. N., on pieces 3^ in 
square ; aud 33^ ins, or 7 sides long. 

“ Untempered. 100100 to-104000.... 

“ Heated to light cherrv red, then plunged into oil of 82° Fah, 

173200 to 199200....'. 

“ Heated to light cherry red, then plunged into water of 79° 
Fah ; then tempered on a heated plate, 325400 to 340800.... 
“ Heated to light cherry red, then plunged into water of 79° 

Fah, 275600 to 400000. 

“ Elastic limit, 15 to 27 tons. 

“ Compression, within elas limit averages abt 
1 part in 13300, or .1 of an inch in 111 ft per ton per sq inch ; 
or .1 of an inch in 5.3 ft under 21 tons per sq inch. 

Best steel knife e«l«es, of large R R weigh scales 

are considered safe with 7000 lbs pres per lineal inch of edge; and 

solid cylindrical steel rollers under bridges, and 

rolling on steel, safe with Vdiam in ins X 3 100 000, in lbs per lineal 
inch of roller parallel to axis. And per the same, for 


Pounds per 
sq. inch. 


Tons per 
sq. inch. 


85000 to 125000 38 to 56 


.175000. 

22400 to 35840 

.. 29120. 


78.1 
10 to 16 
13 


....165000. 

....117000. 

....103000. 

.15500. 

• ••••.. / 350. 


.102050. 

.186200. 

.333100. 


.337800. 

..47040. 


73.6 

52.2 

46.0 

6.92 

3.28 


45.5 

83.1 

148.7 

150.8 

21 


Solid east Iron wheels rolling on wrought iron, y/l)iam ins X 352 000. 

“ “ “ “ “ “ cast iron, j/Diam ins X 222 222. 

Solid steel “ “ “ steel, ]/i)iam ins x 1300000. 

“ “ 11 “ “ wrought iron, ]/Diam ins X 1024 000. 

“ “ “ “ “ cast iron, |/Diam ins X 850 000. 

From ‘‘Specifications for Iron Drawbridge at Milwaukee,” by Don J. Whittemore, C. E. 












































STRENGTH OF PILLARS. 


4oTF 


STRENGTH OF PILLARS. 


The foregoing remarks on crushing or compressive strength refer to that of 
pieces so short as to be incapable of yielding except by crushing proper. Pieces 
longer in proportion to their diameter of cross section are liable to yield by 
bending sideways. They are called pillars or columns. 

The law governing the strength of pillars is but imperfectly understood; and 
the best formulae are rendered only approximate by slight unavoidable and un¬ 
suspected defects in the material, straightness and setting of the column. A 
very slight obliquity between the axis of a pillar and the line of pressure may 
reduce the strength as much as 50 per cent; and differences of 10 per cent or 
more in the bkg load may occur between two pillars which to all appearances 
are precisely similar and tested under the same conditions. Hence a liberal 
factor of safety should be employed in using any formulae or tables for pillars. 
See “ factor of safety ” p 442, and at foot of p 446. 

In our following remarks on this subject, the pillars are supposed to sustain 
a constant load ; and the ultimate or breaking load referred to is that one which 
' would, during its first application, cripple or rupture the pillar in a short time, 
i But struts in bridges etc often have to endure stresses which vary greatly in 
amount from time to time. Their ultimate load is then less. For such cases 
see p 435. 

Long pillars with rounded ends, as in Fig 1, have less strength than 
those with flat ends, whether free or firmly fixed. See table p 442, which 
also shows that in short columns the difference in this respect is 
but slight. 

In iron bridges and roofs, the ends of the pillars and of oblique 
struts are frequently sustained by means of pins or bolts passing 
through (across) them, at either one or both ends, as at p. Fig 1. 
See also p 612. These we will call hinged ends. Our table 
1. p 442, shows that pillars so fixed are about intermediate in 
strength between those with flat and those with round ends. 
There is much uncertainty about this and all such matters. The 
strength of a given hinged-end pillar is increased to an important 
extent by increasing the diameter of the pin. 

The formula in most general use for the strength of pillars, 
is that attributed to Prof. Lewis Gordon of Glasgow, and 
called by his name. With the use of the proper coefficients for the given case, 
it gives "results agreeing approximately with averages obtained in practice with 
pillars of sucb lengths (say from 10 to 40 diams) as are commonly used. 

It is as follows 



in which 


Breaking load in lbs per sq inch 
of area of cross section of pillar 



1 + 


P 

r 2 a 


f is a coefficient depending upon the nature of the material and (to 
some extent) upon the shape of cross section of the pillar. It is often taken, 
approximately enough, as being the ult crushing strength of short blocks of the 
given material. For good American wrought, iron, such as is'used for pillars, 
40000 is generally used ; for cast iron 80000. Mr. Cleeman* found for mild steel 
(.15 per cent carbon) 52000; and for hard steel (.36 per cent carbon) 83000 lbs. 
Mr. C. Shaler Smith gives 5000 for Pine. See p 458. 


», for wrought iron, is usually taken as follows: a = 

when both ends of the pillar are flat or fixed. 36000 to 40000 

when both ends of the pillar are hinged. 18000 to 20000 


when one end is flat or fixed, and the other hinged... 24000 to 30000 

For cast iron about one eighth of these figures is generally used ; and for pine 
about one twelfth. 

1 is the length of the pillar. If the pillar has, between its ends, supports 
which prevent it from yielding side-ways, the length is to be measured 
. between such supports. See lines 11 to 18, p 457. 

r is the least radius of gyrationf of the cross section of the pillar. I and r 
must be in the same unit.; as both in feet, or both in inches. 


# Proceedings Engineers' Club of Pbila, Nov 1884. 
t See p 440. 





















440 


STRENGTH OF PILLARS. 


Radius of gyration. Suppose a body free to revolve around an axis which 
passes through it in any direction; or to oscillate like a pendulum hung from a point 
of suspension. Then suppose in either case, a certain given amount of force to be 
applied to the body, at a certain given dist from the axis, or from the point of sus¬ 
pension, so as to impart to the body an angular vel; or in other words, to cause it 
to describe a number of degrees per sec. Now, there will be a certain point in the 
body, such that if the entire wt of the body were there concentrated, then the same 
force as before, applied at the same dist from the axis, or from the point of suspen¬ 
sion as before, would impart to the body the same angular motion as before. This 
point is the center of gyration ; and its dist from the axis, or from the point of sus¬ 
pension, is the Radius of gyration, of the body. In the case of areas, as of cross- 
sections of pillars or beams, the surface is supposed to revolve about an imaginary 
axis; and, unless otherwise stated, this axis is the neutral axis of the area, which, 
passes through its center of gravity. Then 

Radius of gyration = q/Moment of inertia -t- Area 

Square of radius of gyration = Moment of inertia -v- Area 

For moment of inertia, see p 486. 

In a circle, the radius of gyration remains the same, no matter in what direc¬ 
tion the neutral axis may be drawn. In other figures its length is different for 
the different neutral axes about which the figure may be supposed to be capable 
of revolving. Thus, in the 1 beam Fig 18, p 521, the radius of gyration about the 
neutral axis X Y is much greater titan that about the longer neutral axis W Z. 
In rules for pillars the least radius of gyration must be used. 

The following formulae enable us to find the least radius of gyration, and the 
square of the least radius of gyration, for such shapes as are commonly used for 
pillars. 


Shape of cross section Least radius 
of pillar of gyration 



Solid square 


side 

1 / 12 * 



Hollow square of uniform 
thickness 


Solid rectangle 


I r>2 + 

A/ 12 


least side 


Square of 
least radius 
of gyration 

side 2 
12 ~ 


D 2 + (P 
12 


least side 2 
12 ' 



Hollow rectangle of uniform 
thickness 


/ C 3 A — c 3 a 
A 12 (CA — ca) 



Solid circle 


diameter 

4 


C 3 A — c 3 a 
12 (C A — c^tj 

diameter 2 



* V 12 = about 3.4611. 



































STRENGTH OF PILLARS. 


441 


The following are only approximate: 


Shape of cross section 
of pillar 


r F— 


Least radios 

of gyration J,,. |* ££££ 



Phoenix column. (See p 449) D X -3636 D«X .1322 




K-E--4 


trT77^yA 


zmz* 


I beam. (See p 521) 


Channel. (See p 521) 


Deck beam. (See p 521) 


4.58 


3.54 


F2 

21 


F2 

12.5 


F* 

36^5 



Ancle, with equal legs 

(See p 525) 


T, with F — f 


(See p 525) 


Cross, with F — / 


F 

5 


F / 


4.74 


F2 

25 


F 2/2 


(See p 525) 2 .6 (F + /) 13 (F2 + P) 


F2 

22.5 


32 

























442 


STRENGTH OF IRON PILLARS 


Table of approximate average ultimate loads in lbs per 
square inch, as found by experiment with carefully prepared specimens. In 
practice, allowance must be made for the rougher character of actual work, for 
jarrings etc etc. 


C«-4 

o 

CO 

3 

Pencoyd Angles, Tees, I beams and Channels.* 

See pp 521 to 527. 

Phoenix 
columns.! 
See p 449. 

adius of 

U J 

-*-* 5 
cn 

e& — 

Steel. 






t» o 

c3 •— 
o> ^ 

a> r3 

•—» J-. 

+ $s 

Hard; .36 
per cent 
carbon 

Mild; .12 
per cent 
carbon 


Iron 

• 


Iron. 

r— 

+ 

bO 

-♦j 

to 

G 

o> 

"8> 

a 

o 

A 

Flat 

ends 

Flat 

ends 

Fixed 

ends 

Flat 

ends 

Hinged 

ends 

Round 

ends 

Flat 

ends 

3 

17 

20 

100000 

70000 

•••••••• 

46000 

46000 

46000 

44000 

57200 

50400 

48000 

3 

17 

20 

30 

74000 

51000 

43000 

43000 

43000 

40250 

40000 

30 

40 

62000 

40000 

40000 

40000 

40000 

36500 

37000 

40 

50 

60000 

44000 

38000 

38000 

38000 

33500 

37000 

50 

60 

58000 

42000 

36000 

36000 

36000 

30500 

37000 

60 

70 

55500 

40000 

34000 

34000 

33750 

27750 

37000 

70 

80 

53000 

38000 

32000 

32000 

31500 

25000 

36000 

80 

90 

49700 

36000 

31000 

30900 

29750 

22750 

35000 

90 

100 

46500 

34000 

30000 

29800 

28000 

20500 

35000 

100 

120 

40000 

30000 

28000 

26300 

24300 

16500 

34500 

120 

140 

33500 

26000 

2.5500 

23500 

21000 

12800 


140 

160 

28000 

22000 

23000 

20000 

16500 

9500 


160 

200 

19000 

14800 

17500 

14500 

10800 

6000 


200 

300 

8500 

7200 

9000 

7200 

5000 

2800 


300 


The following simple formula, by Mr. D. J. Whittemore, was found t 
agree very closely with the results of the experiments on Phoenix columns :f 

Breaking load in lbs 525000 

per sq inch of area = [(1200 — H) X 30] -f _- a - 
of cross section of pillar 


where H — 


length of pillar 
diam D, fig p 449 


both in the same unit. 




See also p 443. 

Mr. Christie* adopts the following formula for obtaining the proper faeto 
of safety for pillars of wrought iron or steel: 


For flat and fixed ends, 


For binged and round ends, 


Factor of safety = 3 -f ( .01 ;-—) 

\ least rad of gvr/ 

Factor of safety = 3 -f (.015 - - .. -] 

\ least rad of gyrj 


It will be noticed that the factor of safety, as found by these formula, i 
creases with the ratio of the length of the pillar to the least, radius of gvratit 
of its cross section; and is creater for round and hinged ends than for flat ai 
fixed ends. See foot of p 446. 

- : --- 


* See “■ Wrought Iron ami Steel in Construction”, by Pencoyd Iron "Works ; published by Jo 
Wiley k Sons, New Yorft, 1884. 

t See Transactions, American Society ol Civil Engineers; Jan, Feb and March 1882. 

























































STRENGTH OF IRON PILLARS 


443 


»ee!i»n stri '!'K tU * lbs per sq inch of metal 

m d JdnrSf hv Chi ^ 7 Wr c M, ?T 1 '\* rwn i >i,,ars below. These formulas 
aro-p niiior« r /£ h ri' 4 Shale . r Sn ? lth > from many tests by G. Bouscaren, C. E„ of 

ulf ones hv F f t S m n rX ^ a “ rp The lower TabIe is an abridgment of the 

uJl ones by C. L. Gates, C. E., in the Trans. Am. Soc. C. E., Oct., 1880. 

i _ length between end bearings 

least diameter d both in the same measure; and is to be squared, 
lor safety take irorn X / A to according to circumstances. 





llltimate and sa i® l<»a«ls in lbs per sq inch, of the above four pillars, with 
at ends, and equally loaded. Coef of Safety = 4 + .05 II. By C. L. Gates, C. E. 


. 

i H 

A. Square Col. 

B. Phoenix Col 

C. American Col. 

f 

Ult. 

Safe. 

Ult. 

Safe. 

Ult. 

Safe. 

5 

37067 

7822 

40476 

8521 

34434 

7249 

6 

86876 

7683 

40212 

8377 

34167 

7118 

8 

36470 

7443 

39645 

8091 

33597 

6856 

0 

36024 

7205 

39030 

7806 

32982 

6596 

2 

35544 

6970 

38373 

7524 

32327 

6338 

o 

34767 

6622 

37317 

7110 

31285 

5959 

0 

33344 

6063 

35424 

6440 

29435 

5352 

5 

31806 

5531 

33406 

5810 

27512 

4789 

0 

30198 

5033 

31352 

5226 

25584 

4264 

5 

28562 

4570 

29310 

4690 

23701 

3792 

0 

26932 

4143 

27321 

4203 

21900 

3369 

5 

25333 

3728 

25415 

3765 

20203 

3004 

0 

23787 

3398 

23611 

3373 

18621 

2660 


D. Common Col. 


Ult. 

83693 

33339 

32589 

31790 

30952 

29639 

27375 

25108 

22919 

20857 

18952 

17214 

15643 


Safe. 

7093 

6946 

6651 

6358 

6069 

5646 

4977 

4367 

3820 

3337 

2916 

2550 

2235 

































































































444 


STRENGTH OF IRON PILLARS 


TABLE OF 


BREAKING LOADS OF IRON PILLARS, 


in ton8 per square inch of metal area. Deduced from Gordon. The ends are 
supposed to be planed to form perfectly true bearings; and all parts to be equally 
pressed. The last is rarely the case in practice. If the pillar is rectang - **' 
lar instead of square, use the least side for a measure of length. (Original.), 


Hollow 


Hollow 


Round. 


Breakg loads per 
sq inch of metal 
area of 

transverse section. 


Length 
measd 
iu sides 
or 

diarns. 


Square. 


Breakg loads per 
sq inch of metal 
area of 

transverse section. 


Solid 

Round. 


Breakg loads per 
sq inch of metal 
area of 

transverse section. 


Length 
measd 
in sides 
or 

diarns. 


Solid 

Square. 


Breakg loads per t 
sq inch of metal 
area of 

transverse section. 


Cast. 

Wrt. 


Cast. 

Wrt. 

Cast. 

Wrt. 


Cast. 

Wrt. 

Tons. 

Tons. 


Tons. 

Tons. 

Tons. 

Tons. 


Tons. 

Tons. 

35.7 

16.1 

1 

35 7 

16.0 

35.6 

16.1 

1 

35.6 

16.1 

35.5 

16.1 

2 

35.5 

16.0 

35.2 

16.1 

2 

35.4 

16.0 

35.2 

16.1 

3 

35.3 

16.0 

34.6 

16.0 

3 

34.9 

16.0 

31.8 

16.0 

4 

35.0 

16.0 

33.9 

15.9 

4 

34.3 

16.0 

34.3 

16.0 

5 

34.6 

16.0 

33.0 

15.9 

5 

33.6 

16.0 

33.7 

16.0 

6 

34.2 

16.0 

31.9 

15.8 

6 

32.8 

15.9 

33.0 

16.9 

7 

33.7 

16.0 

30.6 

15.7 

7 

31.8 

15.8 

32.3 

13.8 

8 

33.1 

15.9 

29.3 

15.6 

8 

30.8 

15.7 

31.5 

15.8 

9 

32.4 

15.9 

28.0 

15.5 

9 

29.7 

15.7 

30.6 

15.7 

10 

31.7 

15.8 

26.7 

15.4 

10 

28.6 

15.6 

29.7 

15.6 

11 

31.0 

15.8 

25.4 

15.2 

11 

27.4 

15.5 

28.8 

15.5 

12 

30.3 

15.7 

24.1 

15.1 

12 

26.3 

15.3 

27.9 

15.5 

13 

29.5 

15.7 

22.8 

14.9 

13 

25.1 

15.2 

27.0 

15.4 

14 

28.7 

15.6 

21.6 

14.8 

14 

24.0 

15.1 

26.0 

15.3 

15 

27.9 

15.5 

20.5 

14.6 

15 

22.9 

15.0 

25.1 

15.2 

16 

27.1 

15.5 

19.4 

14 4 

16 

21.8 

14 8 

24.2 

15.1 

17 

26.3 

15.4 

18.3 

14.2 

17 

20.7 

14.7 

23.3 

15.0 

18 

25.4 

15.3 

17.2 

14.0 

18 

19.7 

14.5 

22.3 

14.9 

19 

24.6 

15.2 

16.2 

13.8 

19 

18.8 

14.3 

21.4 

14.8 

20 

23.8 

15.1 

15.3 

13.6 

20 

17.9 

14.2 

20.6 

14.7 

21 

23.0 

15.0 

14.5 

13.4 

21 

17.0 

14.0 

19.8 

14.6 

22 

22.2 

14.9 

13.7 

13.2 

22 

16.2 

13.8 

19.0 

14.4 

23 

21.5 

14.8 

12.9 

13.0 

23 

15.4 

13.6 - 

18.3 

14.3 

24 

20.8 

14.7 

12.2 

12.8 

24 

14.6 

13.5 

17.5 

14.2 

25 

20.1 

14.6 

11.6 

12.6 

25 

13.9 

13.3 

16.8 

14.0 

26 

19.4 

14.5 

11.0 

12.4 

26 

13.3 

13.1 

16.2 

13.9 

27 

18.7 

14.4 

10 4 

12.1 

27 

12.7 

12.9 

15.5 

13.7 

28 

180 

14.3 

9.90 

11.9 

28 

12.1 

12.7 „ 

14.9 

13.5 

29 

17.4 

14.2 

9.41 

11.7 

29 

11.5 

12.6 

14.3 

13 4 

30 

16.8 

14.0 

8.93 

11.5 

30 

11.0 

12.4 'f 

13.8 

13.2 

31 

16.3 

13.9 

8.50 

11.3 

31 

10.5 

12.2 

13.2 

13.1 

32 

15.7 

13.8 

8.10 

11.1 

32 

10.0 

12.0 

12.7 

12.9 

33 

15.2 

13.6 

770 

10.8 

33 

9.59 

11.8 

12 3 

12.8 

34 

14.6 

13.5 

7.36 

10.6 

34 

9.18 

11.6 

11.7 

12.6 

35 

14.1 

13.3 

704 

10.4 

85 

8.79 

11.4 

11.3 

12.5 

36 

13.6 

13.2 

6.71 

10.2 

36 

8.42 

11.2 

10.9 

12.3 

37 

13.2 

13.1 

6.41 

10.0 

37 

8.07 

11.0 

10.5 

12.2 

38 

12.7 

13.0 

6.11 

9.79 

38 

7.75 

10.8 

10.1 

12.0 

39 

12.3 

12.9 

5.85 

9.59 

39 

7.44 

10.7 

9.75 

11.9 

40 

11.9 

12.7 

5.62 

9.39 

40 

7.14 

10.5 

9.40 

11.7 

41 

11.5 

12.6 

5.40 

9.20 

41 

6.86 

10 3 

9.07 

11.6 

42 

111 

12.4 

5.19 

9.00 

42 

6.60 

10.1 

8.76 

11.4 

43 

10.8 

12.3 

4.99 

8.81 

43 

6.35 

9.95 

8.46 

11.3 

44 

10.4 

12.2 

4.79 

8.64 

44 

6.11 

9.77 

8.16 

11.1 

45 

10.1 

12.0 

4.59 

8.46 

45 

5.89 

9.59 

7.88 

10.9 

46 

9.80 

11.9 

4.42 

8.27 

46 

5.68 

9.42 

7.61 

10.8 

47 

9.50 

11.7 

4.26 

8.10 

47 

5.48 

9.25 

7.36 

10.6 

48 

9.20 

11.6 

4.11 

7.94 

48 

5.28 

9.09 

7.13 

10.4 

49 

8.94 

11.4 

3.97 

7.76 

49 

5.10 

8.92 

6.91 

10.3 

50 

8.66 

11.3 

3.84 

7.60 

50 

4.92 

8.77 

5.90 

9.61 

55 

7.47 

10.7 

3.22 

6.86 

65 

4.16 

8.00 

5.10 

8.92 

60 

6.49 

10.0 

2.75 

6.18 

60 

3.57 

7.30 

4.44 

8.29 

65 

5.68 

9.43 

2.37 

5.59 

65 

3.09 

6.671 

3.90 

7.69 

70 

5.01 

8.84 

2.06 

5.06 

70 

2.70 

6 10 

3.44 

7.14 

75 

4.44 

8.30 

1.80 

4.59 

75 

2.37 

5.59.: 

3.08 

6.64 

80 

3.97 

7.77 

1.59 

4.18 

80 

2.10 

5.13 

2.46 

5.73 

90 

3.21 

6.88 

1.27 

3.49 

90 

1.68 

4.34 

2.02 

4.99 

100 

2.65 

6.02 

1.04 

2.95 

100 

1.37 

3.71: 


















































STRENGTH OF IRON PILLARS, 


445 


Table 1. HOLLOW CYLIND CAST IRON PILLARS. 

Breaking; loads, flat, ends, perfectly true,and firmly fixed; 
imi the loads pressing; equally on every part of the top. 

!y Gordon’s formula. 

For dianss or lengths intermediate of those in the table, the 

>ads may be found near enough by simple proportion. 

For thicknesses less than those in the table, the breaking loads 

lay safely be assumed to diminish in the same proportion as the thickness, while the outer diam 
.•mains the same. But for greater thicknesses than those in the table, the loads do not increase 
s rapidly as the new thickness. Still, in practice, they may be assumed to do so approximately, 

f the new thickness does not exceed about % part of the 
>uter diam. 


, 9 

S'" 


CAST IKON. 

THICKNESS yi INCH. (Original.) 


ngth in 
feet. 

Outer Diameter in inches. 

-3 

2 

2* 

2H 


3 

S X , 

4 

4« 

5 

5^ 

6 



Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


1 

45 2 

52.5 

59.8 

66.8 

74.1 

88.5 

103.0 

117.1 

131.3 

145.5 

159.8 

1 

2 

36.2 

43.4 

51.0 

58.6 

66.5 

81.5 

96.7 

111.3 

125.9 

140.5 

155.2 

2 

3 

27.2 

34.1 

41.4 

48.8 

56.7 

71.9 

87.6 

102.7 

117.8 

132.9 

148.1 

3 

4 

20.2 

26.3 

32.9 

39.7 

47.0 

61.9 

77.5 

92.9 

108.3 

123.7 

139.1 

4 

5 

15.1 

20.2 

25.9 

32.0 

38.6 

52.6 

67.4 

82.6 

97.9 

113.4 

129.1 

5 

6 

11.6 

15.8 

20.7 

25.9 

31.6 

44.3 

58.2 

72.7 

87.7 

103.0 

118.7 

6 

7 

9.1 

12.5 

16.6 

21.0 

26.1 

37.4 

50.1 

63.7 

78.1 

92.9 

108.3 

7 

8 

7.3 

10.1 

13.5 

17.3 

21.7 

31.6 

43.2 

55.8 

69.3 

83.4 

98.4 

8 

9 

5.9 

8.2 

11.1 

14.3 

18.2 

26.9 

37.3 

48.8 

61.5 

74.9 

89.2 

9 

10 

4.9 

6.9 

9.3 

12.0 

15.4 

23.1 

32.4 

42.9 

54.6 

67.1 

80.7 

10 

11 

4.1 

5.8 

7.9 

10.3 

13.2 

20.0 

28.3 

37.8 

48.6 

60.3 

73.0 

11 

12 

3.5 

5.0 

6.8 

8.9 

11.4 

17.4 

24.9 

33.5 

43.3 

54.2 

66.1 

12 

13 

3.0 

4.2 

5.8 

7.6 

9.9 

15 2 

21.9 

29.7 

38.8 

48.8 

60.0 

13 

14 

2.6 

3.6 

5.1 

6.7 

8.7 

13.5 

19.5 

26.6 

34.9 

44.1 

54.5 

14 

15 

2.3 

3.2 

4.5 

6.0 

7.7 

11.8 

17.4 

23.9 

31.4 

39.9 

49.7 

15 

16 

2.0 

2.8 

4.0 

5.3 

6.9 

10 7 

15.6 

21.5 

28.4 

36.4 

45.6 

16 

18 

1.6 

2.3 

3.2 

4.2 

5.5 

8.6 

12.7 

17.5 

23.4 

30.2 

38.0 

18 

20 

1.3 

1.8 

2.6 

3.4 

4.5 

7.1 

10.5 

14.7 

19.7 

25.6 

32.3 

20 

25 

...... 

••••... 

1.7 

2.3 

3.0 

4.7 

7.0 

9.8 

13 3 

17.6 

22.5 

25 

30 

..... • 

...... • 

1 2 

1.6 

2.1 

3.2 

50 

7.0 

9.4 

12.6 

16.1 

30 

35 





1.5 

2.4 

3.7 

5.2 

7.1 

9.4 

12.2 

35 

40 


. 



1.2 

1.9 

2.8 

4.0 

5.4 

7.2 

9.5 

40 


Weight of 1 foot of length of pillar in pounds. 

4.31 | 4.91 | 5.53 | 6.13 | 6.75 | 7.97 1 9.22 | 10.4 | 11.7 | 12.9 | 14.1 


Area of ring of solid metal in square inches. 

1.38 | 1.57 | 1.77 | 1.96 | 2.16 l 2.55 1 2.95 | 3.34 | 3.73 1 4.12 | 4.52 


3 

+ . 

CAST IRON. THICKNESS X INCH (Original.) 

ength in 
feet. 

Outer Diameter m inches. 

3 

5 

0% 

6 


7 

7>4 

8 

8X 

9 

10 

11 

12 

►3 


Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


2 

239 

268 

297 

325 

354 

383 

412 

440 

469 

526 

583 

640 

2 

4 

205 

235 

266 

296 

326 

356 

387 

416 

445 

504 

563 

622 

4 

6 

166 

196 

227 

257 

288 

319 

350 

380 

410 

471 

532 

593 

6 

8 

131 

159 

188 

218 

248 

279 

310 

340 

371 

432 

494 

557 

8 

10 

103 

127 

154 

182 

210 

240 

270 

300 

330 

391 

454 

517 

10 

12 

82 

103 

126 

151 

177 

205 

233 

262 

292 

351 

413 

475 

12 

14 

66 

84 

104 

126 

149 

174 

200 

227 

255 

313 

372 

434 

14 

16 

54 

69 

87 

106 

126 

149 

173 

198 

224 

277 

334 

394 

16 

18 

45 

58 

73 

90 

108 

128 

149 

172 

196 

246 

300 

357 

18 

20 

37 

48 

62 

77 

93 

111 

130 

151 

173 

219 

270 

323 

20 

22 

32 

41 

53 

66 

80 

96 

113 

132 

151 

194 

241 

292 

22 

25 

25 

34 

43 

54 

65 

79 

93 

109 

126 

164 

206 

252 

25 

30 

18 

24 

31 

39 

48 

59 

70 

83 

96 

126 

160 

199 

30 

35 

13 

18 

24 

30 

36 

44 

53 

63 

73 

98 

126 

459 

35 

40 

10 

14 

18 

23 

28 

35 

42 

50 

59 

78 

102 

129 

40 

45 

8 

11 

15 

19 

23 

28 

34 

41 

48 

64 

84 

107 

45 

50 

6 

9 

12 

15 

19 

23 

28 

33 

39 

53 

70 

89 

50 

60 

4 

6 

9 

11 

14 

17 

20 

24 

28 

38 

50 

65 

60 

70 

3 

4 

6 

8 

10 

12 

15 

18 

21 

29 

38 

50 

70 

80 

3 

4 

5 

6 

8 

9 

11 

13 

16 

22 

30 

38 

80 


Weight of 1 foot of length of pillar, in pounds. 

22.1 | 24.5 | 27.0 | 29.4 | 31.9 | 34.4 | 36.9 | 39.4 | 41.9 | 46.6 | 51.6 | 56.6 


Area of ring of solid metal, in square inches. 

7.071 7.85 8.64 I 9.43 | 10.2 | 11.0 | 11.8 i 12.6 | 13.4 | 14.9 I 16.5 | 18.1 




















































































446 


STRENGTH OF IRON PILLARS, 


Table 1. HOLLOW CYLIND CAST IRON PILLARS. 

BREAKING LOADS.—(Continued.) • By Gordon’s Rule. 


J3 U 
tC , 


CAST IRON. THICKNESS 1 INCH. (Original.) 


0 ■*- 
<v 

12 

13 

14 

15 

16 


Tons. 

Tons. 

Tons. 

Tons. 

Tons 

4 

1188 

1301 

1415 

1530 

1645 

6 

1138 

1253 

1368 

1484 

1601 

8 

1065 

1184 

1303 

1423 

1543 

10 

989 

1110 

1231 

1355 

1475 

12 

909 

1030 

1152 

1275 

1399 

14 

829 

949 

1071 

1195 

1320 

16 

756 

873 

992 

1114 

1237 

18 

683 

796 

913 

1034 

1155 

20 

618 

727 

840 

958 

1077 

22 

559 

663 

772 

887 

1002 

24 

508 

606 

709 

818 

929 

26 

459 

553 

651 

756 

863 

28 

418 

506 

598 

697 

800 

30 

380 

462 

549 

644 

743 

32 

347 

424 

506 

595 

689 

34 

318 

390 

467 

552 

641 

36 

292 

359 

432 

511 

596 

38 

268 

331 

400 

475 

556 

40 

247 

305 

370 

441 

518 

42 

229 

283 

344 

411 

484 

44 

212 

263 

321 

363 

452 

46 

197 

246 

299 

358 

423 

48 

183 

229 

280 

335 

397 

50 

170 

213 

261 

314 

373 

55 

144 

181 

222 

269 

320 

60 

124 

157 

192 

233 

278 

65 

105 

134 

165 

202 

242 

70 

93 

118 

146 

178 

213 

80 

73 

92 

113 

139 

168 

90 

58 

73 

91 

112 

136 

100 

48 

60 

75 

92 

112 


Outer Diameter in Inches. 


18 


Tons, 

1874 

1833 

1779 

1716 

1644 

1566 

1484 

1401 

1320 

1241 

1163 

1090 

1020 

954 

893 

836 

783 

734 

687 

645 

605 

569 

536 

505 

438 

383 

335 

297 

235 

191 

157 


20 


Tons. 

2103 

2066 

2015 

1957 

1889 

1813 

1733 

1651 

1568 

1486 

1404 

1326 

1250 

1178 

1110 

1046 

984 

928 

874 

825 

776 

734 

694 

656 

573 

503 

443 

394 

315 

257 

213 


22 

24 

27 

30 

Tons. 

Tons. 

Tons. 

Tons. 

2330 

2557 

2896 

3236 

2295 

2525 

1 2866 

3208 

2247 

2479 

2824 

3170 

2193 

2430 

1 2780 

3128 

2129 

2369 

2723 

3076 

2056 

2300 

2659 

3016 

1979 

2226 

1 2589 

2951 ‘ 

1899 

2147 

2515 

2879 

1817 

2066 

2437 

2805 

1734 

1982 

2356 

2726 1 

1650 

1899 

2272 

2644 I 

1570 

1816 

2188 

2560 

1489 

1733 

2103 

2475 

1412 

1H53 

2020 

2392 

1338 

1575 

1938 

2308 

1268 

1500 

1857 

2225 

1199 

1427 

1779 

2143 

1136 

1361 

1704 

2063 

1076 

1296 

1630 

1984 

1019 

1232 

1560 

1909 

963 

1169 

1491 

1834 

915 

1114 

1428 

1765 

869 

1060 

1367 

1696 

824 

1008 

1306 

1627 

725 

893 

1172 

1474 

641 

795 

1054 

1333 

568 

709 

951 

1211 

507 

636 

853 

1099 

408 

516 

705 

914 

335 

426 

586 

768 

279 

356 | 

491 

651 


36 


108 | 


Weight of one foot of length of pillar, in pounds. 

118 | 128 | 138 | 147 | 167 | 187 | 206 | 226 I 255 I 


Tons 

3915 

3890 

3860 

3823 

3778 

3726 

3668 

3604 

3536 

3464 

3387 

3308 

3226 

3143 

3059 

2974 

2889 

2804 

2720 

2636 

2552 

2474 

2396 

2319 

2135 

1964 

1808 

1664 

1414 

1209 

1040 


6 

8 

10 
12 
14 . 
16 
18 
20 
22 
24 
26 
28 
30 
32 
34 


36 

38 

40 

42 

44 

46 

48 

50 

55 

60 

65 

70 

80 

90 

100 


34.6 | 


37 7 I J 4 n e a a ° 1 f -^ 0 } i< ? o metal - in square inches. 

37.7 I 40.8 1 44.0 | 4,. 1 | p3,4 | 59.7 | 66.0 j 72.2 | 81.7 | 


285 | 344 


91.1 | 110.0 


3 g 

be w 

a 


10 

15 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

125 

150 


CAST IRON. THICKNESS TlNCHES (Original.) - 


Outer Diameter in Feet. 


3 


Tons. 

10806 

10453 

9996 

8884 

7688 

6554 

5553 

4704 

3998 

3417 

2940 

2547 

1950 

1532 


3% 


Tons. 

12878 

12564 

12150 

11102 

9907 

8702 

7575 

6570 

5699 

4953 

4321 

3788 

2954 

2350 




Tons. 

14908 

14628 

14250 

13275 

12113 

10888 

9690 

8576 

7570 

6683 

5909 

5238 

4400 

3353 


Tons. 

16967 

16712 

16369 

15459 

14344 

13126 

11892 

10702 

9595 

8588 

7665 

6888 

5864 

4547 


Tons. 
18986 
18754 
18440 
17593 
16531 
15341 
14100 
12870 
11692 
10594 
9588 
8677 
7483 
5900 


5^ 


Tons. 

21038 

20825 

20533 

19743 

18735 

17580 

16349 

15097 

13873 

12706 

11614 

10606 

9257 

7417 


Tons. 

23052 

22855 

22585 

21847 

20891 

19780 

18570 

17319 

16070 

14856 

13700 

12613 

11132 

9058 


972 | 


Weight 

1150 | 1325 I 


Tons. 

27143 

26972 

26737 

26084 

25224 

24107 

23049 

21824 

20566 

19304 

18065 

16869 

14650 

12701 


8 


Tons. 

31196 

31045 

30837 

30257 

29476 

28532 

27460 

26287 

25048 

23791 

22521 

21270 

19448 

16668 


10 


Tons. 

39254 

39133 

38962 

38486 

37838 

37037 

36103 

35058 

33924 

32725 

31481 

30213 

27664 

25182 


12 


Ton 8 . 
47375 
47273 
47131 
46729 
46175 
45484 
44666 
43737 


beg 

a C 


10 

15 

20 

30 

40 

50 

60 

70 


42712 I 80 


of one 

1503 | 


foot of length of pillar, in pounds. 

_1678J_1856 | 2031 | 2388 | 2740 I 3444 


41670 

40438 

39220 

36692 

34127 


90 

100 

110 

125 

150 


I 4153 


Coefficients of safety foe hollow east iron pillars. 


^. w,; should not take the safe load at morethan n end bearin ?s, 

‘°? ’' f . the P ,llars ar « roughly cast, aud the end^ not nerfecUv *r 0 fne i ^ °, f . Hod Arson's breakin, 
when they are so, and the loads about equally distributed ^ P ine<i aud ad J us ted; aud one-fiftl 














































































































































gth in 


STRENGTH OF IRON PILLARS. 


447 


HOLLOW CYIilNTORICAL, WROUGHT IRON PILLARS. 

Table 4, of breaking loads in tons of hollow cylindrical 
y rought iron pillars, with flat ends, perfectly true, and 
irmly fixed, and the loads pressing equally on every part 
if the top. Calculated by Gordon’s formula. No pains have been taken to have 
he last figure of the loads perfectly correct in every case. 

_(Original.) 


a 

WROUGHT IRON. THICKNESS X INCH. 

2 

-d . 

-*-* -4J 

bf) <d 

Outer diameter in inches, 

JS 

S £ 


% 

1 

IX 

IX 

IX 

2 

234 

2 X 

m 

3 

*—s 





BREAKING LOAD 







Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


1 

3.64 

5.27 

6.88 

8.50 

10.1 

11.7 

13.2 

14.8 

16.4 

18.0 

1 

2 

2.94 

4.64 

6.32 

8.00 

9.6 

11.2 

12.8 

14.5 

16.1 

17.8 

2 

3 

2.30 

3.86 

5.57 

7.28 

8.9 

10.6 

12.2 

13.9 

15.6 

17.3 

3 

4 

1.77 

3.13 

4.74 

6.36 

8.1 

9.9 

11.6 

13.3 

15.0 

16.7 

4 

5 

1.36 

2.51 

4.07 

5.66 

7.3 

9.1 

10.8 

12.5 

14.2 

J6.0 

5 

6 

1 04 

2.03 

3.46 

4.91 

6.6 

8.3 

9.9 

11.6 

13.4 

15.2 

6 

7 

.81 

1.65 

2.91 

4.24 

5.7 

7.4 

9.1 

10.8 

12.6 

14.4 

7 

8 

.61 

1.36 

2.46 

3 67 

5.1 

6.7 

8.3 

9.9 

11.7 

13.5 

8 

9 

.50 

1.05 

2.03 

3.18 

4.5 

6.0 

7.5 

9.1 

10.8 

12.6 

9 

10 

.41 

.95 

1.75 

2.77 

4.0 

5.4 

6 9 

8 4 

10.1 

11.8 

10 

11 

.34 

.81 

1.52 

2.41 

3.6 

4.8 

6.2 

7.7 

9.3 

11.0 

i: 

12 

.29 

.70 

1.34 

2.14 

3.2 

4.3 

5.6 

7.0 

8.6 

10.2 

12 

13 

.24 

.60 

1.16 

1.88 

2.8 

3.9 

5.2 

6.5 

8.0 

9.5 

13 

14 

.21 

.53 

1.03 

1.69 

2.5 

3.5 

4.7 

6.0 

7.4 

8.9 

14 

15 

.19 

.47 

.91 

1.50 

2.3 

3.2 

4.3 

5.5 

6.9 

8.3 

15 

16 

.18 

.42 

.84 

1.38 

2.1 

2.9 

4.0 

5.1 

6.4 

7.7 

16 

18 

.14 

.33 

.67 

1.11 

1.7 

2.4 

3.4 

4.4 

5.6 

6.8 

18 

20 


.27 

.55 

.91 

1.4 

2.0 

2.8 

3.7 

4.7 

5.8 

20 

25 



.. • 

. 

.9 

1.4 

2.0 

2.6 

3.4 

4.2 

25 



Weight of one foot of length of pillar, in pounds. 




.820 

1.15 

1.47 

| 1.80 

| 2.13 

| 2.45 

2.78 

| 3.11 

| 3.43 

) 3.77 




Area of ring of solid metal, in 

square inches. 





.246 

.344 | 

.442 

| .540 

| .638 

| .736 

| .835 

j .933 

1.03 

i 1.13 




1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 
18 
20 
25 
30 
35 
40 
45 
50 


Weight of one foot of length of pillar, in pounds. 

4.60 | 5.23 | 5.90 [ 6.53 | 7.20 | 8.50 | 9.83 j 11.1 | 12.4 | 13.7 | 15.0 



WROUGHT 

IRON 

THICKNESS X INCH. 



3 




Outer diameter in inches. 
















a v- 

2 

2X 

2X 

2X 

3 

1 3X 

1 4 

iX 

\ 5 

\ $x 

1 6 

o> 

♦J 





BREAKING LOAD. 






Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


21.9 

25.4 

28.3 

31.4 

34.5 

40 

47 

53 

60 

66 

72 

i 

21.1 

24.3 

27.6 

30.7 

33.9 

40 

47 

53 

60 

66 

72 

2 

19.9 

23.1 

26.4 

29.7 

33.0 

39 

46 

52 

59 

65 

71 

3 

18.6 

21.8 

25.3 

28.5 

31.9 

38 

45 

51 

58 

64 

71 

4 

17.0 

20.4 

23.5 

27.3 

30.7 

37 

44 

50 

57 

63 

70 

5 

15.4 

18.8 

22.1 

25.7 

29.2 

36 

43 

49 

56 

62 

69 

6 

13.9 

17.3 

20.5 

23.8 

27.8 

34 

41 

47 

54 

61 

68 

7 

12.5 

15.6 

19.1 

22.3 

25.9 

32 

40 

46 

53 

60 

67 

8 

11.2 

14.2 

17.5 

20.6 

24.3 

30 

38 

44 

51 

58 

65 

9 

10.0 

13.0 

16.1 

19.1 

22.7 

29 

37 

43 

50 

57 

64 

10 

9.0 

10.7 

15.7 

17.6 

21.1 

27 

35 

41 

48 

55 

62 

11 

8.1 

10.6 

13.5 

16.4 

19.6 

26 

33 

40 

46 

54 

61 

12 

7.3 

9.6 

12.4 

15.1 

18.2 

24 

31 

38 

44 

52 

59 

13 

6.6 

8.8 

11.3 

14.0 

17.0 

23 

30 

36 

43 

51 

57 

14 

6.0 

8.0 

10.4 

12.9 

15.8 

21 

28 

34 

41 

49 

55 

15 

5.5 

7.3 

9.5 

12.0 

14.6 

20 

27 

83 

40 

47 

54 

16 

4.5 

6.0 

8.0 

10.3 

12.7 

18 

24 

30 

37 

43 

50 

18 

3.8 

5.1 

6.8 

8.7 

11.0 

16 

21 

27 

34 

40 

47 

20 





7.9 

12 

16 

91 

97 

33 







13 

17 

22 

27 

32 

30 







10 

14 

IB 

22 

27 

35 







14 

IB 

23 

40 









11 

15 

19 

45 









8 

12 

16 

50 


1.38 ' 1.57 


Area of ring of solid metal, in square inches, 
j 1.77 ! 1.96 | 2.16 | 2.55 | 2.95 | 3.34 i 3.73 I 4.12 1 4.51 









































































448 


STRENGTH OF IRON PILLARS. 


HOLLOW CTUNDRICAL WROUGHT IRON PILLARS. 


Table 4, (Continued.) (Original.) 


* 


2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

25 

30 

35 

40 

45 

50 

60 

70 

80 

90 

100 


WROUGHT IRON. THICKNESS J* INCH. 


Outer diameter in inches. 


5 I | 6 | 6% | 7 | 7J4 | 8 [ | 9 | 10 | 11 j 12 


BREAKING LOAD. 


Tons. 

112 

110 

106 

101 

95 

89 

82 

76 

70 

61 

58 

52 

42 

34 

27 

23 

19 

15 

11 

9 

7 

6 


Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

125 

139 

152 

166 

177 

189 

201 

214 

238 

263 

123 

136 

149 

163 

174 

186 

199 

212 

237 

262 

119 

132 

145 

158 

171 

184 

197 

210 

235 

261 

114 

127 

140 

154 

167 

181 

194 

207 

232 

258 

108 

123 

136 

149 

162 

176 

189 

203 

228 

254 

102 

116 

129 

143 

157 

171 

185 

199 

224 

250 

95 

108 

122 

137 

151 

165 

179 

194 

219 

245 

89 

103 

117 

131 

145 

160 

173 

187 

213 

240 

83 

97 

110 

124 

138 

153 

166 

180 

207 

235 

77 

91 

104 

117 

131 

145 

159 

173 

201 

227 

70 

83 

96 

109 

123 

138 

151 

165 

192 

220 

64 

76 

89 

102 

115 

129 

143 

157 

183 

212 

52 

63 

74 

87 

100 

113 

127 

141 

167 

195 

43 

53 

64 

75 

87 

99 

112 

125 

151 

178 

35 

44 

53 

64 

75 

86 

98 

110 

135 

163 

30 

38 

46 

55 

65 

76 

87 

98 

123 

148 

24 

32 

38 

47 

56 

66 

76 

87 

109 

133 

19 

24 

29 

36 

43 

51 

60 

69 

88 

109 

14 

18 

23 

28 

34 

40 

48 

56 

73 

91 

11 

14 

18 

22 

27 

32 

37 

44 

57 

74 

9 

u 

14 

18 

22 

26 

31 

36 

49 

63 

7 

9 

12 

15 

18 

22 

26 

30 

41 

53 


Weight of one foot of length of pillar, in pounds. 
23.6 | 26.2 | 28.8 [ 31.4 | 34.0 | 36.6 | 39.3 | 42.0 | 44.7 | 49.7 | 55.0 


Area of ring of solid metal, in square inches. 

7.07 | 7.85 | 8.64 | 9.43 | 10.2 | 11.0 | 11.8 | 12.6 | 13.4 | 14.9 


Tons. 

290 

289 

288 

284 

280 

276 

272 

268 

263 

257 

250 

241 

224 

207 

190 

174 

158 

132 

111 

93 

78 

66 


to 5 
c *- 
S 
►J 


2 
4 
6 
8 
10 
12 
14 
16 
18 
20 
22 
25 
30 
35 
40 
45 • 
50 
60 
70 
80 
90 
100 


60.3 


16.5 | 18.1 


to . 
c- 

a. 


1 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

125 

150 

175 

200 


Tons. 

603 

588 

543 

479 

415 

355 

300 

256 

215 

185 

157 

134 

111 

82 

62 

49 


126 


Table 4, (Continued.) (Original.) 
WROUGHT IRON. THICKNESS 1 INChT 


Outer diameter in inches. 


13 1 14 1 15 I 16 I 17 I 18 I 20 I 22 I 24 | 26 I 28 | 30 


Tons. 

653 

638 

595 

538 

470 

405 

348 

300 

255 

222 

190 

162 

135 

101 

78 

60 


Tons. 

704 

691 

651 

594 

528 

462 

400 

348 

298 

261 

225 

193 

162 

122 

95 

74 


Tons. 

753 

742 

702 

645 

584 

516 

452 

398 

344 

303 

262 

227 

192 

145 

112 

89 


BREAKING LOAD. 

Tons. Tons. Tons. Tons. Tons. 

805 854 955 1056 

795 846 949 1049 

759 810 913 1016 

699 758 866 973 

636 691 806 912 

570 627 740 848 

505 559 669 781 

448 499 606 715 

392 440 543 649 

347 392 489 590 

303 845 436 532 

264 302 386 474 

225 259 336 416 

171 198 262 328 

133 155 208 266 

106 124 168 216 


1157 

1149 

1120 

1077 

1027 

961 

891 

824 

757 

694 

631 

568 

505 

405 

331 

269 


Tons. 

1257 

1248 

1223 

1186 

1130 

1067 

1005 

936 

868 

800 

735 

666 

598 

485 

400 

328 


Tons. Tons. 


1357 

1354 

1327 

1289 

1237 

1179 

1115 

1046 

978 

910 

843 

770 

697 

574 

478 

395 


1458 

1457 

1430 

1394 

1348 

1294 

1228 

1160 

1092 

1023 

955 

877 

799 

666 

560 

467 


3- 


Weight of one foot of length of pillar, in pounds. 

I 136 | 147 | 157 j 168 | 178 | 199 | 220 | 241 | 262 | 283 | 304 


l 

10 

•20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

125 

150 

175 

200 


37.7 


Area of ring of solid metal, in square inches. 

| 40.81 44,0 | 47.1 | 50.3 | 53.4 | 59. 7 | 66.0 | 72.3 | 78.5 | 84.8 | 91.1 


The breaking: loads for less Uiirkni>H«o« „ ,, , 

diminish at the same rate as the thickness. CKlieSSeS may safely be assumed to 






























































































STRENGTH OF IRON PILLARS. 


449 


Table of 


rolled-iron 

Iron 



G columns have 


segment-columns of the Phoenix 

do, 410 Walnut St, Philada. For their 
strengths, seepp 442, 443, orformula, p439, with the 
least radius of gyration, as given below, or as obtained 
by multiplying D by .3036. The dimensions given 
are subject to slight variations which are unavoidable 
in rolling iron shapes. The weights of columus given 
are those of the 4, 6, or 8 segments, of which they are 
composed. The shanks of the rivets used in joining them 
together, of course, merely make up tlie quantity of metal 
punched or drilled out, in making the holes; but the rivet- 
heads add from 2 to 5 per cent to the weights given. The 
rivets are spaced 3, 4, or 6 ins apart from cen to 
cen. Prices of the finished columns (188s;, from 4 to 
5% cts per pound, at the works, according to specifica¬ 
tions, and varying with the quantity ordered, the length, 
the thickness, and the amount of extra work required. 

Any desired thickness between the min and 
max for any given size, can be furnished. We give the 
dimensions, wts, &c, corresponding to the principal thick- 
8 segs. £, 6 segs. All others, 4 segs. 


• 

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Diameters, ins. 

One column. 

Size of 
Rivets. 

d 

D 

D/ 

Area 

of cross 
sec, 
sq ins. 

Wt per 
ft run, 
lbs. 

Least 
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X “ 



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5H 


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44 


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33.2 

110.6 

3.34 

3 

X “ 

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3% 

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56. 

4.18 

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% 

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12 


26.4 

88. 

4.36 

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44 

12% 

16* 

37.8 

126. 

4.55 

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i 

44 

13 

16% 

49.8 

166. 

4.73 

3 

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i A 

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13% 

17 A 

61.8 

206. 

4.91 

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X “ 

0 

A 


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24. 

80. 

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1 ./ 

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36. 

120. 

5.59 

2% 

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% 

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15% 

19% 

52. 

173.3 

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2% 

x% 

44 

1 

44 

16% 

20% 

68. 

226.6 

5.95 

3 

X “ 

“U% 

44 

17% 

21 

92. 

306.6 

6.23 

3% 

X “ 















































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454 ROLLED I BEAMS AS PILLARS. 


iilesoent breaking loads In short Ions (2000 lbs) of rolled I beams as pillars or struts, from Car- 
i, (Union Iron Mills, Pittsburgh. Penn,) “ Tables and Information on Wrought Iron.” Example 1 of use. The table 

‘SdofawShSS ron< Piar -T l f 3 / ft long ’, und fixed, just give way edgeways (in the direction of the length of 

,*?*£ ot 283 short tons; or sideways (across the web) with /8 short tons. And the same if it is 24 ft long, with one end fixed 

5 or if it is 16 ft long and both ends hinged. Example 2 of use. The table shows that if the first beam 32 ft 

re -!! ure abo - u . t 2 ?n ? hort to , I J s *° breuk lt , even sideways, which shows that if the 32 ft one is perfectly braced at 
against y lelding sideways, it will be equally strong in both directions. This principle of course applies to every load in the 
ee troni one-third to one-tenth of the breaking load may be used according to the case. * 

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Table of qi 

negie Bros & Cc 
shows that a hei 
the. web) with a 
and the other er 
long, had been t 
about every 7 ft 
table. In praeti 

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Heavy....9 

Light.9 

Heavy... .8 

Light .8 

Heavy... .7 
Light . 7 

Area. 

Sq 

Ins. 

CO 

r-4 O O CO LO -H- 
g »c CO ^ O 05 

Ths 
of web. 
Ins. 

tflHXOO-H 
X H (» lO O H 1 IO 

5® . 

bfl a: 

. 32 a 
< o M 

-r: O — CO CO — 
OOiO® CO CO GO 

*-H f-H 

x3 aJ . 

^ tfior! 

!S *33 a 

ift CO SO 

111 O -h lO X X ifi 

icj ifj tfj ^ '■** ^ 

J . 

*» *- M 

fe u£> 

OJ — 

o. 

r-» © O M tfO — 00 
CO IO CO ^ CO CO CO 

Depth. 

Ins. ! 

i 

Heavy....15 

Light.15 

Heavy.... 12 

Light.12 

Heavy... .10^ 

Light.10J^ 

Heavy....10 






























































































456 


ROLLED CHANNEL BAR PILLARS 


Table of quiescent breaking- loads in short tons (2000 lb* 

« r . w i e “ channel iron LJ , as pillars or struts. From Carnegi 

Bros <s Cog. (Union Iron Mills, Pittsburgh, Penn) “ Tables and Information on Wrought Iron.” 

For manner of use see the paragraph above the preceding table. Hgd, fxd, mean hinged, fixed. 

The headings 12 Hy, 12 mm. &c, mean 12 inch heavy, and 12 inch niediui ' 
channels. Sideway means across the web; Edgeway means along th , 

web. 


Length of Strut. 

12 

Hy. 

Both 


Both 

Side 

Edge 

ends 

1 fxd. 

ends 

Way. 

Wav. 

flxd. 

1 hgd. 

hgd. 

Sht 

Sht 

Ft. 

Ft. 

Ft. 

'Ions. 

Tons. 

i 



268 

270 

2 

1.5 

1 

260 

268 

4 

3. 

2 

235 

266 

6 

4.5 

3 

200 

264 

8 

6. 

4 

168 

262 

10 

7.5 

5 

138 

260 

12 

9. 

6 

114 

258 

14 

10 5 

7 

95 

256 

16 

12 

8 

79 

254 

18 

13.5 

9 

66 

253 

20 

15. 

10 

56 

252 

22 

16.5 

11 

48 

249 

24 

18. 

12 

41 

246 

26 

19.5 

13 

36 

243 

28 

21 . 

14 

32 

239 

30 

22.5 

15 

28 

235 

32 

24. 

16 

25 

230 

34 

25.5 

17 

23 

226 

36 

27. 

18 

21 

222 

38 

28 5 

19 

19 

218 

40 

30. 

20 

17 

213 




9 Hy. 

1 



160 

162 

2 

1.5 

1 

154 

161 

4 

3. 

2 

134 

160 

6 

4 5 

3 

110 

159 

8 

6. 

4 

90 

158 

10 

7.5 

5 

72 

157 

12 

9. 

6 

58 

154 

14 

10.5 

7 

47 

150 

16 

12. 

8 

38 

147 

18 

13.5 

9 

32 

143 

20 

15. 

10 

27 

140 

22 

16.5 

11 

23 

138 

24 

18. 

12 

20 

136 

26 

19.5 

13 

17 

134 

28 

21. 

14 

15 

131 

30 

22.5 

15 

13 

129 

32 

24. 

16 

12 

126 

34 

25.5 

17 

11 

122 

36 

27. 

18 

10 

118 

38 

28 5 

19 

9 

115 

40 

30. 

20 

8 

111 




7 I 


1 

1.5 


105 

106 

2 

1 

103 

106 

4 

3. 

2 

87 

105 

6 

4.5 

3 

70 

104 

8 

6 . 

4 

55 

104 

10 

7.5 

5 

43 

103 

12 

9. 

6 

34 

100 

14 

10.5 

7 

27 

98 

16 

12 . 

8 

23 

96 

18 

13.5 

9 

18.2 

94 

20 

15. 

10 

15.5 

92 

22 

16.5 

11 

13.5 

89 

24 

18. 

12 

12 

86 

26 

19.5 

13 

10.5 

83 

28 

21. 

14 

9 

80 

HO 

22.5 

15 

7.5 

77 

32 

24. 

16 

6.9 

75 

34 

25.5 

17 

6.2 

72 

36 

27. 

18 

5.5 

69 

88 

28.5 

19 

5.0 

66 

40 

30. 

20 

4.5 

63 


12 mm. 

10 

Hy. 

lO mm. 

i 

Side 

Edge 

Side 

Edge 

Side 

Edge 

f 

Way. 

Wav. 

W ay. 

Way. 

Way. 

Way 


Sht 

Sht 

Sht 

Sht 

Sht 

Sbt 


Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Ton* 

1 

160 

160 

187 

189 

121 

124 

I 

154 

160 

181 

189 

118 

123 


135 

160 

161 

188 

104 

123 


112 

160 

135 

187 

87 

122 


90 

159 

111 

186 

71 

122 


72 

159 

90 

185 

57 

121 


60 

159 

75 

183 

46 

121 


47 

158 

60 

181 

38 

120 


38 

156 

50 

178 

31 

118 


32 

155 

41 

176 

26 

116 


27 

153 

35 

174 

22 

115 


23 

151 

31 

171 

19 

113 


20 

149 

27 

167 

16 

111 


17 

147 

24 

163 

14 

109 


15 

145 

20 

159 

12 

107 


13 

143 

17 

156 

11 

105 


11 

140 

15 

153 

10 

103 


10 

138 

14 

149 

9 

100 


9 

136 

13 

146 

8 

98 


8 

134 

11 

143 

7 

96 


7 

132 

10 

140 

6 

91 


9 min. 

8 Hy. 

8 min. 


96 

97 

134 

135 

85 

86 


93 

97 

130 

134 

83 

85 


80 

96 

112 

133 

70 

85 


65 

96 

91 

132 

57 

84 


53 

95 

72 

131 

45 

84 


42 

95 

58 

130 

36 

83 


33 

94 

46 

127 

28 

83 


27 

93 

30 

125 

23 

82 


22 

92 

25 

123 

19 

81 


18 

91 

21 

120 

15 

80 


15 

90 

19 

118 

13 

78 


13 

88 

17 

115 

11 

76 


11 

86 

15 

112 

10 

74 


9 

84 

13 

109 

8 

72 


8 

82 

12 

105 

7 

70 


7.3 

80 

10 

102 

6 

68 


6.6 

78 

9 

99 

5.5 

66 


5.9 

76 

8 

96 

5 

64 


5.2 

74 

7.4 

93 

4.5 

62 


4.5 

72 

6.7 

89 

4 

60 

i 

3.8 

70 

6.0 

86 

3.5 

58 


7 mm. 

6 Hy. 

5 Hy. 


75 

76 

54 

55 

53 

54 


72 

75 

51 

54 

50 

54 


61 

75 

43 

53 

41 

53 


49 

74 

33 

52 

31 

52 

| 

39 

74 

25 

51 

24 

51 


30 

73 

20 

50 

18 

50 


24 

72 

16 

49 

14 

49 


19 

71 

13 

48 

11 

47 


16 

70 

10 

47 

8.8 

45 


13 

68 

8 

46 

7.2 

43 


11 

66 

7 

45 

6 

41 


9 

64 

6 

44 

5 

40 

i 

8 

62 

5 

42 

4.3 

38 


7 

60 

4.5 

40 

3.7 

36 


6 

58 

4. 

38 

3.2 

34 


5 

56 

3.6 

37 

2.8 

32 


4.6 

54 

3. 

36 

2.5 

31 


4.2 

52 

2.6 

35 

2.2 

30 


3.7 

3.3 

50 

48 

2.3 

2.1 

34 

33 

2.0 

1.8 

28 

26 


3.1 

46 

1.9 

30 

1.6 

24 







































STRENGTH OF IRON PILLARS. 


457 


In arches of cast iron for bridges, &c, it is usual among English 
engineers not to allow more than 2^ tons (5600 lbs) of compression, or thrust, per sq 
inch. Brunei never subjected cast-iron pillars to more than 1% tons (3360 lbs) per 
sq inch. C. Slialer Smitli employs as maximum working strains, 1 of the calcu* 
■Gated breaking strain for such hollow chords and posts of bridges as are 1 inch or 
( more in thickness, and not more than 15 diams long. For posts, only j ; when not 
less than % inch thick, nor more than 25 diams long: or from to X when V 
- thick or less, and more than 25 diams long. To ’ 

The young- engineer must bear in mind that the breakg and the 
sate loads per sq inch, of pillars of any given material, are not constant quantities; 

, hut diminish as the piece becomes longer in proportion to its diam. If a very long 
piece or pillar be so braced at intervals as to prevent its bending at those points, 
t “® n i® bec °™s virtually diminished, and its strength increased. Thus, if a 

‘ P llIar 100 ft lo ng be sufficiently braced at intervals of 20 ft, then the load sustained 
may be that due to a pillar only 20 ft long. Therefore, very long pillars used for 
bridge piers, <fcc, are thus braced; as are also long horizontal or inclined pieces, 
exposed to compression in the form of upper chords of bridges; or as struts of any 
kind m bridges, roofs, or other structures. 

Mistakes are sometimes made by assuming, say 5 or 6 tons per sq inch, as the safe 
compressing load for cast iron ; 4 tons for w rought; 1000 pounds for timber; without 
any regard to the length of the piece. 

But although the final crushing loads, as given in tables of strengths of materials, 
are usually those for pieces not more than about 2 diams high, they will not be much 
less for pieces not exceeding 4 or 5 diams. 

Cautions. Remember a heavily loaded cast-iron pillar is easily broken by a 
i side blow. Cast-iron ones are subject to hidden voids. All are subject to jars and 
vibrations from moving loads. It very rarely happens that the pres is equally dis¬ 
tributed over the whole area of the pillar; or that the top and bottom ends have per- 
i * e ^t bearing at every part, as they had in the experimental pillars.f Cast pillars are 
seldom perfectly straight, and hence are weakened. 

Hollow pillars intended to bear heavy loads should not be cast 
with such mouldings as a a; or w'ith very 
projecting bases or caps such as g , Fig 19. 

It is plain that these are w T eak, and w’ould 
break off under a much less load than 
would injure the shaft of the pillar. When 
snch projecting ornaments are required, 
they should be cast separately, and be at¬ 
tached to a prolongation of the shaft, as 
: d , by iron pins or rivets. 

Ordinarily, it is better to adopt a more 
limple style of base and cap, which, as at 
>, can be cast in one piece with the pillar, 
vithout weakening it. 


Hodginson states that while the quantity of material is the same 
•i both pillars, no strength is gained in hollow ones by making 
ho diams greater at the middle than at tlae 
•nds ; but that in solid ones, with rounded ends, there is a gain 
fabout l-th part; and in those with flat ends, of about ith or 
th part, by making the diam at the middle about 1*^ or 2 times 
hat at the ends. Also that a uniform round pillar has the same 
trength as a moderately tapering one whose diam at half-way up 
i equal to the uniform diam of the cylindrical one. 

Also, that when a flat-ended pillar, Fig. 2, is so irregularly 
xed, that the pressure upon it passes along its diagonal a a, it 
)ses two-thirds of its strength. Hence the necessity for equalizing, as far as possi- 
le, the pressure over every part of the top and bottom of the pillar; a point very 
iffieult to secure in practice. 


t In important cases, hoi li ends should he planed perfectly true; 

s is doue in iron bridges, &c. 



O 


[a] 


Fig. 2 





33 
























458 


STRENGTH OF WOODEN PILLARS. 


Steel pillars. Mr. Kirkaldy experimented with a small steel pillar or tube 
of Shortridge, Howell & Co’s homogeneous metal. Its leugth was 4 It, or 2 o .6 diams; 
outer diam 1% inch: inner diam 1%; thickness % inch. Area °f cross-section 1 >4 
sq ins Flat ends. Under 67300 lbs, or 30 tons total pressure, or 17.14 tons per sq 
inch of solid metal section, it bent very slightly. On increasing the pressure con¬ 
siderably, the pillar sprang out from under the load. Our table, page 444, gi ves 24 % 
tons total, or 13 tons per sq inch, as the ultimate load for a wrought-iron tube of the 


same size. „ _ . , . 

Mr. M. G. Love, Paris, as the result of a trial with small steel rods, about .4 
inch diam. and which had a tensile strength of 48 tons per sq inch,suggests that the 


comparative strength of pillars of wrought iron, cast-iron, and steel, are prol>a!»ly 
about as follows: At from 1% to 5 diams in length, steel and cast-irou ones have 



this needs confirmation. Now that powerful and accurate testing machines are 
coming more into use, we may hope that the doubts at present existing on such 
subjects will be set at rest. 

Mr Stoney advises that until then steel pillars should not be trusted 
with more than 1.5 the loads of wrought iron ones. 


WOODEN PILLARS. 


The strengths of pillars, as well as of beams of timber, depend much on their de¬ 
gree <»I seasoning. Hodgkinson found that perfectly seasoned blocks, 2 diami 
king, required, in many cases, twice as great a load to crush them as when only 
moderately dry. This should be borne in mind when building with green timber. 

In important practice, timber should not be trusted with more than % to %of it; 
calculated crushing load; and for temporary purposes, not more than % to * 4 . 

Mr. Hilaries Slialer Smith. C. E., of St. Louis, prepared th< 
following formula for the breaking loads of either square or rectangulai 
pillars or posts, of moderately seasoned white, and common yellow pine, with fia 
ends, firmly fixed, and equally loaded, based upon experiments by himself. 

It is Gordon’s formula adapted to those woods; and gives results considerably 


smaller than Hodgkinson’s, 


It is therefore safer. 


Call either side of the square, or the least side of the rectangle, the breadth. 
Breakg load in lbs, per ^ _ 5000f 


Ther 


Rule. 


sq inch of area, of a 
pillar of W or Y pine 


1 + 


'sq of length in ins 


X .004 


,sq of breadth in ins 

Or in words, square the length in ins; square the breadth in ins; div the first squat 
by the second one; mult the quot by .004 ; to the prod add 1; div 5000 by the sum 

Ex. Breakg load per sq inch, of a white pine pillar 12 ins square, and 30 ft, or 3( 
Ins long. Here the sq of leugth iu ins is 360 2 = 129600. The square of the breadth 

.0 ... . 129600 50( l 

122 = 144; aud = 900; and 900 X -004 = 3.6; and 3.6 + 1 = 4.6. Finally, 


144 


4.t 


= 1087 lbs, the reqd breakg load per sq in. As the area of the pillar is 144 sq in 
the entire breakg load is 1087 X 144 = 156528 lbs, or 69.9 tons. 

Recent experiments on wooden pillars 20 ft long, and 13 ins square, by M 
Kirkaldy, of England, confirm the far greater reliability of Mr. Smith’s formul 
Hence we present the following new set of original tables based upon it. 

For solid pillars of cast iron and of pine, whose heights ran; 
from 5 to 60 times their side or diam, we may say, near enough for practice, that 
cast iron one is about 16% times as strong as a pine one; but no such approxima 
ratio holds good between wrought iron and pine, or between cast and wrought ire 


t The breaking load in lbs per sq inch in short blocks, by Hr. Smith. 













STRENGTH OF WOODEN PILLARS, 


459 


Table of breaking: loads in tons of square pillars of half 
seasoned white or common yellow pine firmly fixed and 
equally loaded. By C. Shaler Smith’s formula. (Original.) 


tfc V 

xB 


X 

H 

2 

x 

x 

x 

i 

X 


X 

1 

X 

1 

X 

) 

X 

) 

X 

I 

X 


Side of square pine pillar, in inches. 


IK I IX | lh I 2 | 2X I ‘*X I 2Ji I 3 \ 3X l %X \ 8% | 4 


BREAKING LOAD. 


Tons. 

Tot>s. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

1.42 

2.54 

3.99 

5.73 

7.80 

10.1 

12.8 

15.7 

18.9 

22.3 

26.1 

30.1 


1.17 

2.22 

3.59 

5.26 

7.25 

9.6 

12.2 

15.1 

18.3 

21.7 

25.4 

29.2 

33.7 

.97 

1.93 

3.19 

4.80 

6.74 

9.0 

11.6 

14.5 

17.6 

21.0 

24.7 

28.6 

33.0 

.81 

1.66 

2.81 

4.35 

6.19 

8.4 

10.9 

13.7 

16.8 

20.2 

23.9 

27.8 

32.1 

.68 

1.44 

2.48 

3.92 

5.66 

7.8 

10.2 

12.9 

15.9 

19.3 

23.0 

26.9 

31.2 

.57 

1-24 

2.19 

3.53 

5.17 

7.2 

9.6 

12.3 

15.2 

18.5 

22.0 

25.8 

30.1 

.49 

1.07 

1.93 

3.16 

4.70 

6.7 

8.9 

11.5 

14.3 

17.6 

21.1 

24.9 

29.1 

.42 

.93 

1.71 

2.85 

4.29 

6.2 

8.3 

10.8 

13.5 

16.7 

20.1 

23.8 

28.0 

.36 

.82 

1.52 

2.55 

3.89 

5.6 

7.6 

10.0 

12.7 

15.8 

19.2 

22.9 

27 0 

.28 

.63 

1.21 

2.07 

3.23 

4.8 

6.6 

8.8 

11.3 

14.2 

17.4 

20.9 

24.8 

.22 

.50 

.98 

1.70 

2.70 

4.0 

5.7 

7.7 

9.9 

12.7 

15.7 

19.0 

22.7 

.18 

•40 

.81 

1.42 

2.28 

3.4 

4.9 

6.7 

8.8 

11.4 

14.1 

17.2 

20.7 

.15 

.34 

.68 

1.19 

1.94 

3.0 

4.2 

5.8 

7.7 

10.0 

12.6 

15.5 

18.8 

.12 

.28 

.57 

1.02 

1.67 

2.6 

3.7 

5.1 

6.8 

8.9 

11.3 

14.0 

17.1 

.10 

.24 

.49 

.86 

1.44 

2.3 

3.3 

4.6 

6.1 

8.0 

10.2 

12.7 

15.6 

.09 

.21 

.43 

.74 

1.26 

2.0 

2.9 

4.1 

5.4 

7.2 

9.2 

11.6 

14.2 

.08 

.18 

.37 

.66 

1.11 

1.8 

2.6 

3 6 

4.9 

6.5 

8.3 

10.5 

12.9 

.07 

.16 

.33 

.59 

.98 

1.6 

2.3 

3.3 

4.4 

5.9 

7.6 

9.6 

11.8 

.06 

.14 

.29 

.52 

.87 

1.4 

2.0 

2.9 

3.9 

5.2 

6.8 

8.7 

10.8 

.05 

.12 

.26 

.47 

.78 

1.2 

1.8 

2.6 

3.5 

4.8 

6.2 

7.9 

9.9 

.05 

.11 

.23 

.42 

.71 

1.1 

1.6 

2.3 

3.2 

4.3 

5.6 

7.2 

9.1 


. 10 

.21 

.67 

.64 

1 0 

1 5 

2.1 






. 

.09 

.19 

.34 

.58 

.93 

1.4 

2.0 

2.7 

3.6 

4.7 

6.1 

7.8 

. 

. 

.17 

.31 

.53 

.86 

1.3 

I 1.8 

2.5 

3.4 

4.4 

5.7 

7 2 



.16 

.28 

.48 

.79 

1.2 

1.7 

2.3 

3 1 

4.1 

5.3 

6 7 



.14 

.26 

.44 

.72 

1.1 

1.5 

2.1 

2.9 

3.8 

4.9 

6.2 



.13 

.24 

.41 

.65 

1.0 

1.4 

2.0 

2.7 

3.4 

4.5 

5.8 




.21 

• 6 o 

.03 

.84 

1.2 

1.7 

2.3 

3.1 

4.0 

5.0 




.18 

.31 

.46 

.70 

1.0 

1.4 

2.0 

2.7 

3.5 

4.4 





.27 

.41 

.63 

.91 

1.2 

1.7 

2.4 

3.1 

3.9 





.24 

.37 

.57 

.78 

1.1 

1.5 

2.1 

2.7 

3.5 







.50 

70 

1 0 










.45 

.66 

92 

1 3 









.76 

1 0 

1 4 

L. £> I 













U> 0> 

"3** 

a a 


X 

X 

% 

2 

X 

X 

H 

3 

X 

4 

X 

5 

X 

6 

X 

7 

X 

8 

X 

9 

X 

9 

X 


10 

i 

if 


12 

13 

14 

15 

16 

17 

18 
20 


X 


II 



Side of square pine pillar, in inches 

• 



= .2 

*X 1 

iX 1 


5 

5X 

5X 

1 5X 

1 6 

1 

1 6X 

1 

1 1 

1 ix 

D 

x .2 






BREAKING 

LOAD. 







Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


2 

35.8 

40.6 

45.7 

51.1 

56.8 

62.8 

69.0 

75.5 

82.3 

89.4 

96.8 

104.5 

112.4 

2 

3 

31.4 

36.1 

41.1 

46.2 

51.8 

57.7 

63.8 

70.2 

76.9 

84.0 

91.3 

98.9 

106.7 

3 

4 

26 8 

31.2 

35.9 

40.8 

46.1 

51.7 

57.6 

63.8 

70.4 

77.3 

84.5 

92.1 

99.9 

4 

0 

22.6 

26.5 

30.8 

35.4 

40.5 

45.8 

51.5 

57.4 

63.7 

70.3 

77.2 

84.5 

92.1 

5 

6 

18.9 

22.5 

26.4 

30.5 

35.2 

40.1 

45.4 

51.0 

57.0 

63.3 

69.9 

76.8 

84.1 

6 

7 

15.8 

19.0 

22.6 

26.2 

30.5 

35.0 

39.9 

45.0 

50.5 

56.5 

62.8 

69.4 

76.3 

7 

8 

13.3 

16.1 

19.2 

22.5 

26.3 

30.4 

34.9 

39.7 

45.9 

50.4 

56.2 

62.4 

69.0 

8 

9 

11.3 

13.7 

16.5 

19.5 

22.9 

26.6 

30.7 

35.0 

39.9 

44.8 

50.2 

56.0 

62.1 

9 

0 

9.7 

11.8 

14.2 

16.9 

19.9 

23.2 

26.9 

30.9 

35.2 

39.9 

44.9 

50.3 

56.0 

10 

1 

8.3 

10.2 

12.4 

14.8 

17.5 

20.4 

23.8 

27.4 

31.3 

35.6 

40.2 

45.1 

50.4 

11 

2 

7.2 

8.8 

10.7 

12.9 

15.4 

18.0 

21.1 

24.3 

27.9 

31.8 

36.0 

40.6 

45.5 

12 

3 

6.2 

7.7 

9.4 

11.4 

13.6 

16.0 

18.8 

21.7 

24.9 

28.5 

32.4 

36.6 

41.1 

13 

1 

5.5 

6 8 

8.3 

10.1 

12.1 

14.2 

16.7 

19.4 

22.4 

25.7 

29.2 

33.1 

37.3 

14 

5 

4.8 

6.0 

7.4 

9.0 

10.8 

12.7 

15.0 

17.5 

20.2 

23.2 

26.4 

30.0 

33.9 

15 

6 

4.4 

5.4 

6.7 

8.1 

9?8 

11.5 

13.6 

15.8 

18 3 

21.0 

24.0 

27.3 

30.8 

16 

1 

4.0 

4.9 

6.1 

7.3 

8.8 

10.4 

12.3 

14.3 

16.6 

19.1 

21.9 

24.9 

28.3 

17 

8 

3.6 

4.4 

5.5 

6.6 

8.0 

9.4 

11.2 

13.0 

15.1 

17.4 

19.9 

22.7 

25.8 

18 

9 

3.3 

4.0 

5.0 

6 0 

7.3 

8.6 

10 2 

11.9 

13.8 

16.0 

18.3 

20.9 

23.7 

19 

0 

3.0 

3.7 

4.6 

5.5 

6.6 

7.8 

9.3 

10.9 

12.6 

14.6 

16.8 

19.2 

21.8 

20 

2 

2.5 

3.0 

3.8 

4.6 

5.6 

6.6 

7.9 

9.2 

10.7 

12.4 

14.3 

16.3 

18 6 

22 

1 

2.1 

2.6 

3.2 

3.9 

4.7 

5.6 

6.7 

7.9 

9.1 

10.6 

12.2 

14.1 

16.0 

24 

5 

1.8 

2.2 

2.8 

3.4 

4.1 

4.9 

5.8 

6.8 

7.9 

9.2 

10.6 

12.2 

13.9 

26 

3 

1.5 

1.9 

2.4 

2.9 

3.5 

4.2 

5.1 

5 9 

6.9 

8.0 

9.3 

10.7 

12.2 

28 

i 

1 3 

1.7 

2.1 

2.6 

3.1 

3.7 

4.4 

5.2 

6.1 

7.1 

8.2 

9.4 

10.8 

30 

i 

1.2 

1.5 

1.9 

2.3 

2.7 

3.2 

3.9 

4.6 

5.4 

6.3 

7.3 

8.4 

9.6 

32 

i 

1.1 

1.3 

1.7 

2.0 

2.4 

2.9 

3.5 

4.1 

4.8 

5.6 

6.5 

7.5 

8.6 

34 

> 

1.0 

1.2 

1.5 

1.8 

2.2 

2.6 

3.1 

3.7 

4.3 

5.0 

5.8 

6.7 

7.7 

36 

t 

.9 

l.t 

1.3 

1.6 

2.0 

2.4 

2.8 

3.3 

3.9 

4.5 

5.3 

6.1 

7.0 

38 

i 

.8 

1.0 | 

1.2 

1.5 1 

1.8 

2.1 

2.6 

3.0 1 

3.5 

4.1 

4.8 

5.5 

6.3 

40 


Continued on next page. 


































































































460 


STRENGTH OF WOODEN PILLARS 


Table of breaking: loads in tons of square pine pillars, with 
flat ends firmly fixed, and equally loaded. (Contiuued.) 


Original. 



Side of square pine pillar, in inches. 

7M [ 7* | 8 | 8* | 8* \"fSH I 9 l»XI I 10 I 10 * 1 10 * 



Tons.' 

Tons. 

Tons. 

Tons. 

2 

120 6 

129.1 

137.9 

147.0 

4 

107.9 

116-3 

125.0 

133.9 

6 

91.7 

99.7 

108.0 

116.5 

8 

75.9 

83.2 

90.8 

98.7 

10 

62.0 

68.5 

75.3 

82.4 

12 

50.7 

56.5 

62.5 

68.7 

14 

41.8 

46.7 

51 9 

57.3 

16 

34.7 

38.9 

43.4 

48.1 

18 

29.1 

32.7 

36.6 

40.7 

20 

24.6 

27.7 

31.1 

34.7 

23 

19.6 

22.1 

24.8 

27.7 

26 

15.8 

17.8 

20.1 

22.5 

29 

13.1 

14.8 

16.7 

18.7 

32 

10.9 

12.3 

13.9 

15.7 

35 

9 3 

10.6 

11.9 

13.4 

38 

8.0 

9.1 

10.2 

11.5 

41 

6.9 

7.9 

8.9 

10.0 

44 

6.0 

6.9 

7.8 

8.8 

50 

4.7 

5.4 

6.1 

6.9 


BREAKING LOAD, 


Tons. 

Tons. 

Tons. 

Tons. 

156.3 

165.9 

175.8 

186-0 

143.0 

152.6 

162.4 

172.5 

125.3 

134.5 

143.5 

153.0 

106.8 

115.5 

124.4 

133.6 

89.7 

97.7 

105.8 

114.3 

75.1 

82.2 

89.6 

97.2 

62.9 

69.2 

75.7 

82.4 

53.0 

58.5 

64.3 

70.3 

45.0 

49.8 

54 9 

60.1 

38.5 

42.7 

47.2 

51.8 

30.9 

34.4 

38.0 

41.9 

25.2 

28.1 

31.1 

34.4 

20.9 

23.4 

25.9 

28.6 

17.6 

19.7 

21.8 

24.2 

15.0 

16.8 

18.7 

20.7 

12.9 

14.6 

16.3 

18.0 

11.2 

12.6 

14.1 

15.6 

9.8 

11.0 

12.3 

13.6 

7.7 

8.8 

10.0 

11.3 


Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

196.4 

207.2 

218.2 

229.5 

241.0 

182.8 

193.4 

204.2 

215.3 

226.6 

162.8 

173.5 

184.4 

195.6 

207.0 

143.0 

153.0 

163.2 

173.7 

184.4 

123.0 

132.3 

141.8 

151.6 

161.6 

105.0 

113 5 

122.2 

131.2 

140.5 

89.4 

97.1 

105.0 

113.2 

121.6 

76.5 

83.3 

90.4 

97.7 

105.3 

65.6 

71.7 

78.0 

84.6 

91.4 

56.7 

62.0 

67.6 

73.5 

79.6 

46.0 

50.5 

55.2 

60.2 

65.4 

37 9 

41.6 

45.6 

49.8 

54.3 

31.6 

34.9 

38.2 

41.9 

45.6 

26.7 

29.5 

32.3 

35.5 

38.7 

22.8 

25.2 

27.7 

30.5 

33.3 

19.7 

21.8 

23.9 

26.3 

28 8 

17.2 

19.1 

21.0 

23.1 

25.2 

15.0 

16.7 

18.4 

20.2 

22.1 

12.6 

13.7 

14.9 

16.2 

17.5 


■= Z 

O' „ 

K.2 


2 

4 

6 

8 

10 

12 

U 

16 

18 

20 

23 

26 

29 

32 

35 

38 

41 

44 

50 


2 £ 
to S 


Side of square pine pillar, in inches 



s 5 

tt .2 

io* 1 

n 1 u* 1 

11 * 1 

11 * 1 

12 | 

1 io*| 

11 1 

11 * 1 

11 * 1 

11 * 

12 

S3.2 



BREAKING LOAD. 



BREAKING LOAD. 




Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 



Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


4 

238.9 

251.0 

263.2 

275.9 

288.8 

302.1 



26.5 

28.6 

31.1 

33.8 

36.7 

39.9 

42 

6 

218.8 

230.6 

242.7 

255.2 

267.9 

281.0 



24.2 

26.4 

28.7 

31.2 

33.9 

36 8 

44 

8 

195.5 

207.0 

218.5 

230.4 

242.5 

255.1 



22.4 

24.3 

26.4 

28.7 

31.2 

33.9 

46 

10 

172.2 

183.0 

194.1 

205.6 

217.3 

229.5 



20.7 

22.6 

24.5 

26.7 

28.9 

31.4 

48 

12 

150.3 

160.3 

170.7 

181.5 

192.5 

204.0 



19.2 

20.9 

22 7 

24.7 

26.8 

29.2 

50 

14 

130.5 

139.8 

149 4 

159.4 

169.6 

180.2 



17.8 

19.5 

21.1 

23.0 

25.0 

27.2 

52 

10 

113.2 

121.7 

130 4 

139.6 

149.0 

158.8 



16.6 

18.2 

19.7 

21.5 

23.3 

25.4 

51 

18 

98.7 

106.2 

114.1 

122.5 

131.0 

140.0 



15.5 

17.0 

18 4 

20.1 

21.8 

23.7 


20 

86.2 

92.9 

100.0 

107.6 

115.4 

123.6 



14.5 

15.9 

17 2 

18.8 

20.4 

22.2 

58 

22 

75.6 

81.7 

88.1 

95.0 

102.0 

109.5 



13.6 

14.9 

16.2 

17.7 

19.2 

20.9 

60 

24 

66.7 

72.2 

77.9 

84.1 

90.5 

97.3 



11.8 

12.9 

13.9 

15.2 

16 5 

18.0 

65 

26 

59.1 

64.0 

69.2 

74.9 

80 6 

86.8 



10.1 

11.1 

12.0 

13.2 

14.3 

15.6 

70 

28 

52.6 

57.1 

61.8 

66.9 

72.1 

77.7 



8.9 

9.8 

10.6 

11.6 

12.5 

13.7 

75 

30 

47.0 

51.1 

55.3 

60.0 

64.7 

69.9 



7 8 

8.6 

9.4 

10.2 

11.1 

12.1 

80 

32 

42.1 

46.0 

49.9 

54.0 

58.4 

63.0 



7 0 

7.7 

8.4 

9.1 

9.9 

10.7 

85 

34 

38.2 

41.5 

45.0 

48.8 

52.8 

57.1 



6.2 

6.8 

7.4 

8.1 

8.8 

9.6 

90 

36 

34.6 

37.7 

40.9 

44.3 

48.0 

51.8 



5.6 

6.1 

6.7 

7.3 

8.0 

8.7 

95 

38 

31.5 

34.2 

37.2 

40.4 

43.9 

47.4 



5.1 

5.6 

6.1 

6.6 

7.2 

7.8 

100 

40 

28.8 

31.3 

1 34.0 

1 37.0 

1 40.1 

43.5 


1 3.5 

3.9 

4.2 

1 4.6 

5.0 

5.5 

120 


Continued on next page. 

Remarks. Mr ltirkaldy found for Rigra and Rantzic firs.( 

20 ft long, and 13 ins square, (or 18% sides high,) 148 and 138 tons total; or .87<, 
and .817 tons, (1963 and 1829 lbs,) per sq inch. Mr Smith’s rule gives for comnioij 
pine, 160 tons total; or .947 ton, or 2121 lbs, per sq inch. Ilodgkinson would give', 
for Riga about 297 tons total. 

Each of Mr Kirkaldy’s 20-ft pillars shortened about .6 of an inch total; or .01 
inch per ft; or % of an inch in 4 ft 2 ins, under a mean of 1900 lbs per sq inch. 

The writer has known 8 unbraced pillars of hemlock, tolerably seasoned i 
12 ins square, and 42 ft high, to be gradually loaded each with 32 tons, or 7168c 
lbs total; (or .2222 ton, or 498 lbs per sq inch) without appreciable yielding. As, 
suming their strength and stiffness to be about as for Mr Smith’s pine, (as in al 
our tables,) they should by him yield at 39.9 tons total. With these same dn/r, 
but with Hodgkinson’s formula, they should yield at 69.3 tons; and wit] 
Hodgkinson’s own data, for seasoned red deal, at 91.6 tons. See Remarks, p ME 







































































STRENGTH OF WOODEN PILLARS 


461 


Table of breaking- loads in tons of square pillars of half* 
seasoned white or common yellow pine, with flat ends 
firmly fixed, and equally loaded. By C. Shaler Smith's formula. 
(Continued.) 

As this table was partly made by interpolation, the last figure is not always pre¬ 
cisely correct. 


Original. 


-g *5 

*3 ** 

Side of square pine pillar in inches. 

bc.S 

S5 .5 

13 

1 

1 15 

1 16 

1 17 

1 18 

1 19 

| 20 

1 21 

| 22 

| 23 

| 24 

ES 2 






BREAKING LOAD. 







Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


4 

358 

418 

482 

552 

625 

703 

786 

872 

964 

1060 

1161 

1265 

4 

6 

335 

394 

456 

526 

599 

676 

760 

847 

938 

1033 

1134 

1236 

6 

8 

308 

367 

429 

500 

572 

649 

732 

818 

910 

1005 

1106 

1208 

8 

10 

281 

339 

400 

466 

537 

612 

694 

780 

870 

964 

1064 

1166 

10 

12 

252 

307 

365 

432 

502 

576 

656 

740 

829 

922 

1022 

1124 

12 

14 

225 

277 

333 

397 

464 

536 

614 

696 

784 

876 

973 

1074 

14 

16 

201 

250 

303 

363 

428 

497 

573 

652 

739 

829 

925 

1024 

16 

18 

179 

224 

274 

331 

392 

458 

531 

608 

692 

780 

873 

972 

18 

20 

160 

201 

248 

301 

359 

422 

492 

566 

647 

732 

822 

919 

20 

22 

143 

182 

224 

274 

329 

388 

455 

526 

604 

686 

773 

866 

22 

24 

127 

163 

203 

249 

301 

357 

421 

488 

563 

642 

726 

816 

24 

29 

115 

148 

184 

226 

275 

328 

389 

453 

523 

599 

680 

767 

26 

28 

103 

133 

167 

206 

252 

302 

359 

420 

490 

560 

638 

721 

28 

30 

93 

121 

152 

189 

231 

278 

332 

389 

453 

522 

597 

677 

30 

32 

84 

109 

138 

173 

212 

256 

307 

361 

421 

487 

558 

635 

32 

34 

76 

99 

126 

159 

196 

237 

284 

335 

392 

455 

523 

597 

34 

36 

69 

91 

116 

146 

180 

219 

264 

312 

366 

426 

490 

560 

36 

38 

63 

84 

107 

134 

166 

203 

245 

290 

341 

397 

458 

525 

38 

40 

58 

77 

99 

124 

154 

188 

227 

270 

318 

372 

429 

494 

40 

42 

54 

71 

91 

115 

143 

175 

212 

253 

298 

349 

403 

465 

42 

44 

50 

66 

84 

107 

133 

163 

198 

236 

280 

328 

380 

438 

44 

46 

46 

61 

78 

99 

123 

152 

185 

221 

263 

308 

358 

413 

46 

48 

43 

57 

73 

92 

115 

142 

173 

207 

247 

290 

337 

389 

48 

50 

40 

53 

68 

86 

107 

133 

162 

194 

231 

272 

317 

367 

50 

52 

37 

50 

64 

81 

101 

124 

152 

182 

217 

256 

300 

347 

62 

54 

35 

47 

60 

76 

95 

117 

144 

172 

205 

242 

283 

328 

54 

56 

33 

44 

56 

71 

89 

no 

135 

162 

193 

228 

267 

310 

56 

58 

31 

41 

52 

67 

84 

103 

127 

153 

182 

215 

253 

294 

58 

60 

29 

38 

49 

63 

79 

98 

120 

144 

172 

204 

240 

280 

60 

65 

25 

33 

43 

55 

69 

86 

105 

126 

151 

179 

211 

246 

65 

70 

22 

29 

37 

48 

60 

74 

92 

111 

134 

159 

187 

218 

70 

75 

19 

25 

33 

42 

53 

66 

82 

98 

118 

141 

166 

195 

75 

80 

16 

22 

29 

37 

46 

58 

72 

87 

105 

125 

148 

174 

80 

85 

14 

19 

26 

33 

41 

52 

65 

78 

94 

112 

132 

156 

85 

90 

13 

17 

23 

30 

37 

46 

58 

70 

85 

102 

120 

141 

90 

95 

12 

16 

21 

27 

33 

42 

53 

64 

77 

93 

108 

127 

95 

100 

11 

14 

19 

24 

30 

38 

48 

58 

70 

84 

99 

117 

100 

110 

10 

12 

16 

20 

26 

33 

40 

48 

58 

70 

82 

97 

no 

120 

9 

11 

14 

17 

22 

28 

34 

41 

49 

60 

71 

83 

120 

130 

7 

9 

12 

14 

18 

23 

29 

36 

43 

52 

61 

72 

130 

140 

6 

8 

10 

12 

16 

20 

26 

31 

37 

44 

53 

62 

140 

150 

5 

7 

9 

11 

14 

18 

22 

27 

32 

88 

46 

54 

150 

160 

5 

6 

8 

10 

13 

16 

20 

24 

29 

34 

41 

48 

160 

170 

4 

5 

7 

9 

11 

14 

17 

21 

25 

30 

36 

43 

170 

180 

4 

5 

6 

8 

10 

12 

15 

19 

22 

27 

32 

38 

180 

190 

3 

4 

5 

7 

9 

11 

14 

17 

20 

24 

29 

34 

190 

200 

3 

4 

5 

6 

8 

10 

12 

15 

18 

22 

26 

31 

200 






































462 


STRENGTH OF WOODEN PILLARS. 


Table of breaking loads in tons, or in lbs, per sq inch of 
cross section of half seasoned square pine pillars, whose 
heights are measured by one of their sides. 


Height 
in sides. 

BR LOAD PER SQ IN. 

- ® 
'StS 

*.5 

BR LOAD PER SQ IN. 

Height 

in sides. 

BR LD PER SQ IN. 

I 

CD 

BR LD PER SQ IN. 

1 

Tons. 

2.2232 

Lbs. 

4980 

26 

Tons. 

.6027 

Lbs. 

1350 

51 

Tons. 

.I960 

Lbs. 

439 

76 

Tons. 

.0924 

Lbs. 

207 

2 

2.1969 

4921 

27 

.5697 

1276 

52 

.1888 

423 

77 

.0902 

202 

3 

2.1544 

4826 

28 

.5398 

1209 

53 

.1826 

409 

78 

.0879 

197 

4 

2.0978 

4699 

29 

.5116 

1146 

54 

.1763 

395 

79 

.0862 

193 

5 

2.0290 

4545 

30 

.4853 

1087 

55 

.1705 

382 

80 

.0839 

188 

6 

1.9513 

4371 

31 

.4607 

1032 

56 

.1647 

369 

81 

.0S21 

184 

7 

1.8665 

4181 

32 

.4379 

981 

57 

.1598 

358 

82 

.0799 

179 

8 

1.7772 

3981 

33 

.4165 

933 

58 

.1545 

346 

83 

.0781 

175 

9 

1.6857 

3776 

34 

.3969 

889 

59 

.1496 

335 

84 

.0763 

171 

10 

1.5942 

3571 

35 

.3781 

847 

60 

.1451 

325 

85 

.0746 

167 

11 

1.5040 

3369 

36 

.3599 

809 

61 

.1406 

315 

86 

.0728 

163 

12 

1.4165 

3173 

37 

.3447 

772 

62 

.1362 

305 

87 

.0714 

160 

13 

1.3317 

2983 

38 

.3295 

738 

63 

.1321 

296 

88 

.0696 

156 

14 

1.2513 

2803 

39 

.3152 

706 

64 

.1286 

288 

89 

.0683 

153 

15 

1.1745 

2631 

40 

.3018 

676 

65 

.1250 

280 

90 

.0670 

150 

16 

1.1027 

2470 

41 

.2889 

647 

66 

.1210 

271 

91 

.0656 

147 

17 

1.0353 

2319 

42 

.2772 

621 

67 

.1179 

264 

92 

.0638 

143 

18 

.9723 

2178 

43 

.2661 

596 

68 

.1147 

257 

93 

.0625 

140 I 

19 

.9134 

2046 

44 

.2554 

572 

69 

.1112 

249 

94 

.0616 

138 

20 

.8585 

1923 

45 

.2455 

550 

70 

.1085 

243 

95 

.0603 

135 

21 

.8076 

1809 

46 

.2357 

528 

71 

.1054 

236 

96 

.0589 

132 

22 

.7603 

1703 

47 

.2268 

508 

72 

.1027 

230 

97 

.0576 

129 

23 

.7165 

1605 

48 

.2183 

489 

73 

.1000 

224 

98 

.0567 

127 

24 

.6755 

1513 

49 

.2100 

472 

74 

.0973 

218 

99 

.0554 

124 

25 

.63S0 

1429 

50 

.2031 

455 

75 

.0951 

213 

100 

.0545 

122 

- - 


Remark.—Gordon and Hodgkinson compared. The difference 
between them is greater in wooden pillars than in hollow cast iron ones. More¬ 
over, in the latter, Gordon is sometimes greater, sometimes less, than Hodgkin¬ 
son, as seen per lower table. Mr. Smith’s assumed strength and stiffness of pine 
may safely be taken at about one-fourth less than Hodgkinson’s for his red deal; 
and with this assumption, Hodgkinson’s rule would make the strength of Smith’s 

E ine (in all our tables for wooden pillars) greater, to the extent shown 
y the multipliers in the following table. The truth is probably between 
the two. See Remark, p. 460. 


Ht in 
sides. 

Multr. 


Ht in 
sides. 

Multr. 


Ht in 
sides. 

Multr. 

i Ht in 

1 sides. 

Multr. 

Ht in 
sides. 

Multr. 

5 

1.04 


12.5 

1.23 


20 

1.44 

85 

1.72 

50 

1.67 

7.5 

1.09 


15 

1.30 


25 

1.54 

40 

1.76 

60 

1.63 

10 

1.15 ! 

17.5 

1.37 


30 

1.64 

45 

1.71 

80 

1 59 


Gordon’s and Hodgkinson's hollow cylindrical cast iron 

pillars compared. 

The thickness is usually from to of the outer diameter; and for these i 
limits, the column G, (Gordon), and H, (Hodgkinson), show the proportions of 
the breaking loads. 


Thickness = ^of the outer diameter. 


Ht in 
Dianis. 

G. 

H. 

Ht in 
Diams. 

G. 

H. 

Ht in 
Diams. 

G. 

H. 

Ht in 
Diams. 

G. 

H. 

Ht In 
Diams, 

G. 

H. 

5 

1 

1.25 

10 

1 

1.11 

20 

1 

.97 

40 

1 

.90 

70 

1 

.90 

8 

1 

1.12 

15 

1 

1.02 

30 

1 

.94 

50 

1 

.88 

100 

1 

.97 



Thickness = 

14 of the outer diameter. 



5 

1 

1.23 

10 

1 

1.08 

20 

1 

.92 

1 40 

1 

.81 

70 

jl 

| .82 

8 

1 

1.10 

15 

1 

.98 

30 

1 

.88 

1 50 

1 

.80 

100 

|1 

1 .88 
















































































































STRENGTH OF MATERIALS. 463 


Art. 4. Ultimate average tensile or cohesive strength of 

Timber, 

Being tlie least weights in pounds which, if attached to the lower end of a vert rod 
one inch square, firmly upheld at its upper end, would break it by tearing it apart. 
For large timbers we recommend to reduce these constants to ^ part. 


The strengths in all these tables may 
readily be one-third part more or less 
than our averages. 

Lbs per 
sq. inch. 

Alder . 

14000 

Ash, English... 

16000 

“ American (author) abt. 

Birch.. 

“ Amer’n black. 

16500 

15000 

7000 

Bay-tree. 

12000 

Beech, English. 

11500 

Bam bo# .. . 

6000 

Box . 

20000 

Cedar, Bermuda. 

7600 

“ Hnadalonpe . 

9500 

Chestnut. 

13000 

horse. 

10000 

Cyprus .. 

6000 

Elder. 

10000 

Elm . 

6000 

“ Canada . 

13000 

Fir, or Spruce . 

Hawthorn. . 

10000 

10000 

1 Ih.hr I .. . 

18000 

1 lollv . 

16000 

Hornbeam. 

20000 

Hickory Amer’n .. 

11000 

I.igtmm Vitap Amer’n. 

11000 

] A f\x\ now o oil. 

23000 

T.areh Scotch. 

7000 

Locust . 

18000 

Maple . 

10000 


Lbs per 
sq. inch. 


Mahogany, Honduras. 

“ Spanish. ... 

Mangrove, white, Bermuda.... 

Mulberry. 

Oak, Amer’n white. 

“ “ basket. 

“ “ red.. 

“ Dantzic, seasoned .... ■ 

“ Riga . 

“ English. 

“ live, Amer’n. 

Pear... 


8000 

16000 

100U0 

12000 


10000 


10000 


Pine, Amer’n, white, red, t 
and Pitch, Mernel, Riga... ) 

Plane. . 

Plum. 

Poplar.... 

Quince. . 

Spruce, or Fir. 

Sycamore. 

Teak. 

Walnut. 

Yew. 


10000 

11000 

11000 

7000 

7000 

10000 

12000 

15000 

8000 

8000 


Across the grain. Oak.. 

“ “ “ Poplar. 

“ “ “ Larch,900 to 

“ “ “ Fir, & Pines 


2300 

1800 

1700 

550 


These are averages. The strengths vary much with the age of the tree; the 
locality of its growth; whether the piece is from the center, <>r from the outer por¬ 
tions of the tree; the degree of seasoning; straightness of grain; knots, Ac, Ac. Also, 
inasmuch as the constants are deduced from experiments with good specimens of 
small size, whereas large beams are almost invariably more or less defective from 
knots, crookedness of fibre, Ac, it is advisable in practice to reduce these constant* 
as recommended above. 






































































464 


STRENGTH OF MATERIALS. 


Art. 5. Average ultimate tensile or cohesive strength of 

Metals, per square inch.* 


The ultimate tensile or pulling load per square inch of any 
material is frequently called its constant, coefficient, or modulus of 
tension, or of tensile strength. 


Antimony, cast. 

Bismuth, cast. 

Brass, cast 8 to 13 tons, say 18000 to 29000 Ibi.. 

“ wire, unannealed or hard, 80000. Annealed. 

Bronze, phosphor wire, hard, 150000. Annealed. 

Copper, cast 18000 to 30000. 

u bolts,’28ooo*to^ 

“ wire (annealed 16 tons); uuannealed.. 

Gold, east. 

“ wire, 25000 to 30000. 

Gun metal of copper and tin, 23000 to 55000. 

“ “ cast iron, U. S. ordnance, 36000 to 40000. 

Iron, cast, English.13400 to 22400. 

“ “ ordinary pig.. 13000 to 16000. 

American cast iron averages one-fourth more than the above. 
Average cast iron, when sound, stretches about .00018; or 1 part 
in 5555 of its length; or % inch in 57.9 ft, for every ton of ten¬ 
sile strain per sq inch, up to its elastic limit, which is at about 
% its break-strain. The extent of stretching, however, varies 
much with the quality of the iron; as in wrought-iron. 

Cast, malleable, annealed 18 to 25 tons... 

Iron, wrought, rolled bars, 40000 to 75000, the last exceptional”' 

“ “ “ “ ordinary average. . 

“ “ “ “ good “ . 

“ “ “ superior... 

“ “ “ best American, (exceptional). 

“ “ “ “ Low Moor, English, average. 

“ “ plates for boilers, &c, 40000 to 60000. 

“ “ English rivet irou.55000 to 60000. 

“ “ wire, annealed.30000 to 60000. 

“ “ uuannealed, or hard...50000 to 100000. 


“ “ “ ropes, per sq inch of section of rope . 

“ “ large forgings, 30000 to 40000. 

In important practice, good bar iron should not be trusted per¬ 
manently with more than about 5 tons per sq inch; which will 
stretch it about % inch in from 20 to 25 ft. 

Good bar iron stretches about 1 partin 12000 of its length; or 
about 1 inch in 1000 ft; or % inch in 125 ft, for every ton of 
tensile strain per sq inch of section, up to its elastic limit. 
This limit usually ranges between 8 and 13 tons per sq inch, or 
about half the breaking strain, according to quality. The, 
ultimate stretching of rolled bars is from 5 to 30 per ct of the 
original length; usually 15 k> 20 per cent. Plates and angle 
iron 3 to 17 per cent. Heating, even up to 500° Fah, does not 
weaken bar iron or steel. For stretch by heat see p 212. 

Lead, east, 1700 to 2400.by author... 

“ wire, 1200 to 1600. Pipe 1600 to 1700.“ “ 

Platinum wire, annealed, 32000. Unannealed. 

Steel, plates, range, 60000 to 103000.!!!!!!!!!”!!! 

“ “ of Ilussey, Wells & Co, Pittsburg, Pa, 91500 to 97400 

“ “ Bessemer. 

“ Bessemer tool...!.!!'.'.”.”” 

“ wire, annealed 30 to 50 tons. Unan, 50 to 90 tons!!!.'.'.*””” 


Pounds 

per 

sq.inch. 

Tons 
per 
sq. in. 

1000 

.45 

3200 

1.4 

23500 

10.5 

49000 

22 

63000 

28.1 

24000 

10.7 

30000 

13.4 

33000 

14.7 

60000 

26.8 

20000 

8.9 

27500 

12.3 

39000 

17.4 

38000 

17 

17900 

8 

14500 

6.47 

48160 

21.5 

57500 

25.7 

44800 

20 

50400 

22.5 

60000 

26.8 

76100 

34 

60000 

26.8 

50000 

22.3 

57500 

25.7 

45000 

20.1 

75000 

33.5 

38000 

16.8 

35000 

15.6 

2050 

.95 

1650 

.7^ 

56000 

25 

81500 

36.4 

94450 

42.2 

98600 

44 

112000 

50 

156800 

70 i 


'’Large bars of metal bear less per sq inch than small ones. In cast iror i 

ones 1, 2 and 3 ins sq, the strengths per sq inch were about as 1, .85 and .66: and wrought iron nroh 1 
ably averages about the same. See top of next page. 1 t 

Iron bars re-rolled cold have tensile strength increased 25 to 50 per ot. with no increase «t 
density. They are said to lose this strength if reheated. 


























































465 


STRENGTH OF MATERIALS. 

Average ultimate tensile, or cohesive strength of Metals 
_ per square inch. (Continued.) 



100000 lbs per sq inch, but stretched considerably. 

feteel, cast, Bessemer ingots, average. 

“ “ best American Bessemer ingots 

rolled and hammered, 120000 

u .. to 130000. 

« « homogeneous, Cammell & Co, England, No 1.. .. 

“ “ “ No 2 

« U u .. ^. 

u No 3 

“ puddled bars, rolled and hammered, 65000 to 135000'1"””' 
' teel * Experiments by Lieut. W. S. Shock,U. S N., at Washington 
on steel from the Black Diamond Steel-Works, Pittsburg’ 
Pa. All the pieces were cut from the same bar three’ 
pieces for each exp. They were turned down to a diam 
of .62 ot an inch at the intended point of fracture, by a 
groove, in shape of a circular segment, with a chord of 
about l inch: 

the bar in its original condition, 109500 to 131900. 

heated to light cherry-red, then plunged into oil of 82° 

Fall, 201300 to 227500 . 

heated to light cherry-red, then plunged into water of 79° 
Fah. Then tempered on aheated plate, 152500to 176100.. 
heated to light cherry-red, then plunged into water of 79° 

Fah, 1327 00 to 150500 . 

Tempering in oil usually increases the strength from 40 to 
80 per cent. 

chrome, made at Brooklyn, N. Y., and tested at West Point 
Foundry, N.Y , (specific gr 7.816 to7.956,) 163000 to 199000 

Average of 12 specimens . 

made from very pure Swedish iron, but containing differ¬ 
ent proportions of carbon. The bars were 21)4 ins long, 
with 14 ins of this length turned down to a uniform dT- 
am of 1 inch. The breaking wts, however, in the table 
are per sq inch : 

Mark No. 2, carbon .33 per ct, stretched 1.37 ins 
.43 
.48 
.53 
.58 
.63 
.74 
.84 
1.00 
1.25 


No. 
No. 
No. 
No. 
No. 
No. 10 
No. 12 
No. 15 
No. 20 


137 
1.25 
1.12 
0.81 
1.00 
0.69 
1.12 
1.00 
0 62 


'ith more than about 1.5 per ct of carbon the tensile strength of 
steel diminishes. A bar of the above No. 15, which broke at 
60 tons per sq inch, when turned down for 14 ins of its length ; 
broke with 79% tons per sq inch when turned down at one 
point only. This is owing to the fact that the last could not 
stretch as much as the first, and therefore its diam could 
not be diminished as much before breaking. All its fibres 
pulled more unitedly. It will bo observed that the steel of 
greatest strength stretched the least before breaking. This 
stronger steel would break under a suddenly applied force, or 
impulse, more easily than a weaker one would ; because the 
weaker one, by its stretching, gradually breaks the force of the 
impulse, on the same principle as a spring. Hence the steel, 
iron, &o, which is strongest against a gradually applied force 
or strain, may be unfit for uses where the strain comes upon it 
iddeuly. The average ultimate tensile strength of steel is about 
vice that of wrought iron. Its deflection as a beam within the elastic 
mitsisabout fthatof wrought, or % that of cast iron. Its average stretch 


Pounds 

per 

sq.inch. 

Tons, 
per 
sq. in. 

63000 

281 

86600 

38.6 

125000 

55.8 

58240 

26 

71680 

32 

76160 

34 

100000 

44.6 

120700 

53.9 

214400 

95.7 

161300 

73.3 

141600 

63.2 

180000 

80 

68100 

30.4 

76160 

34 

84000 

37.5 

95200 

42.5 

92960 

41.5 

100800 

45 

101920 

45.5 

123200 

55 

134400 

60 

154560 

69 














































466 


STRENGTH OF MATERIALS, 


is about .1 inch in 111 ft for every ton per sq inch of load, up to its elastic limit, which generally 
ranges at between H and % of its breaking streugth ; the latter being for the harder, stronger, and 
less stretchy kinds. A uniform bar of rolled steel, gradually loaded, will stretch from -L to -jl 
of its length before breaking ; or from of an inch to 2.4 ms per foot, according to quality. The 
mean of these is nearly yy of the length, or ly^inch to a foot. When steel, especially if hard, has 
to be heated to softness in order to give it a required shape, it is therebf weakened. 


Average ultimate tensile or cohesive strength of Metals 
per square inch. (Continued.) 



Pounds 

per 

sq. inch. 

Tons 
per 
sq. in 

Silver, cast.:.. 

41000 

18 3 

Tin, English block. 

4600 

2 

“ wire. 

7000 

3 1 

Zinc, cast...3000 to 3700 ; (the last by author). 

3350 

4 



Art. 6. Average ultimate tensile or cohesive strength of 

various materials. 


The strengths in all these 
tables may readily be one-third 
part more or less than our 
averages. 

Pounds 

per 

sq.inch. 

Tons 
per 
sq. ft. 


Pounds 

per 

sq.inch. 

Tons 
per 
sq. ft 

Brick, 40 to 400. 

220 

14.1 

Marble,strong,wh.Italy.* 

1034 

66.5 

Caen stone, 100 to 200. 

150 

9.7 

“ Champlain,varie- 


i 

Cement, hydraulic, Port- 



gated *. 

1666 

107.1 

laud, pure, 7 days 



“ Glenn’s F’lls.N.Y. 



in water. 

300 

19.3 

blk,* 750tol034 

892 

57.4 

“ 6 months old. 

450 

28 9 

“ Montg’y co. Pa, 



“ 1 year old. 

550 

35.4 


1175 

77 6 

Common hyd cements 



“ “ whit®*... 

734 

47.2 

average 1-6 as much. 



“ Lee, Mass.w lute.* 

875 

56.3 

The last, neat, adhere 



“ Manchester, Vt,* 



to brick and stone with 



57.0 to 800. 

675 

43.4 

from 15 to 50 tbs when 



“ Tennessee, varie- 



only 1 month old. 

32 

2 

gated * .. 

1034 

66 5 

At end of 1 year, 3 



Oolites, 100 to 200. 

150 

9/7 

times as much. 

96 

6 

Plaster of Paris, well set. 

70 

4.5 

See “ Cement,” p 675, Ac 



Rope, Manilla, best. 

12000 

1 i 1 

Glass, 2500 to 9000 (p 432) 

5750 

369.6 

“ hemp, best. 

15000 

965 

Glue holds wood together 



Sandstone, Ohio * . 

105 

6 7 

with from 300 to 800... 

550 

35 

“ Pictou, N. S.* 

434 

27.9 

Horn, ox. 

9000 

579 

“ Conn, red.*.... 

590 

37.9 

Ivory .. 

16000 

1029 


2475 

159.1 

Leather belts, 1500 to 



“ Peach bot’m,* 3025 

5000. Good . 

3000 

193 

to 44100 

381^ 

94* 1 

Mortar, common, 6 mos 



Stone, Ransome’s artif.... 

300 

19.3 

old, 10 to 20. 

15 

.96 

Whalebone .... 

7600 

489 


To find the diam iu ins of a round rod to bear safely a given pul 

in lbs. 

Diam _ / g* ven P ult X coef of safety 

in ins = \J ult tensile strength 

^ of material in tbs per sq inch •' 004 * 

Iron is weakened by extreme cold. 

The belief (originating with Styff of Sweden,) is gaining ground that Iron an 
steel are not rendered more brittle by intense cold, but that the great number o 


* By the author’s trials with one of Riehle’s testing machines. Section 

broken 1>$ sq inches. 





























































STRENGTH OF MATERIALS. 


4G7 


breakages of rails, wheels, axles, &c, in winter, is owing to the more severe blows 
incident to the frozen and unyielding nature of the earth at that period of the year 
, ut Sandbergs experiments show conclusively that although these metals may per¬ 
haps bear as much steady force, gradually applied, in winter as in summer vet their 
resistance to impulse, or sudden force , is not more than W or as great in severe 
cold; which renders them less flexible and less stretchy. It is'probable that this 
tact does not receive as much attention as it should, in proportioning iron bridges &c 
no .\ e ex P eriments . wit h good wrought iron showed that even at 23° Fah or only 
9 colder than freezing point, there was a loss of strength of from 2b£ to 4 pei 
cent. * 


» 








s 


% 



4G8 


RIVETS AND RIVETING 


RIVETS AND RIVETING. 


R, Figs 3, p 469, shows the usual shapes of rivets as sold.* 

The weights iu the following table of course include the head; but the lengths, as usual, 
are taken •• under the head; ” or are those of the shanks only. In practice, discrepancies of 5 or 0 
per ct in wt may be expected. 

From Carnegie Bros. & Co's "Useful Information,” by C. L. Strobel, C E. 


Length 
of Shank. 
Ins. 

% 

K 

Diamet 

% 

ers of RI 

% 

vets In lr 

% 

iches. 

1 

VA 

1 X 

- 1 




Weight of 100 Rivets, In pounds. 



l A 

3.0 

8.5 







% 

3.8 

9.9 

17.3 





. 

i 

4.6 

11.2 

19 4 

25.6 

38.9 




l A 

5.4 

12.6 

21.5 

28.7 

43.1 

65.3 

91.5 

123 


6.2 

13.9 

23.7 

31.8 

47.3 

70.7 

9S.4 

133 

k 

6.9 

15.3 

25.8 

34.9 

51.4 

76.2 

105 

142 

2 

7.7 

16.6 

27.9 

37.9 

55.6 

81.6 

112 

150 

l A 

8.5 

18.0 

30.0 

41.0 

59.8 

87.1 

119 

159 

U 

9.2 

19.4 

32.2 

44.1 

64.0 

92.5 

126 

167 

k 

10.0 

20.7 

34.3 

47.1 

68.1 

98.0 

133 

176 

3 

10.8 

22.1 

36.4 

50.2 

72.3 

103 

140 

184 


11.5 

23.5 

38.6 

53.3 

76.5 

109 

147 

193 

U 

12.3 

24.8 

40.7 

56.4 

80.7 

114 

154 

201 

k 

13.1 

26.2 

42.8 

59.4 

84.8 

120 

161 

210 

4 

13.8 

27.5 

45.0 

62.5 

89.0 

125 

167 

218 

l A 

14.6 

28.9 

47.1 

65.6 

93.2 

131 

174 

227 


15.4 

30.3 

49.2 

68.6 

97.4 

136 

181 

236 

k 

16.2 

31.6 

51.4 

71.7 

102 

142 

188 

244 

5 

16.9 

33.0 

53.5 

74.8 

106 

147 

195 

253 


17.7 

34.4 

55.6 

77.8 

110 

153 

202 

261 


18.4 

35.7 

57.7 

80.9 

114 

158 

209 

270 

% 

19.2 

37.1 

59.9 

84.0 

118 

163 

216 

278 

6 

20.0 

38.5 

62.0 

87.0 

122 

169 

223 

287 


21.5 

41.2 

66.3 

93.2 

131 

180 

236 

304 

7 

23.0 

43.9 

70.5 

99.3 

139 

191 

250 

321 


24.6 

46.6 

74.8 

106 

147 

202 

264 

338 

8 

26.1 

49.4 

79.0 

112 

156 

213 

278 

355 

9 

29.2 

54.8 

87.6 

124 

173 

234 

306 

389 

10 

32.2 

60.3 

96.1 

136 

189 

256 

333 

423 

11 

35.3 

65.7 

105 

148 

206 

278 

361 

457 

12 

38.4 

71.2 

113 

161 

223 

300 

388 

491 


The diain of rivet* for bridge work is from % to 1 inch; usually % to 
and for plates more than .5 inch thick, it is about 1.5 times the thickness; 
and for thinner ones about twice; but these proportions are not closely adhered 
to. The common form of rivet* as sold is shown at R, Figs 3, a head 
and the shank in one piece; and S shows the same when after being heated 
white hot it is inserted into its hole, and a second head (conical) formed on it by 
rapid hand-riveting as it cools. When longer than about 6 in* they 
are cooled near the middle before being inserted, lest their contraction in cooling 
should split off tlieir heads. The hemispherical heads often seen, called snap 
head*, are formed by a machine. The two head* alone require aboul 
as much iron as 3 diams length of shank. Length of a head = about 1 
diam of shank; and its width about 2 diams of shank. 


Riveting of Steam and Water Tight Joints. 


Joints for boiler* and water-tight eisterns are usually proportioned abou 
as per the following table by Fairbairn ; and are made as shown either bv Fisr 1 
or Fig 2 Fig 1 is called a single-riveted, and Fig 2 a donble-rivetec 
lap-joint. The dist a a, or c c, is the lap. 

Mr Fairbairn considers the strength of pie single-riveted lap-joint to be abou 
.5b; and that of the double-riveted, about .7 that of one of the full unholet 


Price in Philada, 1888, about 3% cts per lb. 













































RIVETS AND RIVETING. 


469 


lates, when both joints are proportioned as 




a ^ a 

/ 

^ a. 

^-, 

W J 


1-^ 

-w w a 

1 

> 

f 

\ . 

oo o 

, . 

t c 

o 

O 0 

0 o o 

Fig i. 



Fig 2. 


in his following table. But some 
later experimenters consider about 
.5 and .6 as nearer the correct aver¬ 
age. Experiments on the subject 
are quite conflicting; and it is 
plain that no one set of propor¬ 
tions can precisely suit all the dif¬ 
ferent qualities of plate and rivet 
iron. With fair qualities of both, 
there is every reason to rely upon 
.5 and .6 (or about one-seventh 
part less than Fairbairn’s assump¬ 
tion) as safe for practice. These 


,. . , , _ , . , uuu ) its sate lur practice, inese 

roportions include friction (Art 4), without which they would be about .4 and .5. 


airbairn's table for proportioning: the riveting: for steam 
and water-tight lap-joints. 


Thickness of 

Diameter of 

Length of shank 

From center to 

each plate. 

ri vets. 

before driving. 

center of rivets. 

Ins. 

Ins. 

Ins. 

Ins. 

3-16 

% 

% 

IX 

X 

5-16 

% 

IPs 

ll 4 

I 

% 

13-16 

% 

2 

% 

15-16 


2 X 

3 

A 

1 Vs 

3/4 


Lap in single 
riveting. 


Ins. 

i M 

1 H 

2 

2 '4 

2 *% 

3j| 


Lap in double 
riveting. 


Ins. 

2 1 

11> 
2^ 
3 JL 
3'k 


n 

'>'A 


Aiimm 


Riveting of iron girders, bridges, dc. 

1 D r ■ ^ - 1 — 1 N KC 


■XT" 


to 


R 




w 



Figs 3. 


r 

o o o 

o o o 

M 1 

( 

0 O 0 

) 


P l 


Qj—^^-g> .. ^, Q 


0 ' ''' y Q' y V-V' 


-1 - ‘P 

J (> 1 


w 1 


Art. 1. The subject of riveting is abstruse, and involved in 
uch uncertainty; and experimental results are very discrepant. We here pro- 
>se merely to confine ourselves to what is considered the best joint; and for 
fety we shall omit friction; see Art 4. In girder and bridge work the lap- 
tints above described are seldom used. Instead of them, the plates p , Figs 3, to 
! joined, are butted up square against each other, thus forming a butt-joint, 
b Fig D; and are united by either a single covering-plate, cover, 
rapper, fish-plate, or welt e e, Fig K; or the best of all by two of them, 
at A, or o o, o o, Fig B. In what follows; the term plate never includes the 
vers. The single cover, like the lap-joint, allows both plates and cover to bend 
ider a strong pull, somewhat as at W, thus weakening them materially; whereas 
e double cover o o, o o. Fig B, keeps the pull directly along the axis of the plates, 
us avoiding this bending tendency. It also brings the rivets into double shear, 
us doubling their strength. When there is but one cover, it should be at least 
thick as a plate; and when there are two, experience shows that each had bet- 
r be about two-thirds as thick as a plate, although theory requires each to be 
it half as thick as a plate. 








































































470 


RIVETS AND RIVETING. 


The length w w of covers across the joint is equal to that of the joint. 

Buffs require twice as many rivets as laps, because in the lap each 
rivet passes through both the joined plates;, and in the butt through only one. 

Tlie rivets and plate on one side only (right or left)d thejomt- 
line i i of any properly proportioned butt-joint D, represent the full strength 
of the joint, inasmuch as those on one side pull in one direction, against those on 
the other side, which pull in the opposite direction. Therefore in designing such 
joints we need keep in mind only those on one.side, as is done in what follows. 
Thus a single, double, or triple-riveted butt-joint D implies one, two, or three 
rows of rivets on each side of the joint-line i i, and parallel to it. In a Prop¬ 
erly proportioned lap the strength is as all the rivets, because one-halt of them 
do not pull against the other half, but one end of every rivet pulls in one direc¬ 
tion, and its other end in the opposite direction. . .. . ._ 

TIi© not iron* n©t pint©* or n©t joint* is that which is left !>©■ w©©n 

the rivet holes, and outsit!© of the two outer ones, all on a straight line drawn 
through the centers of the holes of one row. Its width and area are called the net 
ones of the joint. That between other rows does not increase the strength. 

In Figs 3, N, and K, the rivets are plainly exposed only to single shear; 
that is, the opposing pulls of the two plates tend to shear each rivet across only 
one circular section; whereas iu Fig B, with two covers, or in 1 lg A, each rnet 
is exposed to double shear, one just above and one just below the joined 


plates. 

Art. 2. Bridge-joints are not required to be steam or water- ; 
tight like those of boilers or cisterns; and, therefore, by increasing the breadth , 
of the overlap, or the length of the covers, the rivets may be placed in several jj 
rows behind each other, as the 3 rows of 3 rivets each in M and D, instead of only | 
one row of 9 rivets, as in L. By this means, without losing any of the strength of L 
the 9 rivets, or of the net iron, we may narrow the width of the plate to an ex- ; 
tent equal to the combined diams (6 in this case) of the holes thus dispensed with 
in the one row. Moreover, by using more than one row we lessen the weakening t 
effect shown at W. This mode of placing the rivets directly behind each other in I 
several rows, as at M, and at the left-hand half of lig I), constitutes Mr lair- jj 
bairn’s chain riveting:; but the joint will be somewhat stronger if the rivets . 
are placed in zigzaging order, as in the right-hand half of Fig D. j 

The dist apart of the rows from cen to ceil should not be less 
than 2 diams. It is questionable to what extent this increase in the number of 
rows may be carried without an appreciable loss of strength in the rivets conse¬ 
quent upon the impossibility of quite equalizing the strains on the separate rows. 
But it is probable that if we'do not exceed 2 or 3 rows in laps, or the same num¬ 
ber on each side of the joint-line in butts, we may in practice assume that each 
row, and each rivet, is nearly equally strained. | 

Rivet-holes are usually of about one-sixteenth inch greater diam than the 
original rivet, so as to allow the hot rivet to be easily inserted The subsequent 
hammering swells the diam of the rivet until it fills the hole. We may either j 
take this increased diam of rivet into consideration, as we have done, in calcula¬ 
ting its shearing and crippling strength, as explained farther on, or with reference 
to increased safety we may omit it. Drilled ri vet-holes are said to be better 
than punched ones, as the drilling does not injure the iron around them; but on 
the other hand their sharper edges are said to shear the rivets more readily 
Hence, such edges are sometimes reamed off. Both these points are, however 
disputed; and both modes are in common use. 

The dist from the edge of a hole to the end of a plate or cover shoulc 
not be less than about 1.2 diams, to prevent the rivets from tearing out. the-enc 
of the plate; nor nearer the side edge of a plate than half the clear dist betweei 
two holes as given by the Rule in Art 5. The first is rather more than Fairbairi 
directs. 

Rivet holes weaken the net iron left between them, not only by th< 
loss of the part cut out, but either by disturbing the iron around them, or perhap I 
by changing the shape of the net line of fracture, which may not. then resis i 
tension as well as while it was a continuous straight line. Some deny both caus' 
and effect entirely, each party basing its opinion on experiments. But the mas 
of evidence seems to the writer to show that, the net iron loses on an averag 
about one-seventh of the strength due to the net width. With a view to safety 
which we consider to be of paramount importance, we shall in what follow 
assume (until the question is definitely settled) that there is such a loss o 
strength in the net iron. 

Riveted joint* for resisting compression should depend, not a j 
might be supposed upon their butting ends, but upon either the shearing or th j 
crippling strength of the rivets; for contraction or bad work may throw th| 







RIVETS AND RIVETING. 


471 


■ressure on the rivets, machine riveting- is somewhat stronger than that 
one (as is assumed in our examples) by hand. The thickness of plates 
ised in girders, tubular bridges, &c, is usually .25 to .5 inch ; with thicker ones 
ip to 1 inch sparingly in large ones. A packing piece, as the shaded piece 

ogether 0 by theYims betWeeU two P lates to prevent their being bent or drawn 

Art. 3. A riveted joint may yield in three ways after being 
•roperly proportioned namely, by the shearing of its rivets; or by the pullin^ 
part of the net plate between the rivet holes; or by the crippling (a kind of 
ompression, mashing, or crumpling) of the plates by the rivets when the two are 
oo iorcibly pulled against each other. It also compresses the rivets themselves 
ransversely, at a less strain than the shearing one; and this partial 
lelding of both plates and rivets allows the joint to stretch, and may thus 
roduce injurious unlooked-for strains in other parts of a structure, considerably 
etore there is any danger of actual fracture. Or in steam and water joints it may 
ause leaks, without farther inconvenience, or danger. For a long time this 
rippling had entirely escaped notice, and it was supposed that t he only important 
omt in designing a riveted joint was that the tensile strength of the net plate 
nd the shearing strength of the rivets should be equal to each other. 

The crippling strength of a, joint is as the number of rivets, in a lap 
r the number on one side of the joint-line in a butt X diam X thickness of joined 
late. This product gives the crippled area of the joint. We shall here call the 
iam X thickness of plate, the crippling area of a rivet. If there are 2 or 
lore plates (not covers) on top of each other at one joint, their united thickness 
; used for finding the crippling area. The ultimate crippling unit, 
y which the above product is to be multiplied for the actual ultimate crippling 
;rength of the joint, may be safely taken at about 60000 lbs, or 26.8 tons per sq 
ich. H 

The diam of a rivet in ins to resist safely a given single-shearing 
>rce is found thus: Mult the shearing force by the coef of safety, that is by the 
umber, 3, 4, or 6, &c, denoting the required degree of safety. Call the product g. 
tult the ultimate shearing strength per sq inch of the rivet-iron, by the decimal 
854. Call the product b. Divide^ by b. Take the sq rt of the quotient. The 
tearing force and the shearing strength must both be in either lbs or tons. 

Or by a formula, 


Diam in ins = 


V 


Shearing force X coef of safety 

Ult shearing strength per sq inch X .7854 


If the rivet is to be double-sheared, first mult only half the shearing 
tree by the coef of safety. Then proceed as before. 

Or, near enough for practice, mult the diam in single shear by the decimal .7. 
The ultimate shearing nnit for average rivet-iron may be taken at 
xiut 45000 lbs, or 20.1 tons per sq inch of circular sheared section. 


Table of ultimate single shearing strength of rivets. 

(market sizes), in single shear; at 45000 lbs or 20.1 tons per sq inch. 

This table is not to be used when as in our “Example,” Art 5, the 
rippling strength of the rivet governs the strength of the joint. 

1 If the rivet is in double shear it will have twice the strength in the 
1 ible. 

i For the diam in double shear to equal the strength in the table, mult 
i ie diam in the table by the decimal .7; near enough for practice ; strictly, .707. 


iam. 

us. 

Diam. 

Ius. 

lbs. 

Tons. 

Diam, 

Ins. 

Diam. 

Ins. 

Sis. 

Tons. 

Joiam. 

Ins. 

Diam. 

Ins. 

lbs. 

Tons. 

Vs 

.125 

552 

.246 


.562 

11183 

4.99 

1 

1.000 

35343 

15.8 


.187 

1242 

.554 

% 

.625 

13806 

6.16 


1.062 

39899 

17.8 

% 

.250 

2209 

.986 


.687 

16705 

7.46 

l 1 /* 

1.125 

44731 

20.0 


.312 

3452 

1.54 

% 

.750 

19880 

8.88 


1.187 

49838 

22.2 

% 

.375 

4970 

2.22 


.812 

23332 

10.4 


1.250 

55224 

24.6 


.437 

6765 

3.02 

Vs 

.875 

27060 

12.1 


1.312 

60885 

27.2 

A 

.500 

8836 

3.94 


.937 

31064 

13.9 

1% 

1.375 

66820 

29.8 








































472 


RIVETS AND RIVETING. 


The tensile strength of a properly proportioned joint is 

equally as either the sectional area of the net plate (not covers) across the cen¬ 
ters of only one row of rivets j or as the shearing or the crippling (as the case 
may be) areas of all the rivets in a lap, or of all the rivets on one side of the 
joint-line in a butt. The tensile strength of fair quality of plate iron, before the 
rivet holes are made, averages about 45000 lbs, or 20.1 tons per sq inch; but we 
shall for safety assume, as stated in Art 2, that the making of the holes reduces 
the strength of the net iron that is left about one-seventh part, or to 38500 lbs, 
or 17.2 tons per sq inch. 

Rem. Even this is considerably too great for laps, or for butts 

with one cover, owing to the weakening of the iron in such by the bending shown 
at W, Figs 3. But we are not speaking of such. 

Art. 4. The friction between the plates in a lap, or between th< 

plates and the covers in a butt, produced by their being pressed tightly togethei 
by the contraction of the rivets in cooling, adds much to the strength of a joint 
while new, perhaps as much as 1.5 to 3 tons per sq inch of circ section of all the 
rivets in a lap, or of all on one side of a single-cover butt; or 3 to 6 tons of all on 
one side of a double-cover butt. In quiet structures, this friction might continue 
to exist, either wholly or in part, for an indefinite period; but in bridges, &c, sub¬ 
ject to incessant and violent jarring and tremor, it is probably soon diminished, 
or entirely dissipated. Hence good authorities recommend not to rely on it, and 
it is, therefore, omitted in what follows. 

Art. 5. We now give rules for finding the number of rivets required for a 
double cover butt-joint (the only kind of which we shall treat), and their 
clear or net distance apart. This dist + one diam is the pitch of the rivets, or 
their dist from center to center. The principle of the rule will be explained 
further on, at Art 7, p 474. 

First, select a diam of rivet either equal to or greater than .85 times the 
thickness of the plate. In practice they are generally 1.5 times for plates x / 2 inch 
or more thick; and 2 for thinner than x / 2 in. 

Second, mult the greatest total pull in pounds that can come upon the entire 
joint by the coef ^3, 4, or 6, &c) of safety, and call the product p. 

Third, multiply the crippling area of the rivet (that is, its diam X the thick¬ 
ness of plate) by 60000. The prod is the ult crippling strength of a rivet. Call it to. 

Fourth, divide p by to. The quotient will be the number of rivets to sustain 
the given pull with the reqd degree of safety. 

Then, Che clear distance apart will be 


Number of rows X Diam X 60000 
38500 ' 

Fifth. The clear dist from either end hole of a row to the side edge of the plate, 
should be not less than half the clear dist between two rivets in a row. 

Example. A double-cover butt-joint in .5 inch thick plate is to bear an actua 
pull of 33750 lbs, with a safety of 4; or not to break with less than 33750 X 4 = 
135000 lbs. How many rivets must it have; and how far apart must they be? 

First, Here .85 times the thickness of the plate is .5 X .85 = .425 inch; there 
fore, our rivets must not be less than .425 inch in diam; but we will take .75 incl 
diam. 

Second, The greatest pull X coef of safety = 33750 X 4 = 135000 ft>3 — p. 

Third, The crippling area of a rivet X 60000 = .75 X .5 X 60000 = 22500 = m 


Fourth, — 


135000 

22500 


= 6 rivets required on each side of the joint-line. 


And the clear space or net width between them will be, if the 6 rivctt 
are in one row: 


Diam X 60000 
38500 


45000 

38500 


= 1.1688 ins. 


And the pitch = net space -|- diam = 1.1688 + .75 = 1.9188 ins, = - 

= 2.56 diams. 

In practice, to avoid troublesome decimals, we might make the net 8pace 1.2 ins 
and the pitch 1.95; but to show farther on the working of the rule, we adhere t 
the more exact ones. 

Fifth, The clear dist from each end hole to the side edge of the plate is half o 
1.1688 = .5844 ins. 

The entire width of net iron is equal to one clear space X number o 
rivets = 1.1688 X 6 = 7.0128 ins; and the entire width of plate is eqtial to on 
pitch X number of rivets, = 1.9188 X 6 = 11.5128 ins. 


li 

T 

01 

t 

( 

f 

n 

P 

fr: 

10 

« 

IS 


m 


!pe> 

tba 

is; 

8t; 

ao 

M 

jWj 

is 

















RIVETS AND RIVETING, 


473 


The area of cross section of unholed plate is 11.5128 X ••*> = 6.7564 sq ins; its ten¬ 
sile strength before the holes are made is 5.7564 x 45000 = 259038 lbs. 


I35000 

The strength of onr joint, omitting friction, is therefore - - -= .52 of that of the 
original unholed plate. 2o9038 

i lf the 6 rivets are in 2 rows of 3 rivets each, the clear dist be¬ 
tween two rivets in one row will be twice as great as before, or twice 1.1688 
, = 2.3376 ins. Pitch = 2.3376 + .75 = 3.0876 ins = 3.0876 = .75 = 4.12 diams. 
' <Clear dist from end hole to side edge of plate = half of 2.3376 =1.1688. 

i Entire width of net iron = 2.3376 X 3 = 7.0128 ins. Entire width 
i)Of plate = 3.0876 X 3 = 9.2628 ins. Area of cross section of unholed 
,plate=9.2628X.5=4.6314sq ins. Ultimate tensile strength, unholed 
= 4.6314 x 45000 = 208413 lbs. Ult strength of riveted joint, omitting 
135000 

friction = —- ■ - - = .65 of that of the unholed plate. 

ZUoilo 

Thus we see that the arrangement with two rows gives the same strength as one 
row, with a less total width and area of plate. It of course requires longer covers. 


If the 6 rivets are in 3 rows of 2 rivets each, the area of cross 
section of the unholed plate is 4.2565 sq ins. Its tensile strength, 

135000 

191542 lbs. Strength of riveted joint = ■ = .7 of that of the unholed plate. 

iy 

The entire width of net iron (7.0128 ins); its area (7.0128 X .5 = 3.5064 sq ins); 
sand its ultimate tensile strength (3.5064 X 38500 = 135000 tbs), are the same in each 
case. The last is the required breaking strength of the joint, as in the beginning 
of our example; and is equal to the combined crippling strength of the six rivets. 

Art. 6. The distance apart of I he rows, from center to center of 
rivets, should not be less than two diameters of a rivet-hole. 

Rem. 1. With our constants for tension, shearing, and compression, the 
rivets will not yield first by shearing in a double-cover butt (and 
of course in double shear), except when the diam is either equal to or less than 
.85 of the thickness of the plate, which will rarely happen. At .85 the crippling 
and shearing strength of a rivet are equal when using our assumed coeffs of crip¬ 
pling, shearing, and tension. 

i Rem. 2. Our example was chosen to illustrate the rule. It will rarely hap- 
i pen in practice that the rule will give a number of rivets without a fraction ; or 
»that may be divided by 2 and by 3 without a remainder. In case of a fraction, it 
|is plainly best to call it a whole*rivet; although the joint thereby becomes a trifle 
^stronger than necessary. Or rivets of a slightly ditf diam may be used. If the 
number of rivets comes out say 7 or 9, we may make 2 rows of 3 and 4, or of 4 and 
(5, &c. Moreover, the width of the plate is frequently fixed beforehand by some 
: requirement of the structure, and we must arrange the rivets to suit, taking care 
in all cases to maintain the calculated area of net iron in one row, &c. 


Rem. 3. We have (as we at first said we should do) confined ourselves to the 

simple butt-joint with 2 covers, and with the 
rivets in either 1, or in 2 or more parallel rows 
on each side of the joint-line; this being the 
strongest and the one in most common use in 
engineering structures. Necessity at times 
calls for less simple arrangements, for which 
we cannot afford space, and the strength of 
which is not so readily calculated. These 
sometimes yield results which appear strange 
to the uninitiated; thus, this lap-joint breaks 
across the net iron of one plate, along either c c or o o, where there is most of it, and 
where, therefore, it might be supposed to be the strongest. 

Rem. 4. The following table shows approximately the comparative 
strengths of the common forms of joints when properly proportioned; varying 
with quality of sheets, and of rivets: 



The original unholed plate. 

Double-riveted butt with two covers 
Double-riveted butt with one cover.. 
Single-riveted butt with one cover.., 

Double-riveted lap. 

Single-riveted lap. 


With 

Without 

friction. 

friction. 

1.00 

1.00 

.80 

.64 

.65 

.52 

.50 

.40 

.65 

.52 

.50 

.40 


34 





















474 


RIVETS AND RIVETING. 


Rom. 5. The above tabular strengths for the lap-joints will be approx¬ 
imately attained by adopting the following proportions, according as the joint is 
double- or single-riveted. 




Double rlv. zigzag. 

Single rlv. 



In thicknesses. 

In diams. 

In thickuesses. 

In diams 

Calling thickness of plate.. . 

1. 

.6 

1. 

.6 

Then make diam of rivet. 

1.67 

1.0 

1.67 

1.0 

it U 

breadth of lap . 

9.0 

5.4 ' 

5.67 

3.4 

it u 

pitch from cen to cen . 

7.0 

4.2 

4.5 

2.7 

U it 

dist from end of plate to 






edge of holes . 

2.0 

1.2 

2.0 

1.2 

it <( 

dist apart of rows from 






cen tp cen . 

3.33 

2.0 




Rem. 6. If two or more plates on top of each other, as the 

four in A B or M H, are to be jointed together so as to act as one plate of the 
thickness c c, the diams of the rivets, and the thickness of the covers c c, e e will 
depend upon whether the junctions of the plates are all in one line with each 
other as at c c, in A B, or whether they break joint with each other as at 0, 1, 2, 3 
in M H. 



It is plain that the two covers c c by means of their connecting rivets convey , 
from A to B, across the joint c c, all the strength that partly compensates for the .■ 
severance of the four plates at that joint; whereas the two covers ee,ee, and 
their rivets in like manner convey from n of one single plate, to o of the adjoining 
one, across the joint between those two letters, only the strength that partly com- i 
pensates for the severance of that single plate; and so with the joints at 1, 2, and ■ 
3. Therefore the covers c c, and their rivets, must be four times as strong as those 
at any one of the four joints 0, 1, 2, 3. The first, c c, are to be regarded as joining 
two solid plates A and B, each of the fourfold thickness c c ; and the others as . 
joining two of the single thickness. The covers c c will, therefore, each be about 
two-thirds of the thickness c c; and the others each about two-thirds as thick as 
a single plate. Thus, suppose each of the 4 plates in A B or M H to be % inch thick ; 
making c c 3 ins. Then each cover, c, is % of 3 ins, or 2 ins thick ; or the two covers’ , 
cc, together 4 ins, which is thus the effective thickness of the joint, cc. But each ’ 
cover, ee, is only % of % inch, or ]/ 2 inch thick; and the effective thickness of joint 
at either 0, 1, 2, or 3, is that of the 3 unbroken plates plus that of the 2 covers or 
(3 X %) + (2 X ^) = 3^ ins. 

Art. 7. Principle of the Rule in Art 5. With our constants for 

shearing (45000 lbs per square inch) and for crippling (60000 lbs per square inch), and 1 
with diameter of rivet equal to, or greater than, .85 times the thickness of the plate, 
as by our rule, the crippling strength of a double cover butt joint will be equal to, or 
less than, its shearing strength. Therefore, to avoid waste of material, either in the 
plate or in the rivets, we must make 


Tensile strength of net plate n . ,. . , 

across one row of rivets Crippling strength of all the rivets. Or, 

Total net 


Tot & I net 

width of V Thickness v Tension _ Crippling area v Crippling .. Total number 
plate of plate x unit “ of one rivet * unit X of rivets. 

Now, by Art 3, the crippling area of a rivet is = diam of rivet X thickness of 
plate. We take the crippling unit at 60000 ftts; and the tension unit at 38500 lbs. 
Therefore (transposing) we must make 


Total net width 
of plate 


Diam of v Thickness .. Total number 

rivet x nf nintu X oouuu x 


of plate 


of rivets 


Thickness of plate X 38500 














































RIVETS AND RIVETING. 


475 


By making the clear dist between each end rivet of a row and the side edge of 
late = half the clear dist between two rivets in a row; and calling the sum of the 
ivo end dists one space, we have 

Number of spaces _ Number of rivets g 0 t^at 
in a row in a row. 

The clear dist between f wo rivets in a row, which is 


ut 


Total net width of plate _ Total net width of plate 

■ ■ - — - ■ ■ - 1 1 Q #1 1 Q /"V ™ — — ~~ — “* 

Number of spaces in a row Number of rivets in a row 


Diam of .. Thickness .. Total number 

rivet X of plate X wuw X of rivets 


Thickness .. v. Number of rivets 

of plate X 138 UU X in a row. 


Total number of rivets 
Number of rivets in a row 


Number of rows. 


herefore, omitting “ thickness of plate,” common to both numerator and denom- 
liator, we have, as in rule in Art 5, 


dear dist 
apart 


Diam of rivet X 60000 X Number of rows 


38500 


But if the diameter of the rivets is less than .85 times the 
hickness of the plates, the shearing strength of a double-cover butt joint 
with our assumed constants for shearing and crippling) is less than its crippling 
trength. In such cases, for the clear dist between two rivets in a row, say 


Circular area of a rivet X Shearing unit 
Clear dist = ——- „ . . . ■ - v. — X 2 


Thickness of plate X Tension unit 


Rem. 1, Butt joints in double shear, or with 2 covers, being the 
nlv ones here considered, and inasmuch as rivets may always be used with a diam 
reater than .85 of the thickness of the plate, we may in practice always use the 
rule in Art 5 for such joints; and, therefore, we gave it alone. 

Bern. 2. When using- these rules for other kinds of joint, 
uch as laps, or butts with single covers, remember that the rivets in such are in 
ingle shear; and, therefore, we can use Rule in Art 5 (for crippling) only when 
[he diam is either 1.7 or more times the thickness of plate. If less, use 
l.ule above for shearing; all on the assumption that our foregoing coefs ot 
rippling and shearing are used. 

Bnt the coef for tension must be changed for each kind of these 
ther joints, to allow for the weakening effects of the bending shown at W, 1-igs 
, as deduced approximately from experiment. The writer believes that the fol- 
iwing tension units will give safe approximate results without friction. I-or 
lonblc-covcr butts, double-riveted, 38500 lbs per sq inch, as adopted above, 
i-or double-riveted laps, or one-cover butts, 28000. For single-riveted 
ips, or one-cover butts, 24000. But, as before remarked, no great certainty is 
ttainable in riveting. 

Item. 3. A joint may fall by crippling without the facts being 
nown or even suspected, for it does not imply that anything breaks, but 
lerely that the joint has stretched ; and this might not be detected even on 
slight inspection of it. Still it might, and probably often has sufficed to endanger, 
nd even destroy both bridges and roofs by generating strains where none were 
rovided for. 


i 











476 


STRENGTH OF MATERIALS 


Art. 7. Breaking' by shearing. Let 

abed , Fig 1, represent a beam, with its ends 
resting on supports SS; with a load l, so heavy 
as to break it by forcing its entire central part, 
oo gg, away from the two end parts adg and ben; 
so that while the two latter remain in their 
places, the central part slides out; or is, as it 
were, punched clean out from between them. 
This peculiar mode of fracture is called shearing, 
or delrusion. The force required to produce it, 
or the resistance which the beam opposes to such 
» force, may practically be assumed to be in pro¬ 
portion to the area of the sheared section. Thus, 
since the area of cross section of a beam 1 ft sq 



is 4 times as great as that of a beam 6 ins sq, the former will present 4 times as great 
a resistance to shearing; or will, in other words, require 4 times as great a load, or 
pres, to shear it across. In Fig 1 the total sheared area is equal to twice the trans¬ 
verse area of the beam. See Eye-bars and Pins, page 612. Bridge chords are ex¬ 
posed to great shearing force where they rest on the abuts, but it becomes less 
toward the center of the span; and so with every equally loaded beam. See p532. 

We have very few experimental data on this subject. 

Tlie shearing strength of white pine, spruce, 
and hemlock, parallel lo the fibres, by the author, 250 to 500 lbs 
per sq inch; oak 400 to 700; and is of use in estimating the 
resistance along the line c c, Fig. 2, at the end of a tie-beam; 
or at the head of a queen-post, Ac. 

Across the tibres the writer found for spruce about 
3250 lbs; white pine and hemlock 2500; yellow pine_4300 to 
5600; white oak 4400. 

Wrought iron is stated at 35000 to 55000 lbs per sq 
inch; cast iron 20000 to 30000; steel 45000 to 75000 lbs; 
copper 33000. 

The shearing strength of steel and wrought iron is about % 
part less than the tensile. The punching of rivet-holes in 
iron or steel plates, is an example of shearing. The rivets in 




Fig. 2'A. 


tubular bridges are frequently sheared in two, in time, by the motion of the plate 
through which they are driven. In punching holes, the area of section is evidently 
found by mult the circumf of the hole by the thickness of the plate in which it if 
punched. If a piece of material be supported as shown in Fig 2V£, its resistance t( 
shearing will be 3 times as great as in Fig 1, where it is sheared across in 2 places 
only; whereas in Fig 2%, shearing would have to occur at 6 places, as per the I 
dotted lines. 


c 


-a 

jh 


Art. 8. Breaking by torsion, or twisting. Let n, Fig 3, be a ver 

cylindrical rod of any material, 1 inch diam, the lower 
end of which is immovably fixed; and let c be a lever 
whose leverage a b, measd from the axis of 

the cylindrical rod, is 1 ft. Suppose that with a spring 
balance attached to the end b of the lever, we apply force 
horizontally, and around the axis of the rod as a center, 
until the rod breaks by being twisted. Then if we mult 
together the leverage a b (1 foot) and the amount of force 
shown by the spring balance in lbs, and div the prod by 
the cube of the diam of the rod in ins the quot will be a 
certain number of foot-pounds ; and will be what is called 


Fig. 3. 


n 




the constant , or coefficient for torsion, for all cylindrical bars of that material. Tf w 
use a square bar, we shall get the coef for square bars; and so w ith any other sbap* 
So that if with any other bar, or shaft, we mult the cube of its diam in inches b 





































STRENGTH OF MATERIALS. 


477 


said constant, and div the prod by the leverage in feet, the quot will be the force in 
tbs which will twist the bar in two. In shape of formulas, 

Breakg 
_ force Also 
in lbs. 


Leverage v Brkg force 
in feet x 


in lbs 


Cube of diam in ins 
Cube of diam 


= Constant. And 


Cube of diam x Constant 
m ins ^ 

Leverage in feet 


in ins 


X Constant 


Brkg force in lbs 


Leverage x Brkg force 

Leverage . in feet _ in lbs 

in feet. And Constant 


Cube of diam 
in ins. 


The constant for solid cylinders of average cast iron is about 600; 
ind for wrought iron 800. For puddled steel about 700 ; cast steel 1000 to 1700. 
brought copper 400. All may vary one fourth part of these more or less. 

For woods, rough averages. W pine or spruce 20 to 25 ft-lbs. ^ pine 35. 

ish 40. W oak 50. Locust 75. Hickory 85. . , . 

To find by the last formula, the diam of a rod to have a safety of 3, 4, 5, &c, 
iir nnst a Liven twisting force in lbs, first mult said force by 3, 4, 5, &c. as the case 
nay be and use the prod as the breaking force. The diam will then be the «>/> one. 

Anv angle described by the force at b, when made to revolve around the axis of 
be rod as a center during the twisting process, is called the angle of torsion, the 
en«-tb of the twisted rod or shaft does not affect the amount of force reqd to produce 
unture• but the longer it is, the greater will be the angle ot torsion; or in other 
.vonls the greater will be the (list through which the force must revolve around the 
ixis be for e^fractu re takes place. On the other hand, a long shaft will twist tluough 
v given angle under a less force than a short one of same diam and material; and u ill 
nore readfly bend under torsion. Authorities say that a working shaft should not 
wist more than 1°. We should not expose it to more than .1 of its ult strain. 

If we know the force in lbs per sq inch reqd for shearing any material, see pre- 
iing ArMhen the force required to break a cylinder of it by torsion, is 
Torsion One-half the shearing y 3 1416 x Cube of rad of 
force _ force in lbs per sq inch x x cylinder in ins 

in lbs — Leverage in inches. 

That of a square shaft is about 1^ times that of a round one whose diam 
s equal to a side of the square; or about A less than that of a round one oi the 
same transverse area. For any solid rectangular shaft 


Breaking 
Torsional force 
in lbs 


One-third of the shearing v The square of x The square of 
force, in lbs per sq in x one side * the other side. 


Square root of the sum ofThe x Leverage 
above two squares in inches. 


Hollow shafts resist torsion better than so lld ,°° e , s . of tlie same areaof 
metal. Calling the outer and inner diams in ms D and d, then 

Breaking (D4 — d 4 ) X Constant 
Torsionalforce = in ft X D 

in . ., 

£ may S,fT- doffing lol 
tfS'of Sir VXwnfbaUiW. weightf.nd will supported * proper 
ntervals, say 8 or 9 ft., by self-adjusting ball and socket hangers. 


Diam of a wrought 
iron shaft in ins. 


-</ 


Horse-powers 

Number of revs 
per minute 


X125. 


Or in words- for the diameter in inches div the number of horse-powers that are 

.f horse-pow™ the less is the torsional strain upon it. This .nay at first seem 
t r „ ™ b ,?t less so when we reflect that a horse-power is made up of pres and d,sl ; 
berSe the faster it moves, the less Is its pressure. Hence many horse-powers re- 
„Mng rapidly will require a less diam than a small number revolving slower in 
uoportiou than its number. 





















478 


STRENGTH OF MATERIALS. 


TRANSVERSE STRENGTH. 


Art. 9. Transverse (or across) Strength.. Sometimes called Relativf 
Strength. 

When either a load l, Fig. 1, or any ether vertical force acts upon a horizontal 



beam f o fixed atone end, the beam becomes a lever, having a tendency to revolve 
about the fixed end,/, as a supporting fulcrum, and in so doing to strain or break 
the beam by forces of tension and compression, acting horizontally or length¬ 
wise of the beam, pulling apart the upper fibres and crushing together the lowei 
ones. This is indicated by the arrows in the figure, those near the load repre¬ 
senting respectively its pull upon the upper fibres, and its push along the lower 
ones, while those near the wall represent the corresponding reactions of tlu 
support. 

The load or other force together with the weight of the beam also tends tc 
break the beam by shearing or cutting it across vertically , as shown at Fig. 1 
p. 532. But. we shall here treat only of the first of these tendencies; namely, th> 
tendency to bend or break the beam (Fig. 1) by compressing the lower fibre, 
and extending the upper ones. 

Such a tendency exists at each cross section as t, in the length of the beam, and 
is called the moment of rupture, or breaking moment, or moment 
of the load, at such section. It is measured (as are all moments, see pp 
335, etc.) in foot-tons, foot-pounds, or inch-pounds, etc.; and is found thus: 

Moment of rupture about t = The load X its leverage about t. The 
leverage about t is the shortest, or perpendicular, distance v s from the section 
t to the line of direction a m of the force by which the beam is strained. When 
this force is a load l, acting vertically as in Fig. 1, the leverage v s is necessarilv 
the horizontal distance between the given section, t, and the load, l. 

The product, or moment, will be in foot-pounds, or inch-tons, etc., according^ 
as the leverage is measured in feet, or in inches, etc., and the load in pounds o*i 
in tous, etc. 

For a given load, the greatest moment, in a beam like that in Fig. 1 i: 
plainly the moment about the point, f of support; for that is the section about 
which the load has the longest leverage. 

If the load, Instead of being concentrated, like /, is distributed in ant 
way along the whole or a part of the beam, its leverage is measured from th« 
given section perpendicular to the line of direction of its center of gravity ; whict 
is plainly the case also with a concentrated load, because its line of direction als< 
passes through its center of gravity. Thus, the weight of that portion, o tor of 
b ig. 1, of the beam itself, beyond the given section, t or /, acts as a distributed load 
having a moment of its own, about t or/, equal to the product of its weight multi 
plied by the horizontal distance from its center of gravity to the given section 
or/- lint for the present we shall, for the sake of simplicity, neglect this momen 
of the weight of the beam itself, and consider only that of th eTextraneous load, l 

Before the beam bends, its leverage is evidently greater than afterwards anc 
it becomes less as the bending increases; but as very little bending is allowet 
in practical cases the leverage may generally be assumed not to change but U 
remain as when the beam is horizontal, 












STRENGTH OF MATERIALS. 479 

Let Fig. 2 represent a rigid beam, incapable of yielding except by rupt ure along 



m 


the vertical plane i e, pulling away from the wall at i «, and crushing into it at 
e v, so as to assume such a position as that indicated by the dotted lines. To do 
this, it must cause a longitudinal stress in all the fibres from top to bottom of 
the beam at the section in e, viz.: a tensile stress in those above n, and a com¬ 
pressive stress in those below n. The greatest strain is at the top and bottom 
fibres; and from them both ways it diminishes until at n it is nothing. This is 
indicated by the varying distances between i e and v s. In an actual beam, 
which must be more or less flexible, the load also stretches or compresses the 
fibres lengthwise at every vertical section along the entire length of the beam, 
more or less according to its leverage and moment at said section; most near the 
fixed end and least near the free end; so that the extent of stretch indicated by 
s i is the sum of the accumulated stretchings that have taken place in the top 
fibres at every point from i to a. 

A beam o o, Pig. 3, supported at only one point, whether at the 


a 

S ' - 


a 

-\s 


V 



center or not, and balanced by two either equal or unequal loads, may plainly be 
regarded as two levers, each of which is essentially in the same condition as that 
in Fig. 2. Whether the loads are concentrated or distributed, their leverages v s t 
v s, are (as before) to be measured from v perpendicularly to the lines ofdirection 
a to, a to, of their centers of gravity as in Fig. 2. Both the 5 ton loads are mani¬ 
festly upheld by the support, which of course reacts vertically upwards againt 
them in a vertical line with their common center of gravity n, and with a force 
of 10 tons. 

Rein. 1. Suppose each lever, v s Fig. 3, to be 4 feet. Then, since each end 
load is 5 tons, the moment of each load about the fulcrum n would be = 5 X 4 = 20 
foot-tons. Hence it might seem that over the support the fibres of the beam 
near n would have to resist a combined moment of 40 foot-tons. But they have 
actually to present a resistance of but 20 foot-tons, on the same principle that if 
two men pull against each other at two ends of a rope, each with a force of, say 
30 lbs, the strain or pull on the rope is not 60 but only 30 lbs. because strain is 
the reaction (pressure or pull) against each other of two equal opposing forces, 
and is equal to only one of them. The two above equal moments are merely two 
forces acting through leverages. 

A beam, supported at botb ends, ami loaded at only one 

point, whether at the center or not, with a concentrated load, as in Fig. 4, 



















480 


STRENGTH OF MATERIALS. 


may also like Fig 3 be regarded as two levers with their common fulcrum at n 
in a vert line with the cen of grav of the load. This however is by no means 
so manifest at first sight as in Fig 3, but needs a little explanation. Let the 

beam bear 10 tons concentrated at its center, then 
evidently 5 tons of it will rest pressing down- 
wards on each end support; and each support 
will therefore press upward or react against an 
end of the beam with a force of 5 tons as per the 
arrows. Now these two 5-ton reactions of the sup¬ 
ports in I ig 4 are to be considered as taking the 
place of the two 5-ton end loads in Fig 3 • while 
in . , * he 10 -ton load in Fig4 takes the place of the 

10-ton reaction of the .rapport in Fig 3, and hence in this view of the c«is no longer 
to be considered at all as load, but merely as a fixture for holding the common ful¬ 
crum n of the two levers in place, or in equilibrium with the upward end reactions 
Being no longer regarded as load, it of course cannot in such cases be assumed to 
have any moment of rupture; that property being now transferred to the end 



Fig . 4. 


.j, i 7 v 1 — J ^ ' unuwnicu tu me ena 

morit r ,, t0 av ? ld awkwardness ot expression we always speak of the mo- 
ment of the load even in such cases, rather than of the moments of the reactions 

of the load. In both Figs 3 and 4 the forces at work are the same in amount but 
plainly reversed in direction . amount, mu 


6 tows 


Rem. If the load is distributed as the 6 tons in Fig 5 instead of 
concentrated as in Iig 4, we still consider the beam as consisting of two levers 

with their common fulcrum n in a vert line w ith 
the cen of grav c of the load. But to find the mo- 
nicnt of the load (or more correctly, the moment 
ot the reactions of the supports) about n we must 
proceed a little differently. Thus let the beam be 
3 It span, and the load uniform, weighing 6 tons 
and being 1 ft long. Find by rule, p 481, how 
much ot this load rests on each support, (4 toils on 
a, and - tons on o.) The upward reactions of the 
supports will therefore also be 4 and 2 tons. Then 
first hud the moment about n of either one of the reac- 

1 . 1 hl4 lrmmpnt will / a a _ v . - «. . ^ 




e 

4- vL 




e n 


i 

4 


Fig 

.5. 

b 

2 


tions, say of the 4-ton one at«. This moment'wBlpl^nVie^ tons'll 4 

mcnt of the 6 -ton load about ». g '' 15 ~ 3 ‘ 25 ft_ton for the mo- 

agloTand part'oflKalto^^n ith ' h ° ho 5 leTO " 

other point than a sea pp 481 tom ““ ° and To find lhe momcut for any 


iw'gtoe'hw majr , sti " b* found by 

longitudinal St,Sins, SJ. to “e, teclTe cSeif th f 

compile,itiou. We confine ourselves therefore to lior beams. ^ , involving much 

In a cantilever. Fig 29»4, a load or portion of a load causes a mo- 
ment or strain in every part of the beam between said load or nor- 



to'u h “,"S t U°/ 8 ‘Imifo™'" J^b^toalatM. When require, 

and add the result to that obtained for the load. * 01 Case 10, p 482 and 483 

The deflection in ordinary cases may be found by the rule on page 506. 

















STRENGTH OF MATERIALS. 


481 


General Rule for moments of rupture In hor cantilevers, no 

natter how irregularly the load or loads may be distributed. Bear in mind that 
inly that part of the load which is beyond (towards the free end from) any as¬ 
sumed point tends to break the beam at that point as a fulcrum, and that it does 
so with a leverage = dist of the cen of grav of that part of the load from the point. 
Hie other part of the load has no moment at that point. Thus the whole load 
) x tends to break the beam at g or i with a leverage = a g or a i as the case may 
>e, a being the cen of grav of the load. And so for the moment at any other 
joint c, Fig 29, as a fulcrum, find the wt of all the load c x between c and the free 
:nd l of the beam. Also find the cen of grav 
of that part of the load. Mult the weight 
ust found by its leverage c s. 

Example 1. We use a uniform load in 
:>rder to illustrate the rule more readily. Let 
he hor yellow pine beam it be 7 ft long; its 
►readth and its depth i e each 6 ins; the whole 
oad oi4 tons; and c the point or fulcrum at 
vhich the moment of the load is reqd. Then the wt of the load between c and t 
s 3 tons; and its cen of grav s is 1.5 ft or 18 ins from c. Hence the load’s moment 
t c = 3 tons X 18 ins leverage = 54 inch-tons. That is, a load of 3 tons tends 
rith a leverage of 18 ins to rupture the beam at c. 


Fig. 29. 



Example 2. Let Fig 30 be a rolled 
roil I beam cantilever of the cross-section 
liown in ins at S, projecting hor 10 ft or 120 
is, and bearing a concentrated load of 2 tons 
t its free end. The moment of the load at the 
jetiou it is = 2 X 120 = 240 inch-tons. 


Fig. 30. 


cc 


120 ' 


4 > 


S O 

2LU* 


] 

1 

? ig. 31. 

m 

ft w, 

m 

1 

4 

K 

8 

rH 

i 6 a sco x n 

tO 

L- 

rH 


General Rule for M ofltup in hor beams supported 
Jit each end, no matter how irregularly the load or loads may be distributed. 
*>i i n, Fig 31, be such a beam of yellow pine 
f 6 ft or 72 ins span, 6 ins square, and loaded 
nth 3 tons. First find the cen of grav c of 
ie v/hole load a x, and what portion (1.25 
nd 1.75 tons) of said load rests on each sup- 
ort i and n, thus, as whole span : whole load 

: either arm : portion at other arm. Con- _ ,, , . , 

der the upward reactions thus found (1.25 and 1.75 tons) of the two supports to 
e two forces acting vert upwards against the ends of the beam at i and n as de- 
oted by the arrows. Let o be any point whatever in the beam at winch as a lul- 
rum the load’s moment is required. Assume either of the upward end forces, say 
le 1.25 tons at i, to be acting at the outer end i of a lever i o (4 ft long) of which 
is the fulcrum. Mult this force (1.25 tons) at i by this leverage t o (48 ins) Call 
le product (60) p. Find the cen of grav ( s ) of the part load a o (2 tons) between 
and o. Mult said part load by the dist o s (12 ins) of its cen of grav (*) from the 
iven point or fulcrum o. Deduct the product (24) from p The remainder (36 
ich-tons) will be the moment at o of the total load a x. The same result will 
>llow if we use n and the 1.75 tons reaction, but with the load x ©. 

Rem 1. If there is no load between i and the fulcrum point, as would 
3 the case if the moment had been reqd at any point between i and a Jnstead of 
; o then the above p by itself is the moment, thus e is 12 ins from i, hence the 
loment at e of the entire load a x is 1.25 X 12 = 15 inch-tons. 




































482 


STRENGTH OF MATERIALS 


Rem. 2. Although in Fig 31 the load is regarded as having no moment at either 
end i or n of the beam, yet it produces shearing forces there equal to the upward 
reactions. See “Shearing,” p 532. 


Although the foregoing general rules apply to all the following cases 
still these last will often expedite calculations. 


32 


a 


CANTILEVERS. 

Case 1. Concent rated load at free end. Fig 32. 


Greatest moment is at o, and = load X on. At any other point 
a it is = load X «• Make o v = greatest moment, join v n; 
then a c is the moment at any point a 




33 


Case 2. Concentrated load at any point a, Fij 


ft /r> 33. 

o \c fy 




•§ 

Greatest moment is at o, and = load X » «. At c it is = 
load X c a. Make o v — greatest moment, join v a. Then c e it 
the moment at any poiut c. The load has no moment between 
a and n. 


Case 3. 


VK „ 

\\G 

N 


34 


1 




Uniform load throughout, Fig 34. Greatest moment is a 
o, and = whole load X half on. At n it is nothing At anj 
point, a it is = load on aw X half an. Make ov — greatest mo 
merit, draw the dotted parabola with its vertex at n. Then a 
gives the moment at any point a. 

If the load is not uniform the greatest momeu 

is = whole load X dist from o to its cen of grav. 

Case 4. Load on one part. Fig 


n 


35. Greatest moment is at o, and = load X ‘list from o to cen 
of grav c of load. At any point t between the load and o, mo¬ 
ment = load X t c. At any point a in the load, moment = 
load on a s X dist a e of the cen of grav of load on a s 
from a. 



(CC S 

Fig. 35 


W 


X 


J 


U 




4- 


y 


m 


c e a g 

Fig. 36 


Case 5. Several loads, wxy. Fig 36. Fin 1 

their centers of grav c, a, *. Greatest moment is at < 
and = {w X c o) -f (x X « «) + (y X * o). Or first fin 
the common cen of grav of all the loads, and mult it 
dist from o by the sum of the three loads. Between tli 
loads the moment at g = y X ff *; at e = {y X * e) 

(x X a e) ; and at n = (y X««) + (x X « n) +(»Xcn 


Case 6. One uniform load and one local 
one. Fig 37. Greatest moment is at o. Find that of the 
uniform one by Case 3; and that of th« local one by Case 
4, and add them together. No moment between a and «. 


Fig 38 



Fig. 3 7 


S 




o 


/ 




BEAMS SUPPORTED AT BOTH ENOS. 

Case 7. Concentrated 
load at center. Fig 38. 

Greatest moment is at center, and = half load X hal 
span. At the supports it is nothing. Make c s — momei 
at center, join s o, s «; then n t = moment at any jioii 
n. Or the moment at any point n = half load X a n, 
being the nearest support. 

Case 8 . Concentrated 
load not at center. Fig 


m 


N 

_j_ 


a 


j 




n 


s 

/v. 


89. Greatest moment is at the load, and is = load X « o 
X e a -j- o a. Make e. s = moment at load, join s o, s a; 
then at any point c the moment is c t. Or at any point 
c, between load and o, moment = load XaeXoc-^-oa. 
At any point m, between load and a, moment = load 
X o e X a wt o a. No moment at o or a. 


0 . 


fit 


A 


\ Fig. 3 

\ 


a 


c e 


m 


i 





































STRENGTH OF MATERIALS. 


483 


VigAO/?\ n 


Case 9. Several concentrated loads x y z, Fig 40. By Case 8 find 

the greatest moment of each load separately, 
and for each of them draw its dotted vertical 
and two inclined lines as in this fig. Then for 
the moment at any point whatever ase, measure 
the vert dists (in this case e o, e a, e c) to the 
sloping lines, and add them together. For it 
is plain that at e we have e o for the moment 
of the load x at that point; e a for that of the 



Case 10. Uniform load 

mi 

» Fig 1 . 4-1 • 


l [j load y ; and e c for that of the load z ; and so at 


any other point. Or make en = eo + ea-\- 
e c ; also make s i and to h respectively equal 
to the three dists above s and to, and joiny h i 
n k. Then at any point along the beam / k the 
vert dist to these upper lines gives the moment. 

from end to end. Fig 41. 


y 



s 

m 

1 

w 

C/Z'A 



0 

e c 

a 


ter is zero or nothing. 


The greatest 

moment is at the center c, and is = half load X quarter 
span. At any other point e moment = half load on e o 
X e a; or to half load on e a X e o. Make cs — half load 
X quarter span, and draw a parabola o s a, then at any 
point e the moment is = e t. Or, moment at any point = 
half what it would be at that point if the whole load were 
concentrated there. 

The shearing or vertical strain at the cen- 
See Art 11, p 535. 


Rein. 1. The weight of the beam itself is usually such a load, but is frequently 
so small compared with the load that in this and other cases it may be neglected. 

Rem. 2. In the case of a uniformly distributed load like that in figure 12, page 
535, the greatest moment of rupture that can occur at any given point on the span is when 
I the load covers the span from end to end; and in beams or trusses of uniform depth 
the hor strains at any given section are then also greater than under any partial 
I load; so that if the chords are then strong enough in every part, they will be strong 
enough for any partial load; which is not the case with web members; any one of 
which is most strained when the longest segment reaching to it is loaded. See p 536. 

Case 11. Uniform load from a support to part way across. 

Fig 42. Find the cen of grav g of the load, and by p 481 what portio'n of it rests 

on each support o and x. Then by General Rule, p 481 
the mom at o or x — nothing. At n or at any point a be¬ 
tween n and x it is = portion or reaction at x X asn (or 
a: a as the case may be). At any point c between n and 
o moment is = reaction at x X % c — (load on c n X 
half c n) or to reaction at o X o c — (load on o c X half 
o c). This plainly applies to unequal loads also.f 


Fig 1 . 42. 


mo, 


r 

y 


r 

n 


a 


5b 


Fig. 43. 






To find the place of g-reatest moment t if the load is uni¬ 
form say, as twice xo: n ounoznt. When the load covers 
the whole beam it becomes Case 10. 


Case 12. Uniform load reaching; to neither 
• support. Fig. 43. From either support proceed as from 

x in Fig 42, except as to greatest moment, -which find by trial.* 


* On this subject see “ Humber’s Strains in Girders,” to which the writer is chiefly indebted for the 
foregoing. 

t Except that then the dist from the given point c to the cen of grav of the part-load c o or c n is 
not necessarily equal to half c o or half c n; and if not, said dist must be used instead of said half 
c o or half c n. 





























484 


STRENGTH OF MATERIALS. 


The moment of rupture at any point t in a bent piece R with a load 
upon or suspended from c, is equal to the load X its leverage 11, perp to c to. 

This moment tends to break 



draw o n, o*tn, also o w vertical 
parallelogram o ew c of forces. 


the piece R at its cross section 
at t by tearing apart the fibres 
to the right of its neutral axis, 
and by compressing those to 
the left of it; and to this mo¬ 
ment the piece R opposes the 
moment of resistance of that 
section as in the case of a beam. 

In an arched piece as 
S loaded at any one point o, 


and equal by scale to the load, and complete the 
Then will o e and oc by the same scale give two 
forces into which the load is resolved, and actiug in the directions o m, on, much 
as the two strings of two bows o a n, o u m. The force o e tends to break the bow 
o u m at any section u with a moment = the force X its leverage u v drawn from 
the point, and perp to o m; and the force o c tends to break o an at any point in 
the same way with its leverage. The section at u or elsewhere resists as in R. 
The weights of the pieces R and S themselves have not been taken into con¬ 
sideration. 


RESISTANCE OF BEAMS. 

Having the moment of rupture of tile load, it is necessary to know whether 
tlic Moment of Resistance, or simply the Resistance, of the beam is 
sufficient to withstand it. 

The foregoing' instructions for finding moments of rupture apply to 
horizontal beams of any form of cross-section, and whether said 
cross-section is solid, as in a common wooden beam ; or open, as in a bridge-truss• 
or whether, as in the rolled I beam or plate girder, the web is solid, but of small 
cross-section as compared with that of the flanges. 

But in treating of the action of the beam in resisting this moment 
of rupture, we shall first consider only those beams in which each 
fibre, throughout the entire cross-section, is to he regarded as opposing' the 
moment of rupture by a horizontal or longitudinal resist¬ 
ance. This is always the case in beams of solid rectangular, cylindrical oval 
Ac, cross-section ; and (strictly speaking) in those with thin solid webs, as I beams 
and plate girders; but in plate girders it is usual, on the score of safety and in view 
of the small cross-section of the web, to neglect its share of the resistance and 
thus to regard the girder in the same light as a truss, or “ open beam. - ’ Tor the 
manner of resistance of such beams, see pp 52S, Ac. 






STRENGTH OF MATERIALS, 


485 


THEORY OF RESISTANCE OF CLOSED BEAMS. 

Scientists give the following, which, however, often differs from experiment in 
beams of composite cross-section, such as I beams, &c. See example, p 489. 

In a closed beam, Fig 2, p 479, each of the fibres throughout the entire 
depth of the yielding section i n e opposes the breaking moment of the load by a 
Resisting' Moment or Moment of Resistance of its own. As the breaking 
moment about n of the load is made up of its gravity-force or weight mult by its 
leverage or perp distance n c from the fulcrum or neutral ax n, so the resisting 
moment about n of each separate fibre, say for instance the one at i, is made up of 
its natural longitudinal resisting force or strength mult by its leverage or perp dis¬ 
tance n i above or below the same fulcrum n; and the sum of all these separate 
moments is the moment of resistance of the cross-section, i n e, of the 
beam. A line, passing through this fulcrum, transversely of the beam, and at right 
angles to the action of the breaking force, is called the neutral axis of the 
given cross-section. 

The longitudinal resisting force or strength, in pounds per square inch, of those 
fibres which are farthest from the neutral axis, is equal to the ultimate tensile or 
compressive strength of the material in pounds per square inch, plus the assistance 
in lbs per square inch, which they receive from their natural adhesion to each 
other. This adhesion resists the longitudinal sliding of the fibres upon each other, 
without which the beam cannot break. This total resistance, in pounds per square 
inch, of the fibres farthest from the neutral axis, is called the Coefficient of 
Resistance (frequently, but less aptly, the “coefficient, constant, or modu¬ 
lus, of rupture”) of the material of which the beam consists, and is usually 
denoted by t4 C.” It is shown (page 489) to be = 18 times the center breaking 
load, iu pounds, of a beam of the given material, 1 inch square X 1 foot span; as 
given iu the table, page 493. 

The other fibres of the beam are of course capable of exerting a resistance equal to 
that of the farthest fibres; but, owing to their less distance from the neutral axis, 
they are less stretched or compressed , when the beam yields. The}’ are therefore 
unable to put forth their entire resisting power. The length, through which any 
fibre in abending beam is stretched or compressed (ie, the extent to which it is 
lengthened or shortened), is plainly (Fig 2, p 479,) proportional to its perpendicular 
distance from the neutral axis; and, inasmuch as the longitudinal resistance which 
it actually exerts is proportional to the lengthening or shortening of it, it follows that 

Perp dist from Perp dist from Coefficient of resistance Longitudinal re¬ 
neutral axis to . neutral axis to . . (or longitudinal resist- . sistance of said 

farthest fibre. • any given fibre • • ance of farthest fibre) in • given fibre, in lb» 

lbs per sq in per sq in 

therefore, 


and 


Coefficient of y Dist from neutral . . Area of 
resistance * axis to given fibx-e ' X> given fibre 

Dist from neutral axis to farthest fibre 
C t' a 


or 


t:t'::C:f; 


Longitudinal resist¬ 
ance of given fibre, = 
iu 1 faster sq in 


or 


Coefficient of 
resistance 


X 


Dist from neutral 
axis to given fibre 


Dist from neutral axis to farthest fibre. 
C t f 

/ = 


t 


Longitudinal resistance __ Its longitudinal resist- .. Its area 
of given fibre, in lbs ance in lbs per sq in insqins 

(F = fa) 










486 


STRENGTH OF MATERIALS. 


The moment with which a given fibre resists the rupture of the beam is plainly 

= Its longitudinal resistance X Its perp dist from the neutral axis 
or r = F V 


Coefficient of Dist from neutral Area of 
resistance X axis to given fU>re X given fibre 

Dist from neutral axis to farthest fibre 


Dist from neutral 
X axis to given fibre 


or 


r — 


C V a 
t 


t' 


Coefficient of resistance Area of 
Dist from neutral axis to K' veu fibre X 
farthest fibre 



Square of dist from neu¬ 
tral axis to given fibre 


It follows from the above that the strengths of solid or “ closed" beams are as the 
squares of their depths; although in “open" beams, page 528,&c, it is simply as the 
depths. 

The Mom out of Resistance of the entire cross-section of the beam is 


Coefficient of resistance 

—- v 

Dist from neutral axis to 
farthest fibre 


rThe sum of 1 

/Area 

Square of \ 

I the products for > c 

,f (ofthe 

X its dist from 1 1 

L all the fibres J 

'fibre 

neutral axis/ J 



This sum. or “ I,” is, for convenience, called the Moment of Inertia of 
the cross-section of the beam about its neutral axis GG. For beams of rectangular 
cross section, such as Figs p 487, it may be found by the rules on that page. For 
irregular sections, as Fig 14V£ above, it cannot be found by ordinary arithmetic, but 
an approximation, sufficiently close for all practical purposes, may readily be made 
thus: Both above and below G G, and parallel to it, draw lines j k, l ?n, &c, dividing 
the section into narrow strips. If these lines are equidistant, the subsequent calcu¬ 
lations will in some cases be easier; but otherwise it is immaterial whether they 
are so or not.. If they are drawn no closer together, proportionally to the size of the. 
.figure, than in Fig 14*^, the approximation will be near enough for practical pur¬ 
poses. The closer they are the more accurate will be the result; but however close 
they may be, it will always he a trifle ton small. Begin by finding the area in so ins 
of the first strip x xj k , below G G. Mult this area by the square of the dist Or to 
the cen of grav of the strip. Then proceed to the next stripy kl ??i; find its area 1 
find mult it b} r the square of the dist o s to its cen of gr s. So with each strip below 
G G. Add all the prods together. If the section has the same shape, size and posi¬ 
tion above G G as below it, (as would be the case with a square. I beam, or circle ) 
mult their sum by 2. The prod will be the reqd moment. But if, as in Fig 14b<, the 
section above G G differs from the portion below it, we must div it also into strips 
and proceed as with the lower part. The sum of all the products on both sides of 
the neutral axis will then be the moment of inertia. 



















STRENGTH OF MATERIALS, 


487 


The position of the hop neutral axis G G, may be found by cutting 
it a correct figure, 4 or 5 ins long, of the section, drawn on thick paper or tin, and 
dancing it over a straight edge. The line at which it balances is G G. When this 
vs been done, the dist o g, in ins, to the farthest fibre, (which may be either above 
' below G G, according to the shape of the Fig,) can be measured in ins. 

tie true moment _ The approx moment , The moment of inertia of each strip 
of inertia found as above l m n p. &c, about its own neutral axis 

ie neutral axis of each strip to be taken parallel to that of the whole figure. 

Or (which amounts to the same thing) 

The moment of inertia of The sum of the moments of 
the whole figure about = inertia of the several strips 
any given neutral axis about the same neutral axis, 

i which 


Moment of inertia of Its moment of in- / » I)ist® from neu-\ 
each strip about the neu- = ertia about its own -j- j , ^ X tral axis of fig \ 
tral axis of the whole fig neutral axis \ area to that of strip / 

This affords a convenient method of finding the moment for figures, like these be- 
•w, made up of rectangles; the moment ot each rectangle about its own neutral axis 
iing = its breath X its depths 12. 


From the above it follows that in any liollow figure, as A, B, D or G, p 495, 
• in figs 1 to 5 below 


a b 


i 



The moment of in Mom of in of the en- Mom of in of 
ert” about an} = * £re - % ( includi »S the _ the missing 
given neutral axis missing parts) about parts about the 

the same axis same axis 

Thus 

Mom of in of Mom of in of Mom of in of 
channel abed = rectangle a b g h — rectangle ede/ 
efg h about G G about G G about G G 


The moment of inertia is plainly independent of the material of which the beam 
insists, of the span, and of the manner in which the beam is supported or loaded; 
id is the same for all beams of a given cross-section. It follows from the foregoing 
iat it is proportional to the cube of the depth of the beam. 

Moments of Inertia of a few well-known figs are given below. Those of 
niilar figs are to each other as their breadths X cubes of depths. 

Square. (Fourth power of side) 12, whether any side or diagonal is vert. 
Parallelogram, rectangular or otherwise ; neutral axis parallel to either two 
' the sides. Breadth X (cube of depth) -t- 12. The breadth must be measured 
irallel to the neutral axis; and the depth at right angles to it. 

Hollow square or rectangle. [(B X D 3 ) — (6 X <7 3 )1 -f-12.* 

Circle. Rad* x .7854. Semicircle. Rad 4 x .1098. 

Ring. (Outer rad 4 — Inner rad 4 ) X .7854. 

Ellipse. Long diam vert. Half short diam X (half long diam) 3 X .7854. 
Elliptic ring. Long diam vert. Let L. S, l, s, be half the long and half 
e short diems. Then [(S X L 3 ) — (.? X 2 3 )J X .7854. 

Triangle. BaseX Perp Ht a -^36. The base is that side which is parallel to 
e neutral axis. This does not apply to hollow triangles. 

<b> 



A 

r > 

"A 



d 

D d'$ p: 


-Y- 

v < 


2 


d 




b' 

Fig 3. 


< b > 

A tr 

JCc 

d , 

(. 

-TV "7\ v 


d' d'E 

> 

V 

b' 


< b' > 

Fio 4. 


Fig 5. 


1. (B.D 3 — 2 b.d 3 ) -=- 12. 2. (B.D 3 + 2 fed 3 ) -=• 12. 3 mid 4. [5.d 3 +■ b'.d' 3 
(b' _ 0) d" 3 ] -f- 3. 5. [b.d 3 — (b — k).{d — c) 3 + b'.d' 3 — ( b' — k).{d’ — d) 3 } + 3. 


I# B and b are respectively the outer and inner dimensions parallel to the neutral axis, whethei 
id axis be lengthwise or crosswise of the figure. D and d, are the outer aud inner dimensions par. 
ndicular to the neutral axis. 








































488 


STRENGTH OF MATERIALS 


From the last formula on page 486, we have 

Coefficient of Resistance 

Moment of Resistance = Dust from neutral axis to X Moment of inc 

farthest fibre 


(—40 


Fop beams of sqnare or rectangular cross-section, this t 

comes 

Coefficient of v Area of cross-sec- v depth 
Moment of Resistance * tion in sq ins A in ins 

Resistance = —- 


6 


(r = c -r) 


In rolled I beams, the moment of resistance is approximately found thu 
all the dimensions in inches. 

moment /Area of cross- % area ofv Depth Elastic limit o 
of resist- = I section of one -1- cross-section! X of X iron in lbs per s 


ance 


of web 




beam inch 


\ flange 

the area of web being = its thickness X extreme depth of beam ; and the area 
one tlange being = (area of whole l>eam — area of web) -i- 2. For average roll 
iron, the elastic limit may be taken at 22400 lbs or 10 tons; or about half the ul 
mate tensile strength. 

When a beam is upon the point of failing, its 

Moment of rupture is = its moment of resistance. 

In other words 

Co-efficient of resistance 


Load in lbs X span In ins 


in lbs per sq in 


lit 


X 


Moment of 
inertia, ins 


Dist from neutral axis to 
farthest fibre, ins 

(“ = t0 

Here m is, according to circumstances, 1, 2, 4, 8, Ac, as follows: 

Wbcu the beam is firmly fixed at one end, and loaded at the other, m = 
“ “ “ “ “ uniformly, m =» 

“ “ merely supported at both ends “ at the center, m = 

“ “ “ “ “ uniformly, m == 

“ “ firmly fixed “ “ at the center, m = 

“ “ “ “ * uniformly, m = 


Therefore, 

Total breaking 
load in lbs 


Moment of inertia X Co-efficient of resistance X m 
Dist in ins from neutral axis to farthest fibre X span in 


or W 


IC TO 
t l 


For the neat, or extraneous, breaking load ; if uniformly loaded deduct the weij 
of the beam itself. If supported at both ends and loaded at the center, or if fixed 
one end and loaded at the other, deduct half the wt of the beam. 




















STRENGTH OF MATERIALS. 


489 


4 ? 3 18 a tabIe of ce ] Dter breaking loads in pounds, for beams 1 inch square 
and ol 1 toot span, supported at both ends. In such beams square, 

Depth 3 X Breadth 1 

12 


Moment of inertia = 


m (p 488) is 4; the dist 
the span in inches is 12 . 


12 


in ins from the neutral axis to the farthest fibre is and 
1 he last formula on page 488 becomes, in such cases," 

Total center break- = Coefficient of resistance X X 4 Coefficient of 
mg load in lbs 1 --- - 


or, in other words, 

Coefficient of 
resistance of any = 
given material 


i X 12 

(w-£ 

V 18 


C 


=18 w) 


18 


18 times the center breaking load of a beam 
of the given material, 1 inch square, 1 foot 
span, supported at both ends 

I>"slte cross-section, such as l.oil.™ ™, Z' f," forlwuiuo com- 

m„ark.d g i, « re3 ,", 8 differing fro,,, those if e* £™US X„s *' “ e “'* y 

nr ^T n V 8 he C6Uter breakin ^ 1<,ad of ' a solid cast-iron beam 4 ins' square and 6 ft 
or 1 2 ins, clear span, supported at both ends ? 11 rt ’ 

Here the moment of inertia is 4 ^ 43 


4X64 
12 

cast-iron 


256 


12 

on p 


= 21.333. The coeff 


. 12 
«L)To SIS o anCe ’ 18 ti,nes olir constant for , JU ,, 

3b4o(). Since the beam is supported at both ends, and loaded 

rhe dist o <7 °f the farthest fibre from the neutral axis must in a square be equaT to 
ot one side; consequently it is here 2 ins. The clear span is 72 ins. Ilence, 

ireakg Mom of In X Coeff of res X m 21.333 X 36450 V 4 31 mas-* 

load — -— — 


493, is 2025 X 18 = 
at the middle, m is 4 . 


— 21600 lbs — 9.64 tons. 


Dist o g of farthest fibre X span 2 X 72 144 

By our table, p 502, a beam of average cast-iron, 4 ins deep, 1 inch broad, and 6 ft 

oreak wit a h k VdW 2 i 41 consequently, four such, or one 4 ins square, would 

jieak with 2.41 X 4 = 9.04 tons; thus confirming the accuracy of the foregoing, 
applied in the same way to solid cylinders, the result corresponds equally well with 
experiment and with our table on p 503. H * 

lint for Jflr Clark’s hollow squares, p 516,the formula gives 3.06 tons in¬ 
stead ot the actual 2.15; and for his hollow cylinders 2.980 instead of 2 287 A 
true Ilodgkiiison beam, p 518, with top flange of 1 by 3 ins, bottom flknge 
.5 by 12 ins, vert web .75 inch thick, total depth 15 ins, clear span 20 tt, has a 
noment of inertia of <80; dist from neutral axis to upper fibre 10.7 ins. and to the 
owest one 4.3 ins. By Hodgkinson it would yield at the lower flange, aud by bis 
ule with a center load of 29.24 tons. By the formula it would be 19.8 tons. Beam 
’ P °20i actually broke with about 52 tons; by the formula it would be 40. 


The CoefT of resistance for average rolled iron is about 45000 lbs or say 
tons per sq inch. Cast-iron 36000 lbs or 16 tons. Good straight-grained well- 
asoned white pine or spruce 8100 lbs or 3.6 tons; yellow pine 9000 or 4 ; good oaks 
KKMJ or nearly 4,n. But as large beams are liable to defects and imperfect season¬ 
al*’ not more than about two-thirds of these constants should be used in practice. 


35 

















490 


STRENGTH OF MATERIALS. 


Comparison between models ami actual structures. Ma y 

practical men imagine that if a model is strong, an actual bridge, roof, Ac. con¬ 
structed with precisely the same proportions, must be equally stiong inprop r ' 
tion to its size. This arises from their ignorance of the fact that the strenfeiu 
of similar beams, trusses, Ac, increases only in proportion to the squares oi tneir 
spans; while their weight increases as the cubes of the spans; so tnatamortel 
5 or 10 ft long may show a great surplus of strength; while the roof or bridge ot • 
or 100 ft span, constructed like it in every respect, may break down under its own 

" \Ve may compare the two in the following manner: Let us suppose a model 4 feet 
long of a bridge truss, its wt 6 lbs, and the extraneous center load reqd to break it 
120 lbs, or 20 times its own wt Then its entire center hreakg load, including nan 
its own wt, is 120 + 3 = 123 lbs. Now suppose we are going to build a bridge truss 
of 200 ft, or 50 times the span of the model. The strength ot the truss will be 50 
or 2500 times that of the model; that is, it will require for its entire center breaKg 
load 50* X 123 = 2500 X 123 = 307500 lbs. Its wt, however, will be 50 , or 12am 
times that of the model, or 125000 X 6 = 750000 lbs; and one-hall of this weight oi 
375000 lbs, must be deducted from its entire center strength, in order to tmu us ex¬ 
traneous center load. But in this case the half weight is greater than tbe entiH 
center strength; consequently the truss would break under its own wt. It, insteal 
of a center load in the model, we had broken it by an equally distributed one, tin 
calculation would plainly he the same, except that in the model the entire weight 
instead of of it, would be added to the extraneous load lor the entire distribute! 
breakg load; and in the truss, its whole weight must be deducted from its breakup 
strength, to get the extraneous distributed load. 

If the break ins; loinl of a mortel is 2, 3 or 4, &c, times as great as it 
weight, then a similar structure 2, 3 or 4, Ac, times as large in every particular wil 
break under its own weight. 





STRENGTH OF MATERIALS. 


491 


PRACTICAL METHODS FOR FINDING STRENGTHS OF 

BEAMS. 


find constants. In 

beams of the same material, and 
exactly alike, except in their 


Fig-. 4. 



breadths, n d , the strengths vary 
in the same proportion as those 

breadths; that is, if one is 2, 3, or — 

10 times broader than the other, its strength will be 2, 3, or 10 times as trreat Tf 

porl theR streneths wHl\Je 1 L C l !^w/l! I l= t +>w?Z. 8 l^”!^_ a, ^ t ^? e,1 .. the Points of sup- 


port, their strengths will be inverse!^ those Tn^’S' ZZts or?0 
t mes longer than the other, it will be but & or ^ part as strong. If they are 

or ot“the!?dM»?h?™®? 8U '' e< * Tq’ their stren S ths wil1 be directl -'> ™ 

■x- t w jH .,i so k„ I p i. ’ at ,s ’ one 2, 3, or 10 times as deep as the other, 
i win also be 4, 9, oi 100 times as strong; or in other words will reanire 4 9 nr inn 
imes as great a load to break it. See Art. 11. It mustbeiSt he 
iow speaking only oi strength, or resistance to breaking; and not of SL orTesist- 
mce to bending, or deflecting. Stiffness follows laws very d Km 
those of strength. See Art 26, &c. y ,rom 

^SlhlTi 00 ?’" 6 ! 811 tb , e tbree fore e°ing elements of size, namely, length, 
jreadth, and depth, we have the fact, that the strength of any beam, of any size', of 


i 

< 


ny given material, is in proportion to its _ breadth X the square of its depth. 


There- 


t . its length 

ore, if we find by actual trial, what center load will break any beam of known size; 

md then find what is the proportion between its ^ e _ adth X 8 d of de P tb , anJ 


•reakg load, said proportion (or, more strictly speaking, ratio ) will also be that 
rhich any similar beam has to iG breakg load, and will therefore serve to calcu- 
ate the breakg load of any other similar beam of the same material. For instance 
f we take any piece of average good white pine, say 6 ins broad, 10 ins deep and 12 

eet clear span, we find that its _ breadtb X sq of its depth L 6 X 100 

length 1 1 12 ' 


7® f r f dual !-7 load , tbi f at its center until it breaks, we shall find that the 
leakg load, including half the wt of the clear span of the beam itself, amounts to 


breadth v sq of depth 

111 ins in in a 


2500 lbs. Therefore, the proportion between the in ins in ins aad tIie 

length in feet 


lull w LR ||| r 

reakg load, is as 50 to 22500; which is the same as 1 to 450; that is, the breakg 
readt^ tb S q (?fd?pth C Udlng half lts owr * wei S bt , may be found by mult it! 

in ,nS ,n " >S b y 45 °- And in this same manner may be found the total 


length in feet 

mter breakg load of any rectangular beam of average quality of white pine. For 
in neat load one-half the wt <<f the clear span must be deducted- It is self-evident 
°-f a 10 beam . assists to break it, as well as the neat load; and the 
(tent to which it does so, is tho same as if onedialf of its unsupported- wt were 
moentrated at its center. Hence the rule. On this principle the rule in Art 12 is 
ised. The ratio thus found for any material, Is called its eoef for ecu 
*y n ..*7 0 ^. its constant for the same, as it does not vary with the 

zo ot the beam. It wo take a piece, all of whose dimensions are 1, as 1 
. ., . . , , , breadth w sq of depth 

eh wide, 1 inch deep and 1 ft span; then the in ins * in ins will be 

X a< l °f 1 , _ , , , long till lilt'et " 

1 ; Rnd tho breakg load (including half its own weight) of such a piece 


; at P. n , c0 ,’ tbe e .^, n , stant r ® ( l d -, Remarks l to 4. In an average piece of white 
ne, this load wuli be iouud to be about 450 tbs; or the same as the constant obtained 
om the large beam. So the student may find them for himself t 
td if he uses materials not included in our table, Art 10.lt will be well to supply 
e deficiency, by inserting his own results. 1 ^ y 

r *ho foregoing directions for finding coefficients may be more briefly expressed 






















492 


STRENGTH OF MATERIALS. 


by a formula, 
one-half the wt 


After finding the neat center breakg load, by experiment, add to It 
of the clear span of the beam, for a total center load ; then the 


Coef for breakg 
strength 


Span v Total load 

in feet A in lb s_ 

Breadth v Square of depth 
in ins in inches. 


In a cylinder, as the breadth and the depth are each equal to the diarn, 
it is plain that B X amouuts to the same thing as diarn*; and it is always so 

expressed „ , , . , . , 

Rem. 1. The variation in strength of equal beams of the same material is so great, 
that it is necessary to experiment with several pieces, in order to find an average for 
a constant. The loads given in the preceding tables are also constants, but tor crush¬ 
ing and tension. They are averages of the strengths of the materials, derived from 
experiment. The actual strength of any particular specimen, if of superior quality, 
may be considerably greater than the average; or on the other band, if of very poor 
quality, it may fall as much below it. We should always keep this in mind when 
referring to any table, of constants; and if we have doubts as to the quality of the 
piece of material which we are about to employ, we should make a corresponding 
deduction from the constant in the table. . 

Rem. 2. If, instead of pine, we had experimented with oak, iron, stone, 
the process for finding the constant would have been precisely the same. If in¬ 
stead of a square beam, we use cylindrical, or triangular ones, or any other 
shape, such as hollow cylinders, H, T, or U beams, Ac, we shall in the same way 
establish constants for either larger or smaller beams of those shapes, and of precisely* 
the same proportions iu every part. See Remark, p 509. Or if, instead of support¬ 
ing the beam at both ends, we secure it firmly at one end, and load it at the other 
end until it breaks, we shall obtaiu the constant far beams fixed at one etui, aiul loaded 
at the other, Ac. Remember that the constants are for loads at 
rest. If they are liable to jars, jolts, vibrations, Ac, a large margin must be 
left for safety. Moreover, the constants given in tables arc generally deduced from 
small specimens free from important defects; whereas large beams of any kind of . 
material usually contain irregularities, which diminish their strength; and on this . 
account larger allowances for safety should be made as the dimensions ot the beam 


increase. 

Rein. 3. It is not necessary that band d he taken in ins. and 

lengths in ft. They may all be in ins, ft, yds, or any other measure, but siuce in 
every-day practice we usually speak of the breadths and depths of beams in ins, and 
of their lengths in ft, it becomes more convenient so to consider t hem. 11 ot her meas- ! 
ures be used, the constant will of course be diff; but it will still be such that if the 
same measure be used for calculating the strength of another beam, the final result 
will be the same as before. In like manner, the loads may all be taken in tons, Ac, 
instead of lbs; but in giving the rule, it must be stated what measures have been 
employed. 

Rom. 4. There are peculiarities in some materials, which les¬ 
sen the reliability of constants derived from experimenting with small pieces 
Thus, a large beam of cast iron will break with a less load in proportion than a small ' 
one • because, in the interior of thick masses of that material, more time to cool it f 
required than in the outer surfaces; in consequence of which, there is a want of uni¬ 
formity in the arrangement of the particles of iron, and this conduces to weakness 
AH we can do in such cases, is to exercise judgment and caution in making sulficient 
allowance for safety. 

Art. 10. Table of constants or coefficients for tlie quiescent 
breaking loads of rectangular beams, supported horizon¬ 
tally at both ends, and loaded at the center; being the average qui 
escent breaking loads in lbs (including one-half the weight of the beams themselves 
for beams 1 inch square, and 1 foot clear length between the supports. For safeti 
in practice , not more than about % to % of these constants should be employed; de 
pending upon the importance of the structure, its temporary or permanent charac '' 
ter, and the degree of vibration to which it will be exposed. Thus a roof will prob 
ably be as safe at %, as a bridge at Even with a pei'fectly safe load, a beam ma; 
bend too much. See Art 26. 

If any of these coefficients be mult by .5S9 (or say .6) it will give that for a cylin 
drical beam whose diarn — side of the square. Or if mult by .71 it will give tlia 
for a square beam with its diagonal vertical. Any of these constant 
may vary one-tliird part either more or less. For any bean 

Cen. Breaks Breadth (ins) X Square of depth (i n*) y 
load iu lbs, ~ clear span iu feet. 















STRENGTH OF MATERIALS. 


493 


One Third part of any ot these constants (except those for wrought 
iron and steel), may be taken in ordinary practice as about the average constant 
lor the greatest center load within the elastic limit. The loads 
here given lor wrought iron and steel, are already the greatest within elastic limits. 


Transverse Strengths, in lbs. See explanation, Art 10, p 492. 


WOODS. 

Ash, English . 

“ Atner White (Author). 

“ Swamp. 

Black. 

Arbor Vitie, Amer. 

Balsam , Canada. 

Beech, . 

“ Amer . 


. * 

. 0 

. 9 

. 9 

.5 

. 2 ? 

Birch , Amer Black.. .^4 

“ Amer Yellow. 3 

Cedar , Bermuda .... .^ J 

“ Guadaloupe.... . - - 

“ Amer White, .) 

or Arbor Vitae..... j ^ 

Chestnut .g o 

Elm, Amer White.c 2: 

“ Bock, Canada.of » 

Hemlock . 

Hickory, Amer.. 

“ “ Bitter nut 

Iron Wood, Canada. 

^Locust . .. . 

L’gnurn Vitie . 

jLarch . 

1 Mahogany . 

Mangrove, White. 

Black. 

\Maple, Black. 

“ Soft. 

Oak, English. 

Amer White (by Author). 
“ Red, Black, Basket.. 

1 " Live — .. 

%Pine, Amer White...(by Author) 
“ Yellow. “ “ 

“ Pitch. .. “ “ 

Georgia.. 

Poplar ... 


o 
o 

sri 

o-a. 

03 S. 

• §.§ 

:*§ 


a 

u 


u 

u 


''turn .... 

•Spruce .(by Author) 

“ Black. 

Sycamore . 

"amarack . 

r eak..... .. 

Walnut . 

Willow . 


METALS. 

trass ... 

ron, cast, 1500 to 2700 ...average 

“ “ common pig . . 

“ “ castings from pig .... 

“ employed in our ta¬ 
bles. 

“ “ for castings 2or 3 

ins thick. . 

j r on, wrought, 1900 to 2G00.av 

1 Wrought iron does not break; 


650 

650 

400 

600 

250 

350 

850 

550 

850 

400 

600 

250 

450 

650 

800 

500 

800 

800 

600 

700 

650 

400 

750 

650 

550 

750 

750 

550 

600 

850 

600 

450 

500 

550 

850 

550 

700 

450 

550 

500 

400 

750 

550 

350 


850 

2100 

2000 

2:100 

2025 

1800? 

2250 


but at about the average of 2250 
lbs its elas limit is reached. 

Steel, hammered or rolled; elas 
destroyed by 3000 to 7000.. 
Under heavy loads hard steel 
snaps like cast iron, and soft 
steel bends like wrought iron. 

STONES, ETC. 

Blue stone flagging, Hudson River 
Brick, common. 10 to 30..average 
“ good Amer pressed, 30 to 

50.average 

Cien Stone . 

Cement, Hydraulic , English Port¬ 
land, artificial, 
7 days in water 
1 year in water 
“ “ Portland, King¬ 

ston, N. Y., 7 
days in water. 
“ “ Saylor’s Port., 7 

days in water. 
“ “ Common U. S. 

cements, 7 dys 

in water. 

The following hydraulic ce¬ 
ments were made into prisms, in 
vertical moulds, u uder a pressure 
of 32 lbs per sq inch, and were 
kept in sea water for 1 year 
Portland Cement, English, pure, 

1 year old... 

Roman Cement , Scotch, pure. 

American Cements, pure, av about 

Granite, 50 to 150.average 

“ Quincy . 

Glass, Millville, N. Jersey, thick 
flooring .(by Author). 
Mortar, of lime alone, 60 days old 
“ 1 measure of slacked lime 

in powder, 1 sand . . 

“ 1 measure of slacked lime 

in powder, 2 sand 
Marble, Italian, White (Author) 

“ Manchester, Vt, “ “ 

“ East Dorset, Vt, “ 

“ Lee, Mass, “ “ 

“ Montg’y Co. Pa, Gray “ 

“ “ “ Clouded “ 

“ Rutland.Vt, Grav “ 

“ Glenn'sFalls,N.Y.Black “ 

“ Baltimore, Md, white, 

coarse.“ 

Oolites, 20 to 50. 

Sandstones , 20 to 70.average 

“ Rod of Connecticut and 

New Jersey. 

Slate, laid on its bed, 200 to450, av 


5000 


125 

20 

40 

25 


30 

50 


30 

26 


64 

23 

25 

100 

100 

170 

10 

8 

7 

116 

95 

111 

86 

103 

142 

70 

155 

102 

35 

45 

45 

325 
























































































494 


STRENGTH OF MATERIALS. 


Art. 11. General facts respecting: the breaks loads of a uni¬ 
form beam of any form ot section. Calling the breakg load, 

When the beam is tirmly fixed at one end. and loaded at the other . 1 

Then when so fixed, and uniformly loaded, it will be. *“ 

When merely supported at both ends, and loaded at the center. 

u u “ “ “ and uniformly loaded . ° 

Firmly fixed, or tightly confined at both ends, and loaded at the center .. 8 

t. J J U ° *. “ “ and uniformly loaded . 12 

A beam is said to be *' fixed” at either end when the tangent to the longitudinal 
axis of the bent beam at that end remains always horizontal. 

Rem. 1. When one beam of any form of section is 2, 3, or 4, &c, times as long, broad, 
and deep, as another, its weight will be 8, 27, or 64, &c, times as great, or as the cubes 
of the linear dimensions; but its breaking load will be only 4, 9, or 16. &c, times as 
great; that is, the strengths of similar beams are to each other as the squares of 
their respective linear dimensions. But in these breakg loads are included the 
weights of the beams themselves; one-half of which must be deducted when the 
load is all at the middle of the beam; or the whole of it when equally distributed. 
When beams are of the moderate dimensions usually employed in buildings, their 
own weight is usually so small in comparison with their loads, that at times it may 
safely be neglected; but as they become longer, since their weight increases much 
more rapidly than their strength, it at hist constitutes too important an item to be 
overlooked; for the beam may become so great as to break under its own weight; 
although proportioned precisely like u small one which may safely bear many times 
its own weight. 

When a square beam is supported on its edge, instead of on a 

side (or in other words has its diag vert), it will bear but about x 7 0 ths a6 g l ' eat a 
breakg load. 

Tlie deflections or bendings of beams (see p505) are directly in 
proportion to the load, and to the cube of the length; and inversely to the breadth, 
and to the cube of the depth, all while within limits of elasticity. 

Rf.m. 2. It is very important to remember that a beam will bear a much greater 
load placed upon it in small amounts at a time; or (in case it is all applied together 
if its pres he allowed to come upon the beam very gradually, than if it be placet 
upon it suddenly, even without any jarring, and without any previous momentum. 
When applied in small amounts, or gradually, the bending takes place slowly and 
with slight momentum; but when applied all at once, the great load descends im¬ 
mediately and rapidly to the full extent of the bending due to the load itself, and in 
so doing acquires a momentum which carries it still further; thus producing a strair 
which authorities maintain to be just twice as great as in the former case. A heavy 
train coming very rapidly upon a bridge, presents a condition intermediate betweei 
the two. The tables of constants always suppose the load to be applied very gradi 
ually, but as this is frequently not the case in practice, an allowance must be mad 
accordingly. 

Art. 12. To find tine quiescent breakg load of a lior square 
or rectangular beam 15. j» 491, of any material, supports 
at both ends, and loaded at tine centre. 

Rule. Mult together the square of its depth d o in ins. its breadth n d in ins, an , 
the coef from the table p. 493. l)iv the prod by the clear length a a between tl. 
supports, in ft. The quot will be the reqd breakg load in lbs: including, faoweve i 
half the wt of that portion of the beam which lies between the supports, and wide 1 
must be deducted in order to get the neat load. 

Ex. What will be the center breakg load of a beam of Connecticut red sandston | 
15 Ins wide, 10 ins deep, and 12 ft long between its supports ? Here 
Depth 5 w Breadth .. Constant 

in ins X in ins X _ _ 100 X 15X45 67500 = 6625 ft)g the breai j 

Clear length in ft 12 12 

load reqd. 

But we must deduct one-half the wt of the clear length of the beam. Now abea 1 
of red sandstone, of 15 ins, by 10 ins, by 12 ft, contains *21600 cub ins — 12% cub 1 - 
and a cub ft of red sandstone weighs about 140 lbs; therefore the beam weig 
12.5 X 140 = 1750 lbs; one-lialf of which, or 875 lbs, must be taken from the 56 I 
lbs of breakg load, leaving 4750 lbs as the actual extraneous, or neat breakg load. • 
Rem. 1. If cylindrical, first find the breakg load of a square beaij 
of which each side is equal to the diam of the cylinder. Mult this load by the dec ! 
or, more correctly, .589. Hence a square one is 1.7 times as strong as a cylindru ; 
one. 



















STRENGTH OF MATERIALS. 


495 


If oval, or elliptic, first find the load for a rectangular beam, whose sides 
are respectively equal to the two diams, and mult it by .6. 

If of wood, and triangular, and its base (whether up or down) hor, 
first find the breakg load for a rectangular beam, whose breadth is equal to the 
base; and its depth equal to the perp height of the triangle; and take one-third 
of the result as an approximation. When the edge is down, the ends must rest in 
triangular notches in the supports; otherwise, they will be crushed when loaded. 
See Art 16. 

For boaniN of such sections as A to G, the following rude rules of 
thumb will often be preferred to more intricate ones, being sufficiently approxi¬ 
mate for ordinary purposes, and for any material. See near end of Art 33, p 516. 

For the closed Fig's A, II, 1), G, (each one supposed to be of equal 
thickness throughout,) first find the load for asolid beam of the same size and shape. 



Then find that of a beam of the size of the hollow part. Subtract the last from the 
irsf. 

For C, (its top and bottom being of equal size,) first find fora rectangular beam 
i ana. Then for two beams corresponding to the two hollows v v. Subtract these 
ast from the first. 

For E or F. find for three separate beams r r, i i, t n, and add them together. 

For angle and T iron, see p 525. 

For U and other shapes in common use, we may use the formula, p 488, 
r experiment with a model made of the given material, and thus find the necessary 
onstant, as directed, p 491. See Remark, p 509 ; also Art 33, p 516. 

I For I beams, see Art 37, and for Hodgkinson beams, see Art 
i 5. 

Rem. 2. In this case we may remove % part of the 
nateriai of the solid square, or rectangular beam, without 
iminishing its breaking strength, although it will bend 
lore. The width may remain uniform, and the depth be re- 


SIDE 


m 




TOP 


TM 

-I 


-I 


n 

Fig. 5. 




■M/M 
ipM 
m 
m 




need either at top or bottom, as shown by the dotted lines 
t rn, Fig 5, strictly two parabolas with bases at load, but the 
iijtraight ones are best in practice. Or the depth may remain 
niform, and the breadth be reduced, as shown by the dots at 
ft , which is a top view of the beam. Theoretically, the dotted 
lies in n might meet at the ends of the beam ; but in prac- 

ce this would not generally leave sufficient material at the ends for the beam to 
,'St upon securely. 

Such reductions of beams are rarely made when they are 
wood; but in iron ones much expense may be saved 
lereby. 

Rem. .3. Load at one end of the beam, 

i 'lg G. the oilier end fixed, imagine the load to 
1 * e at the center, and calculate it by the foregoing rule. SHIC W 

'hen div the result by 4. 

In this case the lower side of the beam may be cut away 
i the form of a parabola, as shown by the dots. To draw 
lis curve, see page 153. Or the depth may be left uni- 
rm, and the sides be cut away, as shown by the dots at t, which is a top view. 

Art. 13. When the load is equally distributed along the 


Fig. 6. 




i 


Fig. 7. 


r 


iitire clear length of a horizontal beam, 
upported at both ends, as in Fig 7, instead of be- 
g all applied at the center, assume it to be at the center, 
id proceed precisely as in the foregoing rule, Art 12. 
mu mult the load by 2. But in this case the wt of the 
tire, clear length of the beam is to be deducted for the 
>at load. 

Ex, What will be the equally distributed breakg load of the beam of sandstone 
the last example ? Here the center breakg load has already been found to be 
































































496 


STRENGTH OF MATERIALS. 


5695 lbs • and 5C25 X 2 = 11250 lbs, the reqd distributed load. From this subtract 
the wt of the entire 12 feet clear length of beam, or 17oO lbs; and the rem, 9o00 lbs. 
is the neat extraneous breakg load. About part of this, or 950 lbs, is quite as 
much as should be trusted upon so variable and treacherous a material as red sand. 

8 Rem. 1. A beam requires twice as much breaking load, equally distributed, as it 
will at its center. In this case the breakg strength of the 
beam will not be diminished if the top be cut away in the form 
of a true semi-ellipse, as shown by the dots in Fig 7. Or it the 
depth must be kept uniform, the sides may be trimmed to two 
parabolas oco,oto, Fig 8. The mode of drawing these figs 
to any span and height will be tound under Mensuiation, 
pp 150, 153; but in practice circular segments will answer. 

Rem. 2 ] Load uniformly distributed along 

THE ENTIRE CLEAR LENGTH, y g. Fig 9, OF A HOR RECTANGULAR 
BEAM, FIRMLY fixed at one end only, assume it to be at tbe cen¬ 
ter as in Art 12,and calculate it by the rule in that Art. Then 
div the result by 2. From the quot deduct entire wt of beam 
tor neat load. 

In this case, theoretically, we may cut off one-lialf the pro¬ 
jecting part v o of the beam, as by the dotted line y o, without 
diminishing its breakg strength. But in practice it will rarely 
be advisable to reduce it to a mere thin edge at o. Or the deptli 
c s of the beam may be left uniform, and the sides be cut away 
as shown by the two semi-parabolas a c, a c, at t, which is tin 
top of the beam, if a c, ac, be even made straight, instead of 
parabolas, it is plain that there would still bo a considerabli 
savins? of exDeuse. if the beam is of iron. 



saving of expense, if the beam is of iron. 

Art. 14. When tbe entire breakg load is applied at anj 
point o. Fiji- 10, not at tlie center : first find by Art 12, what would b. 

r the center breakg load; including half the weigh 

of the beam. Then; making c the center of tli j 



span, and having the lengths a o, og, ac and eg; 


extraneous /said total \ half th 

concentrated_/ center ac X eg \_ weigh) 

breaking — \ breaking x a o X o q / the 

load at o \ load / beam 

Tbis rule is not exact if the loa 

rests upon the beam for a short distance on either, or on both sides of the point < 
but only when it all rests at that very point alone; if it does not the load may be it 
creased. See p 483. 

Rem. 1. As a given load approaches either support, its breaking moment decreases 
but its tendency to shear the beam between itself and the nearest support increase 
See Rem, p 533. 

Rem. 2. This beam will bear to be reduced, as at m or n. Fig 5 ; except, that ir 
stead of reducing from its center, as in Fig 5, we must do so from where the load 
applied. 



ST 


1 

0 \s y 

f 

d> 


Fig. 11. 



Art. 15. When the beam, instead e 
being; bor, is inclined, as in Fig 11, 

any of the foregoing cases, the hor dist o y m 
be taken as its span, instead of the actual cl 
length o c ; and s o, s y instead of a o and a c. T1 
applies also to beams fixed at one end, and wheth 
the inclination is upward or downward from t 
fixed end. 


I" 


Note. The quantity of material in inclined beams may 
reduced, in the same manner as in hor ones. 


Art. 16. Triangular beams oi wood, according to uanows expc 
ments with pine, require about % greater breakg loads with the base up, than wl 
it is down. Or with the base down, about ^ less than when up. Tredgold consul 
them about equally strong in either position; and that to find the center brea 
load, we may first calculate it by Art 12, as if the beam were a rectangular one w 
the same base and perp height as the triangle ; and take %of the result Her 
the triangle is not an economical shape for a beam; for with only 34 the strength 
a rectangular one, it has half as much material. 

Ilodgkiusou, with east-iron triangular beams, base 






























STRENGTH OF MATERIALS. 


497 


made the breakg loads equal to % of those of rectangular bars, as in wood. Rennie’s 
experiments give about the same proportion, with the base up; but with the base 
down, he made the strength nearly twice as great, or about -A that of a rectangular 
beam of the same width and vertical height. The comparative strengths in the two 
positions will vary in diff materials, inasmuch as it is affected by the comparative 
resistances which any given material presents to tension and compression. Within 
the limit of elasticity the beam will be equally strong, whether the edge or base be 
up; and will bend equally in either case; so also with the Iiodgkinson, or any other 
form oi beam. J 


Art. 17. To find Ihe side of a square bor beam supported at 
, ° an ^ reqd to break under a given quiescent center 


Rule. Mult the clear bearing in ft, by the given breakg load in pounds. Div the 
prod by the corresponding constant p 493. Take the cube root of the quot. This 
cube root will be the reqd depth or breadth of the beam, approximately, in ins. 

hen the . 8126 of the beam is so great that its wt must be taken into consideration, 
increase either its breadth,as directed, in Art 20; or its depth, as per Art 21, or, first 
find the approx side as before. Then calculate the wt of a sq beam having that’side. 
Add half this wt to the given ceri load, and with this increased cen load, repeat the 
whole calculation. The resulting side will be the reqd one very approx, hut still a 
nere tritie too small. 

The breakg, or the safe load of a square beam, if mult by .6, will give that of a 
cylinder, whose diam is equal to a side of the square one. 

Art. 18. When the beam is reqd to bear its center load 
iafely, mult the given safe load by the number of times it is exceeded by the 
neakg load. Then find, by Art 17, the side of a square beam to break under this in¬ 
creased load. The beam thus found will evidently he approximately the safe one for 
he actual load; exclusive, however, of the wt of the beam. When this mu»t be in- 
1 tided, increase the dimensions as directed in Art 17. 

If the load is equally distributed, first div it by 2, then proceed precisely as before. 

Art. 19. When the beam is cylindrical, and reqd to break 
mder its center load, to find its diam, mult the load by 1.7, and bv Art 17 
ind the side of a square beam, to break under this increased load. The side thus 
jund will also he approximately the reqd diam. See Rems 1 and 2. 

It to be borne safely, first mult it by the number of times it is to be ex- 
ceded by the breakg load. Then mult the prod by 17, and proceed precisely as 
efore. See Rems 1 and 2. 

Rkm. 1. In neither case, however, is the wt of the beam itself included. When this 
i necessary, first find the approximate diam as before. Then calculate the wt of a 
earn having this diam Add half this wt to the given cen load, in either case; and 
ith this increased center load, repeat the whole calculation. 'I he resulting diam 
'ill be the required one very approximately, but still a mere trifle too small. 

Rem. 2. If the load Is equally distributed, first take one-half of 
; as being a center load, and with this proceed precisely as before. 

Art. 20. To find the breadth of a hor rectangular beam, 
npported at both ends, to break under a given quiescent 
enter load ; mult the center load in lbs by the span in feet. Mult the square 
the depth in ins by the constant p 493. Div the first prod by the last. The 
not will be the breadth approximately. Calculate the wt of a beam having this 
-eadth. Then say, as the center load is to half this wt, so is the breadth found, to 
new breadth to be added to it. It will still be somewhat too small, owing to the 
jglect of the wt of the breadth last added. This may readily be found, and its 
nresponding breadth added. 

Ricm. 1. If Jlie lotul is to be borne safely, (without any regard to the 
mount of deflection,) first mult it by the number of times it is exceeded by the 
reakg load. 

Rem. 2 . If in either case equally distributed, take half of it as if a 
nter load, and proceed precisely as before. 

Art. 21. To find the depth, when the breadth is given, mult • 
e load in lbs by the span in feet. Mult the breadth in ins by the constant p 493. 
v the first prod by the last; take the sq rt of the quot for an approximate 
ipth. Calculate the wt of a beam having the depth just found ; add half of it to the 
ven center load, and with this new load repeat the whole calculation; for a more 
iproximate depth, but still somewhat too small, owing to the neglect of the wt of 
e depth last added. We may find this, and repeat the whole calculation, or we 
ay merely increase the breadth by Art 20. 

Rem. If the load is to be borne safely, or if It is equally distributed, see Remarks, 
rt 20. 








498 


STRENGTH OF MATERIALS. 


Art. 22. To find tlie safe dimensions to be given to n rec¬ 
tangular beam of given span, supported at both ends, amt 
which is at the same time exposed both to a transverse 
strain and to a longitudinal tensile or pulling one, or a 
longitudinal compressive one. Tlie writer is unable to suggest any 
better rules than the following, which are at least safe. Namely, when the longi¬ 
tudinal strain is tensile, find separately the safe dimensions as it for a beam alone ; 
and as if for a tie alone; and add the two resulting areas together. \V hen the longi¬ 
tudinal strain is compressive, find separately the safe dimensions as if tor a beam 
alone; and as if for a pillar alone ; and add the two resulting areas together. 


Example 1. A wrought iron rectangular beam of 10 ft span is to sustain 
with a safety of 6, an equally distributed transverse load of 100000 lbs; and a pulling 
strain of 200000 lbs. Of what size must it be? . . Ari0n ,, „ . 

Here the distributed load of 100000 lbs is equal to a safe center one of 50000 lbs, 
or to a breaking center one of 50000 X 0 = 300000 lbs. . 

Now first we may assume for the beam some probable approx depth, say ins. 
Then we find by Art 20 that its breadth as a beam alone will be 


Breakg load in lbs X spa n in ft _ 300000 X 10 _ 3000000 _ g 03 iug> 


sq of depth in ins X coef, p 493 144 X ^500 


360000 


Again, a bar to bear a pull of 200000 lbs with a safety of 6, should not break with 
less than 1200000 lbs; therefore since fair bar iron breaks with about 50000 lbs per 
sq iucli, we have 1200000 = 50000 = 24 sq ins as the area of bar for the pull alone. 
We may add all of this to the width of the beam, making it 24 = 12 == 2 ins wider 
or 10.33 ins wide in all. Or we may add it all to the depth, thus making the bea n 
24 -s-8.33 = 2.88 ins deeper, or 14.88 ins in all. Or part may be added to the bread* 
and part to the depth. 


Example 2. A wrought iron rectangular beam of 10 ft span, is to sustai 
with a safety of G, an equally distributed transverse load of 100000 lbs, and a com 
pressive strain of 200000 lbs. Of what size must it be? 

Here first assuming some probable approx depth, say 12 ins, we find as before th 
its breadth as a beam only will be 8.33 ins. As to the compressive force, it is plait 
that a pillar for sustaining it should be a hollow one with its sides as wide as poss' 
ble; and this is to be effected by placing it around the outside of our beam. Th 
pillar will therefore have sides of about 8.33 and 12 ins wide; and its breaking lof 
must be 200000 X 6 = 1200000 lbs, or say 536 tons. Now the length of the pills 
measured by its narrowest side is 120 = 8.33 = 14.4 sides; and by table, p. 444, w 
find that a hollow square wrought iron pillar 14.4 sides long, breaks with 15.5 ton 
per sq inch of its metal area, lienee we require 536-7-15.5 = 34.6 sq ins metal are 
for our pillar. Now the circumf of the pillar is 8.33 X 2 -f 12 X ‘3 = 40.6G ir 
Hence its thickness must be 34.6 = 40.66 = .85 of an inch. Hence both the breadt 
and the depth of the beam must each he increased twice that much or 1.7 inch ; th 
making it 10.03 ins broad and 13.7 ins deep. 

It is plain that our pillar is thicker than necessary, becau 
the tabular widths are supposed to be by outside measure, whereas our width of 8.1 
ins is inside measure. The final outer width of 10.03 ins would make the pillar onl 
12 sides long; at which it would require 15.7 instead of 15.5 tons per sq inch I 
break it. Other considerations too abstruse to be explained here, combine to nuik 
the resulting dimensions in both examples somewhat in excess. 















STRENGTH OF MATERIALS, 


499 


Art. 23. Table of safe quiescent loads for horizontal rec¬ 
tangular beams of white pine or spruce, one inch broad, 
supported at both ends, and loaded at the center; together 
with their deflections under said loads. 


The safe load is here one-sixth of the breaking load. 

For the neat loads, deduct % the \vt of the beam itself. The deflections, 
lowever, are the actual ones; the wts of the beams having been introduced in cal- 
ailating them, by the rule in Art 27. 

Loads applied suddenly will double the deflections in the table; as 
vhen, for instance, if a load is held by hand, just touching a beam, the hold should be 
uddenty loosed. 


Caution. Inasmuch as this table was based upon well seasoned, straight 
rained pieces, free from knots, and other defects, we must not in practice take 
aore than about two-thirds of the loads in the table for a safety of 6 in ordinary 
uilding timber of fair quality ; and with these reduced loads should not reduce 
he deflections. 


Observe also that our table is for safe center loads, but it is plain that 
a practice we cannot always apply the term in its utmost strictness; otherwise 
he load would have to be sustained by a mere knife-edge, at the very center of 
he beam. Now, in the instance Rem. p. 500, if we attempted to sustain the center 
>ad of 6075 lbs upon such a knife-edge, it would at once cut the beam in two. If 
e even applied it along 3 or 4 ins of the length, it would cut into it, and we should 
ot have a safety of 6 against crushing the top of the beam until as in the case of 
le ends we distributed the load along full 46 ins of length, or about 32 ins for a 
ifety of 4. 

The safe load is here V R of the breakg one; and the last at 450 lbs at the 
niter of a beam 1 inch square, and 1 foot clear length between its supports. For 
i^iere temporary purposes, ^ P ar t may be added to the loads in the table, thus mak- 
ig them equal to the % of the breakg load. But in important structures, subject 
»"vibration, % part should be deducted from the tabular loads, thus reducing 
iem to % of the breaking load. This is especially necessary if the timber is not 
I ell seasoned. 

With the safe loads in this table a beam may bend too 

inch for many practical purposes. When this is the case, we may, by reducing 
le loads, reduce the deflections in nearly the same proportion; or see table, p 512. 

All the loads in the Table are superabundantly safe against shearing'. 
« gainst crushing at the ends, &c, see “ Cautions ” below the Table, 
500. Original 



epth 

Span 4 ft. 

Span 6 ft. 

Span 8 ft 

Span 10 ft 

Span 12 ft 

Spai 

14 ft 

Span 

16 ft 

Wt. of 

10 ft of 

i 

Ot 

iam. 

load. 

I def. 

load def. 

load 

def 

load 

def. 

load 

def 

load 

def. 

1 oad 

def. 

beam. 


ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins 

lbs. 

1 ins. 

ft>s. 

ins. 

lbs. 

ins. 

tbs. 

ins. 

lbs. 


1 

19 

.39 

13 

.92 

10 

1.8 

8 

3.0 

6 

4.4 





2 


2 

75 

.22 

50 

.45 

38 

.82 

30 

1.3 

25 

1.9 

21 

2.7 

19 

3.7 

4 


3 

170 

-13 

114 

..i0 

85 

.53 

67 

.84 

57 

1.3 

48 

1.7 

42 

2.3 

6 


4 

300 

.10 

200 

.22 

150 

.39 

120 

.63 

100 

.92 

86 

1.3 

75 

1.7 

8 


5 

469 

1 .08 

312 

.18 

234 

.31 

187 

.50 

156 

.72 

134 

1.0 

11711.3 

10 


6 


.06 

450 

.15 

337 

.26 

270 

.41 

225 

.60 

193 

.83 

168 1.1 

12 


7 

919 

.06 

612 

.12 

460 

.22 

367 

.35 

306 

.51 

262 

.70 

230 

.93 

14 


8 

1200 

.05 

800 

.11 

600 

.19 

480 

.31 

400 

.45 

343 

.61 

300 

.81 

16 


9 

1520 

.04 

1014 

.10 

760 

.17 

607 

.27 

507 

.40 

434 

.54 

380 

.72 

18 


10 

1875 

.04 

1250 

.09 

937 

.16 

750 

.24 

625 

.35 

536 

.49 

46,8 

.64 

20 


11 

2270 

.04 

1514 

.08 

11.35 

•14 

907 

.22 

757 

.32 

648 

.44 

567 

.58 

22 


12 

2700 

.03 

1800 

.07 

1350 

.13 

1080 

.20 

900 

.29 

772 

.40 

675 

.53 

24 


14 

3675 

.03 

2450 

.06 

1837 

.11 

1470 

.17 

1225 

.25 

1050 

.34 

918 

.45 

28 


16 

4800 

.02 

3200 

.05 

2400 

.10 

1920 

.15 

1600 

.22 

1.372 

.30 

1200 

.40 

32 

■! 

18 

6075 

.02 

4050 

.05 

3037 

.09 

2430 

.14 

2025 

.20 

1736 

.27 

151S 

.35 

36 


iO 

7500 

.02 

5000 

.04 

3750 

.08 

3000 

.12 

2500 

.18 

2145 

.24 

1875 

.31 

40 

J 

!2 

9075 

.02 

6050 

.04 

4537 

.07 

3630 

.11 

3025 

.16 

2593 

.22 

2268 

.29 

44 


!4 

10800 

.02 

72001 

.04 

3400 

.06 

[4320 

.10 

3600 

.15 

3088 

.20 

2700 

.26 

48 


(Continued on next page.) 













































500 


STRENGTH OF MATERIALS. 




Table, continued. (Original.) 


Depth 

Span 18 ft. 

Span 20 ft. 

Span 25 ft. 

Span 30 ft. 

Span 35 ft. 

Span 40 ft. 

Wt. of 
10 ft of 

OI 

beam. 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

beam. 

Ins. 

fibs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

6 

150 

1.4 

135 

1.8 

108 

2.9 

90 

4.5 

77 

6.5 

67 

9.2 

12 

7 

204 

1.2 

184 

1.5 

147 

2.5 

122 

3.9 

105 

5 8 

92 

7.6 

14 

8 

267 

1.0 

240 

1.3 

192 

2.1 

160 

3.2 

137 

4.6 

120 

6.4 

16 

9 

338 

.92 

304 

1.2 

243 

1.9 

202 

2.8 

174 

4.0 

152 

5.5 

18 

10 

417 

.82 

375 

1.0 

300 

1.7 

250 

2.5 

214 

3.5 

188 

4.9 

20 

11 

505 

.74 

454 

.93 

363 

1.5 

302 

2.2 

259 

3.2 

227 

4.3 

22 

12 

600 

.68 

540 

.85 

432 

1.4 

360 

2.0 

308 

2.9 

270 

3.9 

24 

14 

817 

.58 

735 

.72 

588 

1.2 

490 

1.7 

420 

2.4 

367 

32 

28 

16 

1067 

.50 

960 

.63 

768 

1.0 

640 

1.5 

548 

2.1 

480 

2.8 

32 

18 

1350 

.45 

1215 

.5b 

972 

.90 

810 

1.3 

694 

1.8 

607 

2.5 

36 

20 

1666 

.40 

1500 

.50 

1200 

.79 

1000 

1.2 

857 

1.6 

750 

2.2 

40 

22 

2017 

.37 

1815 

.45 

1452 

.72 

1210 

1.1 

1037 

1.5 

907 

2.0 

44 

24 

2400 

.33 

2160 

.41 

1728 

.65 

1440 

.96 

1234 

1.3 

1080 

1.8 

48 

26 

2817 

.31 

2526 

.38 

2018 

.60 

1684 

.88 

1449 

1.2 

1263 

1.6 

52 

28 

3267 

.28 

2940 

.35 

2352 

.55 

1960 

.81 

1680 

1 1 

1470 

1.5 

56 

30 

3750 

.26 

3375 

.33 

2700 

.50 

2250 

.76 

1928 

1.1 

1687 

1.4 

60 

32 

4267 

.25 

3840 

.30 

3072 

.45 

2560 

.71 

2194 

1.0 

1920 

1.3 

64 

34 

4817 

.23 

4335 

.29 

3468 

.44 

2s90 

.67 

2477 

.92 

2167 

1.2 

68 

36 

5400 

.22 

4860 

.27 

3888 

.43 

3240 

.63 

2777 

.86 

2430 

1.1 

72 


White oak, and best Southern pitch pine will bear loads % 

greater. 

For cast iron, mult the loads in the table by 4.5; and for wrought by 
5.3. For these new loads, mult the dels by .4 for cast; and by .3 for wrought. 

If the load is equally distributed over the span, it may be twice as 
great as the center one, and the defs will be \}/± times those in the table. If the 
loads in the table be equally distributed along the whole beam, the defs wil' 
be but live-eighths as great as those in the table. See Art 26. p 5055. W hen mori 
accuracy is reqd, half the wt of the beam itself must be deducted from the cente, 
load; and the whole of it from an equally distributed load. The wt of the beam, in 
the last column, supposes the wood to be but moderately seasoned, and therefore ti 
weigh 28.8 lbs per cub ft. 

Uses of the foregoing' table. Ex. 1. What must be the breadtl 
of a hor rect beam of wh pine, IS ins deep, supported at both ends, and of 20 ft oleai 
length between its supports, to bear 6afely a load of 5 tons, or 11200 lbs at its center 
Here, opposite the depth of 18 ins in the table, and in the column of 20 feet lengths 

’ * x l|900 

we find that a beam 1 inch thick will bear 1215 lbs; consequently,-- = 9.22 ins 

1*215 

the reqd breadth; for the strength is in the same proportion as the breadth. 

Ex. 2. What will be the safe load at the center of a joist of white pine, 18 ft long 
3 ins broad, and 12 ins deep? Here, in the col for 18 ft. and opposite 12 ins in depth 
we find the safe load for a breadth of 1 inch to be 600 lbs; consequently, 600 X 3 = 
1800 lbs, the load reqd. 

Rkm. Cautions in the use of the above table. For instance, ii 
placing very heavy loads upon short, but deep and strong beams, we must take car 
that the beams rest for a sufficient dist on their supports to prevent all danger froi 
crushing at the ends. Thus, if we place a load of 6075 lbs at the center of a beat 
of 4 feet span, 18 ins deen, and only 1 inch thick, each end of the beam sustains 

/»nn pr 

vert crushing force of —= 3037 lbs, and that sitlewise of the groin, i 

which position average white pine, spruce, and hemlock crush under about 8( 
lbs per sq inch, and do not have a safety of 6 until the pressure is reduced to aboi 
133 lbs per sq inch. Therefore our beam, in order to have a safety of 6 again: 
crushing at its ends, must rest on each support 3037 133 = 23 sq ins; or for 

safety of 4 nearly 16 sq ins. When a pressure is equally distributed sidt 1 
wise (that is, at right angles to the general direction of the fibres) over the entii 
pressed surface of a block or beam (to ensure which, the opposite surface must l 
supported throughout its entire length) the resulting compression might readil 
escape detection unless actually measured. But when a considerable pressure 
applied to only a portion of the surface, as of caps and sills where in contact wit 
the heads and feet of posts, or at the ends of loaded joists or girders, the eon 
pression becomes evident to the eve,-because the pressed parts sink below tl 
unpressed ones, in consequence of tlie bending or breaking of the adjacent fibre 
What in the firstcase (especially if slight) would be called compression, wou 

























STRENGTH OF MATERIALS. 


501 




as 


to be unSe d ^ Called crMshln *5 even wh en neither might be so great 

T^sass^trs^ss^s crashtog " hen app,ied to »■* • •? 

L JJ*® writer lias seen 40 half seasoned hemlock posts, each 12 ins square 
• intervals of 5 ft from center to center, upon similar 12 X 12 inch hem- 

oek sills, to which they were tenoned, and which rested throughout their entire 
stone st . e P 3 ’ Each post was gradually loaded with 32 tons, or equal to 

rifpfr hl^ per sq i L'n - 1 \ &U i their feet a11 criIshed into the sills from % to X inch 
Their heads crushed into the caps to the same extent. In practice the nreL- 

niie at the heads and feet of posts is rarely, if ever, perfectly equable; and the 
same remark applies to the ends of loaded joists, girders, &c., in which a slight 



















502 

















































































































































STRENGTH OF MATERIALS, 


508 













































































































































504 


STRENGTH OF MATERIALS 


STONE BEAMS. 

Table of safe quiescent extraneous loads for beams of goad 
building' granite one inch broad, supported at both ends, and loaded at the 
center; assuming the safe load to be one-tenth of the breaking one; and the , atter 
to be 100 lbs for a beam 1 inch square, and 1 foot clear span. The half weight of 
the beams themselves is here already deducted by the rule in Art 12, p. 404, at 170 
lbs per cub ft. 


a 




CLEAR SPANS IN FEET 





a 

J3 

1 

2 

3 

4 

5 

6 

7 

8 

10 

12 

15 

20 

Q* 

0> 

Q 

Safe center loads in pounds. 

i 

10 

5 











2 

40 

20 

13 

10 









3 

90 

45 

29 

21 

17 








4 

160 

79 

52 

39 

31 

26 

21 






3 

250 

124 

82 

61 

48 

40 

34 






6 

360 

179 

119 

89 

70 

58 

48 

42 

32 




7 

490 

244 

162 

120 

96 

79 

67 

58 

45 

36 

27 

16 

8 

639 

319 

212 

158 

126 

104 

88 

76 

59 

47 

36 

22 

10 

999 

499 

331 

248 

197 

163 

139 

120 

94 

76 

58 

38 

12 

1439 

718 

478 

357 

284 

236 

201 

174 

137 

111 

85 

58 

14 

1959 

978 

650 

487 

388 

322 

274 

238 

188 

153 

118 

81 

16 

2559 

1278 

850 

636 

507 

421 

359 

312 

246 

201 

157 

109 

18 

3239 

1318 

1077 

806 

643 

534 

455 

396 

313 

257 

200 

141 

20 

3999 

1998 

1329 

995 

794 

660 

563 

490 

388 

319 

249 

176 

22 

4839 

2417 

1609 

1205 

961 

800 

682 

594 

470 

387 

303 

216 

24 

5758 

2877 

1916 

1434 

1145 

951 

813 

708 

562 

463 

362 

260 

27 

7288 

3642 

2425 

1815 

1450 

1205 

1030 

898 

713 

588 

462 

332 

30 

8998 

4496 

2995 

2243 

1791 

1489 

1273 

1110 

882 

728 

573 

415 

33 

10888 

5441 

3624 

2714 

2168 

1803 

1542 

1345 

1069 

883 

696 

505 

36 

12958 

6476 

4314 

3231 

2581 

2147 

1836 

1603 

1275 

1054 

832 

606 


If uniformly distributed over the clear span, the safe extraneou 

loads will be twice as great as those in the table. 

For good slate on bed the safe loads may be taken at about 3 times; fo 
good sandstone on bed at about one-half; and for good marble © 
limestone on bed at about the same as those in the table. See table, p 493. 






















































505 


STRENGTH OF MATERIALS. 


LIMIT OF ELASTICITY IN BEAMS. 

Under moderate loads, the deflections of a team are practically proportional to 
the load. When they begin to increase perceptibly faster than the load, the latter is 
said to have reached the elastic limit, or limit of elasticity. It is generally at this 
point that the “permanent set” first becomes noticeable; i. e., after removal 
of the load, the beam fails to return to its original unstrained condition, and remains 
more or less bent. The deflections then also begin to increase irregularly; and to 
continue indefinitely without further increase of load. In short, the beam is in 
danger. Hence, the actual load must never exceed the elastic limit; and should not 
exceed from one-third to two-thirds of it, according to circumstances. 

( The limit of elasticity of a beam of any particular form, or material, is 
determined by experiment with a similar beam, as in the case of constants 
for breaking loads, &c. Thus, load a beam at the center, by the careful gradual 
addition of small equal loads; carefully note down the deflection that takes plaeo 
within some minutes (the more the better) after each load has been applied; in order 
to ascertain when the deflections begin to increase more rapidly than the loads; for 
when this takes place, the load for elastic limit has been reached.* 

It is not the deflections of the whole beam that are to be noted, but those of its 
dear 6pan only. Several beams should be tried, in order to get an average constant, 
'or even in rolled iron beams of the same pattern, and same iron, there is a very 
ippreciable difference of strengths and deflections. 

Then, to get the constant, using the total load applied during the equal deflections, 
ncluding half the weight of the beam itself, 


Constant for elastic limit 


Span in feet X Total load in fi>s. 


Breadth in inches X Square of depth in inches 


The constant, for wooden beams, may be had, near enough for common 
i iractice, by taking one third of the breaking constants in the table, page 493. 

The foregoing constants are those for beams supported at both ends and loaded at 
he center. To find constants for other methods of supporting and 
oading; 


J If the beam is 

supported at both ends and loaded uniformly 
fixed f “ “ “ at center 

“ “ “ “ uniformly 

“ “ one end “ at the other 

“ “ “ u uniformly 


Multiply the 
above constants by 

2 

2 

3 


Said-constant, thus calculated, is the elastic limit of a beam of the given shape 
nrl material, 1 inch broad, 1 inch deep, and of 1 foot "span, supported at both ends 
nd loaded at the center. To obtain from it the elastic limit of any other beam of 
lie same design J and the same material, similarly supported and loaded, but of other 
iineusions, 

Elastic _ y breadth in in ches X square of depth in inches 

.. limit 


span in feet 


* Of course, in practice, it is frequently difficult to ascertain with precision, when, 

■ under what load, the deflections actually do bpgin to increase more rapidly than the 
iccessive loads. For although by theory the; deflections are practically equal for 
pial loads, until the elastic limit is reached, yet in fact they are subject to 
tore or less irregularity ; for no material composing a beam is perfectly uniform 
iroughout in texture and strength. Hebce, instead,of regular increase of deflec- 
on, we shall have an alternation of larger and smaller ones. Therefore, some judg- 
lent is required to determine the final point; in doing which, it is better, in case 
' doubt, to lean to the side of safety. It is assumed always that the load is not 
ibject td jars or vibrations. These would increase the deflections. See also p.434/. 

f A beam is said to be “ fixed ” at either end when the tangent to the longitudinal ■ 
ris of the deflected beam at that end remains always horizontal. 

| The shapes of the two beams need not be similar. For instance, the constant 
educed from experiments upon any rectangular beam is applicable to any other 
ctangular beam, whether square or oblong. _ 

36 - ... 














STRENGTH OF MATERIALS 


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STRENGTH OF MATERIALS. 


5056 


! 


DEFLECTIONS OF BEAMS. 

Art. «6. Deflections, or bendings, of beams, under tbetr loads. 

, 6 f°r e going relates chiefly lo the strength of beams, or their resistance to breaking • 

(bfferent*laws t0 Stl ^ neSa> or resistauce to bending. The two follow very 

Ar^29 d p e *fo Cti0 “ S llmited to a S* ve “ fraction of tbe span, see 

lim ^ OPP ° 8ite gi X. es tho Reflections (in inches) within the elastic 

i?!*-’ an ^ P risrnati c beam (beam of uniform cross section throughout) under 
different arrangements of support and of load 5 also (in the last 
column) the extraneous load which will ptoduce a given deflection 

without assistance from the weight of the beam itself. All the formulae are based 
ujion the assumption that the increase of deflection is proportional to increase of load 
The letters signify as follows: 

d = deflection of beam, in inches (see Figs.) 

W = w-eight of extraneous load, in pounds. 
w = “ “ clear span of beam, in pounds. 

/ = clear span of beam, in inches (see Figs.) 

E = modulus of elasticity f of the material of the beam, in pounds per square 
inch. See pp. 434, etc. 4 ““*° 

I = moment of inertia*f of the cross section of the beam, in inches. See 
pp. 486 and 487. 

From the principles embodied in the opposite table, we find that in beams of 
similar cross section and of the same material, and within the elastic limit, the load 
and deflections (neglecting the weight of the beam it-elf) are as follows- 


With the same 

The deflections under a given extraneous load are 

span 

and breadth 
“ “ depth 

breadth “ “ 

; . 

inversely as the breadths and as the cubes of the depths 

♦< a u 

“ “ breadths 

directly “ cubes of the spans 

With the same 

The extraneous loads for a given deflection are 

span 

“ and breadth 

“ “ depth 

breadth “ “ 

directly as the breadths and as the cubes of the depths 

« a u « (t 

“ “ breadths 

inversely “ cubes of the spans 

It also follows that, within the limit of elasticity, a beam of irregular shape, such 
as a T, or a Hodgkinson beam, a triangle, &c., will bend to the same extent, whether 
its top or its bottom be uppermost. 

After the elastic limit is passed, the deflections increase irregularly, and more 
rapidly than before; and the beam becomes unsafe. 

* In beams of any given cross section, whether rectangular, triangular, I beams, 
etc., etc., of equal or unequal size, the moment of inertia \s proportioned to the breadth 
and to the cube of the depth. 

In any solid rectangular beam, 

Moment of inertia - breafUh X cube of de P th . 

12 

For other shapes of cross section, see p. 487. The moment of inertia is independent 
of the material of the beam. 

fFor other methods, not requiring the use of the modulus of elasticity or the 
moment of inertia, see pp. 506, etc. 




























506 


STRENGTH OF MATERIALS. 


Art. 26 a. The following methods obviate the necessity of 
flnding the modulus of elasticity and the moment of inertia: 

The Constant for Deflection for any given material, within the limit of 
elasticity, is the deflection, in inches, of a beam of that material, 1 inch wide, 1 inch 
deep and of 1 foot span, supported at both ends, and loaded at the center with n. total 
weight (including five eighths of the weight of the clear span of the beam) of one 
pound. Such constants may, like those of transverse strength (see p. 491),be readily 
found by experiment. Thus, at the center of any beam, placed horizontally upon 
supports at each end, place any load that is within its elastic limit, and measure 
the resulting deflection in iuches. Multiply the weight of the span of the beam by 
.625, add the product to the neat load, for a total load. Then the required constant 
for any other beam of the same design * and of the same kind and quality of 
material, whether wood, metal, stone, &c., is, 


observed 

Constant = deflection X 
in inches 


breadth in inches X cube of depth, in inches 
total load in lbs. X cube of span in feet 


We add to the experimental neat load, the .625. or % of the wt of the clear span 
of the beam itself, because the wt of the beam equally distributed throughout its 
span, also aids in producing the del \ and it does so to the same extent that of it 
(would do, if collected at the center of an imaginary beam having the same strength 
throughout as the real one, but having itself no weight. Therefore, in applying the 
constants for def to beams intended for actual use, we must not omit to add % of 
the wt of the span, to the intended center load, for an equivalent total center loadj 
before making the calculations for def. The weights of similar beams (that is, 
beams proportioned exactly alike in every part, but of difl sizes) increase so much 
more rapidly than their dear spans, that although a small one may safely bear a 
load of many times its own wt, a much larger one will break down without any 
load. Having by experiment found the constant of def for any given material, the 
deflection, within the elastic limit, of any other beam of the same design* and of the 
same mateiial, whether larger or smaller, and loaded at the center, may be found 
thus: 

total equivalent ^ cube of span, 

Deflection _ con8tant x _ center load, in lbs . A in feet _ # 

in inches ' ' breadth, in inches X cube of depth, in inches 


* The shapes of the beams need not be similar. For instance, the constant deduced 
by experiment from any rectangular beam, applies to any other rectangular beam, 
whether square or oblong. 




».) * 


\ 















STRENGTH OF MATERIALS. 


507 


Table of constants for the deflections, within the safe, or 
elastic limits, of hor rectangular beams, supported at both ends and loaded 
u , j center. the timbers are supposed to be well seasoned; if not, the constant 
should be increased. 


White oak .00023 * 

Best southern pitch pine, ) nfino - * 
and white ash ............ j - 0002 ' * 

Hickory.00016* 


Cast iron. 

Bar iron... 
Steel, rolled.... 


White pine. 

Ordinary yellow pine.. 

Spruce . .00032* 

Good straight-grained hemlock. 

Ordinary oaks. 

.000018 to .000036.Mean .000027* 

.000012 to .000024.Mean .000018 

.000010 to .000020. Mean .000015 

lull and reliable experiments on the strength and deflections of the various steels 

are much needed. 

■ I 1 ,®V. dent that ■ the stiffer the material is, the smaller will be its constant for bend, 
ing. All these constants vary somewhat with the quality of the metal. The defs also of 
timber of the same kind, vary so much with the degree of seasoning, the age of the 
tree, the part it is cut from, Ac, that the writer considers it mere affectation to pre¬ 
end to assign constants for practical use, more nearly approximate than he lias here 
one. lliey are averages deduced from his own experiments on good pieces, well 
seasoned; and the loads were allowed to remain on tor months, instead of minutes 
as usual. Every structure is more or less exposed to vibrations and jars, which in 
i le increase the deflections. In several instances, our experimental timbers bore 
leir breakg loads for months before they actually gave way. And in all kinds, less 
than of the breakg load produced in a few months a permanent set, or def. 

fJ5h. wdStVSelSr 4 fr ° m 9i " 8 ' e on| y- A " allowance b made 

Rolled iron beams proportioned exactly as the 7-inch Phoenix beam, .0000303+ 

30 lbs, 9 inch, “ “ *0000321+ 

50 “ heavy 9 inch “ “ .0000264 

12 lnch “ “ 0000313+ 

15 inch “ “ 0000365+ 

66 73 lbs, 15 inch “ “ .0000438 


Art. 27. To find tlie <lef in inches, of a hor rectaiifrnlar 
beam, supported at both ends, and loaded at its center, with 

h. n ^^ , ^ en ,i 0 ^ d ^ I ! hin “ S > «>*w«city; mult the weight of the clear 
beam itseff, m lbs, by the decimal 62o. Add the prod to the given center load in lbs. 

? ■ L? e ii 8U “ th ? t0 t ta e 0a<1 ;. Mult t0 Kether this total load, the cube of the span in 
■ ’ and constant from the upper table. Also mult together the breadth in 
ins, and the cube of the depth in ins. Div the first prod by the last one. 

. What will be the def of such a beam of average white pine, 9 ins broad, 12 
ibs? deeP ’ f<3et ° ^ 8 P an ’ aDd weighing 450 lb8 5 with a neat center load of 1218.75 

Here first, 450 X .625 = 281.25 lbs. And 281.25 + 1218 75 = 1500 lbs total load. 
Hence, 

1500 X 21* X -Const. 1500 X 6261 X .00032 4445.2 

= .286 inch; reqd def. 


9 X 12 s 


9 X 1728 


15552 


Rem 1. When the load is all at one point not at the center, 

o, Fig 10 p 496, mult together the two dists n a, o g, from the load to the points of 
support. Mult the prod by 4. Div the result by the clear span. Use the quot as if 
it were the span, in the last rule. The wt of the beam is not here taken into account; 
it will of course somewhat increase the def. 


* Averages near enough for ordinary practice by the writer’s own trials. Call* 

i»i«; the average elastic def of a steel beam, 1 , that of a similar 

average wrought one will be 1.2 ; and that of a cast one 1.8. If that of an average oast beam be 1 
that of a wrought one will be .67; and that of a steel one .56. If that of a wrought one be 1. cast will 
be 1.5; and steel .83. 

t We believe that these four beams have the same proportions, as nearly as the process of making 
them will admit of; so that .000033 may be taken as a near enough average for all four. As before 
remarked, extreme accuracy must never be expeoted iu such matters. Two halves of the same iden¬ 
tical beam will often give differences greater thau this. 






























508 


STRENGTH OF MATERIALS. 


Rem. 2. When the neat load is equally distributed along: 

the span, instead of all being at the center, then for an equivalent total center 
load, add together the neat load, and the entire wt of the clear span of beam ; and 
mult the sum by the dec .625. With the resulting equivalent center load, proceed 
precisely as in the foregoing example. 

Ex. The def of the foregoing beam of white pine, 9 ins broad, 12 ins deep, 21 
feet span, weighing 450 lbs, and bearing an equally distributed load of 1218.75 lbs? 

Here first 450 + 1218.75 = 1668.75. And 1668.75 X -625 = 1042.97 lbs = equivalent 
center load. Hence 


1042.97 X 21 3 X -00032 3090.862 

9 X 12* _ 15552“ 


.1987 ins, reqd def. 


Rem. 3. With an equally distributed load, including the wt of the 
beam, the def is only %, or the .625 part as great as it would be if the same total 
load, including the entire wt of the beam, were all applied at the center. 

Rem. 4. If the beam in any of these, or the following cases, is inclined. Fig. 
11, p 496, use the lior dist o y , instead of the actual span o c. 


Art. 28. Rule 1. To find the neat center load which will (to¬ 
gether with the wt oi'the beam itself) produce any given def 
within the elastic limit of the beam ; find the cul*e of the clear length 
in.feet; mult this cube by the constant from the table on p 507. Also mult the 
breadth in ins, by the cube of the depth in ins. Div the first prod by the last one. 
Div the given def in ins, by the quot, for the total reqd load in lbs. Mult the wt of 
the clear length of the beam in lbs by .625, and deduct the prod from the load so ob¬ 
tained, for the neat load. By formula, 


Total load, 
including = Deflect 
wt of beam in * ns 


Cube of length v Constant 
in feet x in p 507 

Breadth ^ Cube of depth 
in ins * in ins. 


Ex. What center load in lbs will (together with the wt of the beam itself) pro¬ 
duce a def of .286 of an inch, in a beam of white pine, 21 ft span, 9 ins broad, 12 ins 
deep, and which weighs 450 lbs? See table, p 499. 


Cube of 21. Const. 

Here 9261 X -00032 = 2.9635. 


And 


2.9635 

15552“ 


.0001906. 


Breadth. Cube of 12. 

And 9 X 1728 == 15552. 
And J)S5i506 = 1500 


For the neat load we must deduct .625 of the wt of the beam ; or 450 lbs X .625 = 
281.25 lbs; so that the neat load is 1500 — 281.25 = 1218.75 lbs, as in Ex 1, Art 27. 

If the load is uniformly distributed, use precisely the same rule for get- 
ting the total load. Then mult this load by 1.6. Deduct the entire wt of the clear 
length of beam. 


Ex. What equally distributed load will deflect the foregoing beam .1987 ins? 
Here, proceeding as before, the only diff is that instead of .286 def, we have .1987 

1 987 

def to be div by .0001906. And — = 1042.5 lbs, as the equivalent center load. 

•uuu i y uo 

And 1042.5 X 1.6 = 1668 ibs for the total distributed load, including the entire wt 
of the beam, or 450 lbs. Hence 1668 — 450 = 1218 lbs, the neat distributed load reqd; 
agreeing with the preceding example within % of a lb; the diff being owing to a 
neglect of small decimals in the calculation. 

Rule 2. The length, depth, neat center load, and def being 
given, to find the breadth. 


Neat cen load v Cube of length v Constant 
in lbs * in feet * in Art 26 

Cube of depth , . Def 
in ins in ins 


Breadth 
in ins 
approx. 


Or sufficient for the neat load alone. 


Now calculate the wt of a 
.625, then say, as 

Neat center , 
load • 


beam with the breadth already found. Mult this wt 

Breadth .625 of the Additional 

first \ * weight of * breadth 
found the beam reqd. 


by 


Add these two breadths together, and their sum will be the total breadth reqd, more 
approximately; but still somewhat too small, inasmuch as it provides only for the 








STRENGTH OF MATERIALS. 509 


wt of the beam of the breadth first found, and not for that 
breadth. This may readily be calculated and added. 


having the additional 


Rule 3. The length, breadth, neat center load, and def, 
being 1 given, to find the depth. * 

Neat cen load v Cube of length v Constant 

in lbs A in feet x in Art 20 Cube of depth 

^tef-~ = in ins 

X ,,, ina approx. 


Breadth 
in ins 


Take the cube root of this for the depth itself, approximately. Rem. This 
like the breadth given by the preceding formula, is too small, inasmuch as it does 
not allow for the wt of the beam. Therefore, when greater accuracy is required 
proceed thus: Calculate the wt of a beam having the depth just found. Mult this’ 
wt by .625. Add the prod to the neat center load. Consider the sum as a new neat 
center load; and using it instead of the neat center load first given, go through the 
whole calculation again, to obtain a new cube of depth. The cube root of this will 
be more nearly correct; but still a trifle too small, tor the same reason as in the fore¬ 
going case. 


Rem. In experimenting for constants of 

any kind, with beams of irregular cross-sections, this, for in¬ 
stance, it is quite immaterial which breadths and depths are 
measd; thus, for the breadth we may take ab , Im, cd. or o e, 

Ac; and for the depth, either nc, Iv, m d, bo, &c. It is only 
necessary to state what parts actually have been taken, so 
that the corresponding ones may be measd in any other beam 
which is to be calculated by the constant derived from the 
experiment. This remark applies to all constants involving 

the breadth and the depth. The constant itself will of course vary according to 
which dimensions are taken in the experiment; but the results derived from it when 
applied to other beams of similar forms, will not be affected thereby, if the corre¬ 
sponding parts be measd in both cases * 



* tVe may cveu take any single oblique measurement, as ab, lm, nc. ad, Ac. and call it both the 
breadth and the depth. This applies to rectangular, or to any other shaped beams. 















510 


STRENGTH OF MATERIALS, 


Art. 29. Allowable deflection In practice. The extent to which a 
beam may bend under even a perfectly safe load, may bo too great for many purposes 
in every-day practice. Tredgold and others assume, that in order not to be observed, 
or that it may not cause the plaster of ceilings to crack, &c., a beam should not 
deflect at its center more than one four hundred and eightieth of its span, or one 
fortieth of an inch per foot. But one three hundred and sixtieth of the span, or one 
thirtieth of an inch per foot, is now generally regarded as the allowable maximum 
deflection. Thus. If its span be 15 feet, a beam should not bend more than 15 
thirtieths of an inch or \/ 2 an inch, which is also one three hundredth and sixtieth 
of 15 feet. 

Shafts of wheels in machinery should not deflpct more than half of this, nor a 
bridge more than, say one twelve hundredth of its span, or .01 inch per foot, under 
its heaviest load. 

If we call the greatest allowable deflection in inches per foot of span, D,”* 
the formula p. 508 for total center load (including .025 of the weight of the 
clear span of tba beam) becomes 


load* in lbs = 


D X span in feet X depth 3 in inches X breadth in inches 
• Span 3 in feet X constant, p 507 
D X depth 3 in inches X breadth in inches 


Span 2 in feet X constant, p 507 


Extraneous center load 

in lbs 


total load found 
as above 


( weight of clear 
span of beam X 

For the total uniformly distributed load 

load* in lbs — X I> X de pth 3 in inches X breadth in inches 

Span 2 in feet X constant, p 507 


.625 ^ 


Extraneous uniformly __ total distributed load 
distributed load in lbs found as above 

To find the breadth required: 


weight of clear 
spau of beam 


Breadth in inches _ extraneous center load in lbs X span 2 in ft X constant p501 
approximately D X depthMn l^heS- 

For a closer approximation, add to the breadth so found 

said breadth X ; 6 . 25 X weight of c lear spa n of beam 

center load in lbs 

The sum will still be a mere trifle too small. 

If the load is uniformly distributed, use distributed load X .625‘ 
instead of center load, in the first formula for breadth. For a closer approxima¬ 
tion, add to the breadth so found, 

said breadth X of beam 

uniform load 

The sum will still he a mere trifle too small. 

To find the depth required 


Bepth 

in inches 
approximately 


3 /extraneous center span 2 in constant, 
, / load in lbs A fppt. p 507 


_feet 

I) X breadth in inches 


Then calculate the wt of the entire clear span of a beam having this depth 
mult it by .625, and add the prod to the neat center load. Consider the sum a> 
a new neat center load; and using it instead of the one first given go throust 
the whole calculation again, for a new depth. This will be the reqd depth mor< 
approx, but a little too small. M v 

If the neat load is uniformly distributed, first, mult it bv 625 Us< 
the prod as a center load, and by the foregoing formula find first 'the appro? 
depth. Then calculate the wt of the entire clear length of a beam having tha 


* 

513 , 


When P = -fa (i e when the greatest allowable deflection is = span) 


see tables pp 512 



























511 


STRENGTH OF MATERIALS. 


depth. Mult this wt by .625, and add it to the prod used as a center load, con¬ 
sider the sura as a new center load ; and using it instead of the one first used, 
go through the whole calculation again, for a new depth. This will be the reqd 
depth, approx, but a mere trifle too small. H 

To find the side required for a square beam 


Side of sqnare /extraneous center v span® in v constant, 
beam in inches __ \/_ load in lbs A feet A p 507 

approximately V ~~ jj ~ 

The fourth root is = the square root of the square root. 

. For a closer approximation, find the weight of a square beam with the side 
just iound. Multiply it by .625, and add the product to the extraneous center 
load. Employ the formula again, using this increased load instead of extrane- 
oils center load. The side thus obtained will still be a mere trifle too small. 

To find the diameter required for a solid cylindrical 
X)eaDi» 

Diameter 4 /l 7 y extraneous center v span 2 in v constant 
in inches _ \J Ioad in lbs feet A p 507 

approximately \ D “ 

The fourth root is = the square root of the square root. 

For a closer approximation, find the weight of the clear span of a beam with 
the diameter just found. Multiply said weight bv .625. Add the product to the 
original given center load. Then repeat the formula, using the sum last ob¬ 
tained, instead of extraneous center load. The resulting diameter will still be 
a trifle too small. 

The stiffness ofa cylinder is to that of a square beam, whose breadth 
and depth are each equal to the diam of the cylinder, as .589 to 1; or that of the 
square one is to that of the cylinder as 1 to .589, or as 1.698 to 1; in practice we 
may use .6 and 1.7. Hence, the cylinder will bend 1.7 times as much as a square 
one, under the same load. 

When, in any of the foregoing cases, the beam is inclined. Fig 11, 
? 496, take the horizontal distance oy for the span, instead of oc. 









512 


STRENGTH OP MATERIALS, 

















































































































513 


STRENGTH OF MATERIALS. 


krt. 32. Table of greatest center loads of square beams of 
cast iron, supported at both ends, and reqd not to bend 
more than 55 of an inch per foot of clear length, or 
part of the span. For W. Pine div by 12; or in practice by 18. 

Wrought iron will bear about ^ more than cast, with the 
ime safe deflection. But .625-(or %) of the wt of the beam itself must be deducted 
= 'om these center loads. If the load is equally distributed, it will be 1.6 times as 
reat as these tabular center loads ; but in this case the wt of the entire clear length 
f the beam is to be deducted. These deductions are rarely reqd in practice. 


reatest 


Clear Spans, in Feet. (From Tredgold ) 


* Sentre 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

ounds. 

A 




A 



A 

A 


A 

A 

A 

A 


ft 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 

a 


P 

P 

P 

P 

P 

ft 

P 

P 

O 

ft 

P 

P 

P 

P 


In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

112 

1.2 

1.4 

1.7 

1.9 

2.0 

2.2 

2.4 

2.5 

2.6 

2.7 

2.9 

3.0 

3.1 

3.2 

221 

1.4 

1.7 

2.0 

2.2 

2.4 

2.6 

2.8 

3.0 

3.1 

3.3 

3.4 

3.6 

3.7 

3.8 

336 

1.6 

1.9 

2.2 

2.4 

2.7 

2.9 

3.1 

3.3 

3.4 

3.6 

3.8 

3.9 

4.1 

4.2 

448 

1.7 

2.0 

2.4 

2.6 

2.9 

3.1 

3.3 

3.5 

3.7 

3.9 

4.0 

4.2 

4.3 

4.5 

560 

1.8 

2.2 

2.5 

2.8 

3.0 

33 

3.5 

3.7 

3.9 

4.1 

4.3 

4.4 

4.6 

4.8 

1 672 

1.8 

2.2 

2.6 

29 

3.2 

3.4 

3.7 

3.9 

4.1 

4 3 

4.5 

4.6 

4.8 

5.0 

S 781 

1.9 

2.3 

2.7 

3.0 

3.3 

3.6 

3.8 

4.1 

4.2 

4.4 

4.6 

4.8 

5.0 

5.2 

l 895 

2.0 

2.4 

2.8 

3.1 

3.4 

3.7 

39 

4.2 

4.4 

4.6 

4.8 

5.0 

5.2 

5.4 

‘ 1,008 

2.0 

2.5 

2.9 

3.2 

3.5 

3.8 

4.0 

4.3 

4.5 

4.7 

4.9 

5.1 

5.3 

5.5 

■ 1,120 

2.1 

2.6 

3.0 

3.3 

36 

3.9 

4.2 

4.4 

4.7 

4.9 

5.2 

5.3 

54 

5.7 

J 1,232 

2.1 

2.6 

3.0 

3.4 

3.7 

4.0 

4.3 

4.5 

4.8 

5.0 

5.3 

5.4 

5.6 

5.8 

; 1,314 

2.2 

2.7 

3.1 

3.5 

3.8 

4.1 

4.4 

4.7 

4.9 

5.1 

5.3 

5.5 

5.7 

5.9 

‘ 1,456 

2.2 

2.7 

3.1 

3.5 

3.8 

4.2 

4.4 

4.7 

4.9 

5.2 

5.4 

5.6 

5.9 

6.0 

| 1,568 

2.3 

2.8 

3.2 

3.6 

3.9 

4.2 

4.5 

4.8 

50 

5.3 

5.5 

5.7 

6.0 

6.1 

i 1.680 

2.3 

2.8 

3.2 

3.6 

4.0 

4.3 

4.6 

4.9 

5.2 

5.4 

5.6 

5.8 

6.1 

6.2 

; 1,792 

2.1 

2.9 

3.3 

3.7 

4.0 

4.4 

4.7 

5.0 

5.2 

5.5 

5.7 

5.9 

6.2 

6.4 

; 1,904 

2.4 

2.9 

3.4 

3.8 

4.1 

4.4 

4.7 

5.0 

5.3 

5.5 

5.8 

6.0 

6.2 

6.5 

i 2,016 

2.1 

3.0 

3.4 

3.8 

4.2 

4.5 

4.8 

5.1 

5.4 

5.6 

5.9 

6.1 

6.4 

6.6 

! ! 2,128 

2.5 

30 

3.5 

3.9 

4.2 

4.6 

4.9 

5.2 

5.4 

5.7 

6.0 

6.2 

6.5 

6.7 

! ! 2,210 

2.5 

3.0 

3.5 

3.9 

4.3 

4.6 

4.9 

5.2 

55 

5.8 

6.0 

6.3 

6.5 

6.8 

| i 2,800 

2.6 

3.2 

3.7 

4.1 

4.5 

4.9 

5.2 

5.5 

5.8 

6.1 

6.4 

6.6 

6.9 

7.2 

< 3,360 

2.8 

3.1 

3.9 

4.3 

4.7 

5.1 

5.5 

5.8 

6.1 

6.4 

6.7 

7.0 

7.2 

7.5 

! 3*920 

2.9 

3.5 

4.0 

4.5 

4.9 

5.3 

5.7 

6.0 

6.3 

6.7 

6.9 

7.2 

7.5 

7.7 

it 4,480 

2.9 

3.5 

4.1 

4.7 

5.1 

5.5 

59 

6.2 

6.5 

6.8 

7.2 

7.6 

7.7 

8.0 

| 5,600 

3.1 

3.8 

4.4 

4.9 

5.4 

5.8 

6.2 

6.6 

6.9 

7.3 

7.6 

7.9 

8.2 

8.5 

J 6,720 

3.3 

4.0 

4.6 

5.1 

5.7 

6.1 

6.5 

6.9 

7.3 

7.6 

7.9 

8.3 

8.6 

8.9 

7,840 

3.4 

41 

4.8 

5.3 

5.8 

6.3 

6.7 

7.1 

7.5 

7.9 

8.2 

8.6 

8.9 

92 

8,960 

3.5 

4.3 

4.9 

5.5 

6.0 

6.5 

7.0 

7.4 

7.8 

8.2 

8.5 

8.9 

9.2 

9.5 

IIO'ORO 


4.1 

5.1 

5.7 

62 

6.7 

7.2 

7.6 

8.0 

8.4 

8.8 

9.1 

9 5 

9.8 

11,200 


4.5 

5.2 

5.8 

6.4 

69 

7.4 

7.8 

8.2 

8.6 

9.0 

9.4 

9.7 

10.1 

113*440 



5.5 

6.1 

6.7 

7.2 

7.7 

8.2 

8.6 

9.0 

9.4 

9.8 

102 

10.5 

•115*680 



5.7 

6.3 

6.9 

7.5 

8.0 

8.5 

8.9 

9.4 

9.8 

10.2 

10.6 

11.0 

••j 17.920 



5.9 

6.6 

7.2 

7.8 

8.3 

8.8 

9.3 

9.7 

10.1 

10.6 

10.9 

11.3 

T 20 160 



6.0 

6.8 

7.4 

8.0 

8.5 

9.0 

9.5 

10.0 

10.4 

10.9 

11.3 

11.7 

22 100 




6.9 

7.6 

8.2 

8.8 

9.3 

9.8 

10.3 

10.7 

11.2 

11.6 

12.0 

21 610 




71 

7.8 

8.1 

9.0 

9.5 

10.0 

10.5 

11.0 

11.5 

11.9 

12.3 

26 880 




7.2 

7.9 

86 

9.2 

9.7 

10.2 

10.8 

11.2 

11.7 

12.1 

12.5 

i 29*1 '0 




7.4 

8.1 

8.8 

9.1 

9.9 

10.4 

11.0 

11.5 

11.9 

12.4 

12.8 

31 360 




7.5 

8.3 

8.9 

9.5 

10.1 

10.6 

11.1 

11.7 

12.1 

12.6 

13.0 

! 33 600 




7.7 

8.4 

9.1 

9.7 

10.3 

10.8 

11.4 

11.9 

12.3 

12.8 

132 

135 810 




7.8 

8.5 

9.2 

9.8 

10.4 

11.0 

11.5 

12.0 

12.5 

13.0 

13.5 

38 080 




7.9 

8.7 

9.4 

10.0 

10.6 

11.2 

11.7 

12.2 

12.7 

13.2 

13.7 

1 40 320 




8.0 

8.8 

9.5 

10.1 

10.8 

11.3 

11.9 

12.4 

12.9 

13.4 

13 9 





8.1 

8.9 

9.6 

10.3 

10.9 

11.5 

12.2 

12.6 

13.1 

13.6 

141 

44 800 





9.0 

9.7 

10.4 

11.0 

11.6 

12.5 

12.7 

13.2 

13.8 

14.2 

49 280 





9.2 

10.0 

10.7 

11.3 

11.9 

12.8 

13.0 

13.6 

14.1 

14.6 

! 53*760 





9.4 

10.2 

10.9 

11.5 

12.2 

13.0 

13.4 

139 

14.4 

14.9 

, 58 240 





9.6 

10.4 

11.1 

11.8 

12.4 

13.3 

13.6 

14.2 

14.7 

15.2 

62,720 





9.8 

10.6 

11.4 

12.0 

12.7 

13.5 

13.9 

14 4 

15.0 

15.5 

























































































514 


STRENGTH OF MATERIALS. 


lateral motion, \ri n^uffice'hm VaflnTi? 0 *i*c and firm, y traced again 

Fig^’p 8 "®^ 6 ^ UeP - h ca } ,n,ot be Procured, brifcbLms mZyTe*l!sell*see Yan^i 

*?rSj^'uSS£! t *; i :xf entire bridge and ioad at two’ton 6 fiSk 


Span in Ft. 

Size of Beam. 

Span in Ft. 

Size of Beam. 

5 

10 

12^ 

8 X 10 ins. 

9 X 12 “ 

10 X 14 “ 

11 X 16 “ 

-—--- 

15 

17« 

20 

22^ 

12 X 18 

13 X 20 

14 X 22 

16 X 24 

_ 


The greatest dimension to be the depth. The ends should be well bolted down t 

c 


a 

IS 

\ 

V 

p 





^ ■ ~iv I 

'vK':., 


Kg 52 


bolsters. These are long stnut slicks of timber, from 10 to 15 ins sons re iWor. 
mg to the span, I laid1 acr vet > the abuts at the bridge-seat, for the chords to rest o. 
i.„ v „ , , ^ re Q u ootly two are used at each abut, even in small spans- and w 

r mnrf/ 1 °" e ’ Und ? r railroad 8 P aus of 150 feet. Large spans may require tlire 
so, " feet apart, ^ 0 ^ y ^ *“ COUt “ t with ° a ™- •*«« ■«, b 

following dimensions will answer; the total deflection of the n.d beini Jv. 

gir/n •*“ ™ s %x?£s&i £ 

For each brain. 


to 




Span. 

Ft. 


15 

20 

30 


Beam. 

Section of Rod. 

Ins. 

Sq Ins. 

12 X 15 

3^ 

13 X 17 

\Ya 

14 X 18 

6 

15 X 20 

7 


Section of Post. 
Sq Ins. 


25 

33 

42 

50 


It is better to have two rods instead of one under each beam • eaeh imi. , , 
the section here given; and the two placed severaHns aS T 1 g f l? 1 

footing for the post. The ends of the beam should be at rhrht angles hftbfd* 
of the rod; and be provided with ample washes c c of wSLSn 1 the direct,oi 

the pressure from the rod, over the whole area of the ends The ends’oftw,' lbutl, , lt 
ers may extend a few ins each way beyond the sides of the be .m A i S tbese , wasb 
scale at g. This allows the rods to be outside of the beam • SsTead of re” °“ a 
to be bored in the latter, for passing the rods throSth^ ho,ei 

together at the foot of the post ious mrougti them. Ihey may be nearei 


The head of the post may be tenoned into the bottom of the he»m • r . . 

iron straps. To prevent the foot from being worn bv the rods U stTolild n th -o, ted tc U b -’ 

iron shoe, as at s, may be bolted to it: having ribs for keening ^ shod with iron. A cast 

iron shoe ’may be’well secured toil, ’in^i'thfr^ase the ro^ at fshonUi 8t e Ut V cu S bl 

check any tendency in the foot to slide toward r or r under the vibrio^ of uul . ted Jo the shoe ns t< 
this can be most conveniently done bv making each rod e s r i„ ° f pas f ln S loads. Perhaps 

uniting their lower ends to the shoe at s by hooks and Vve's f.r nv !^^ 16 , 16 ! 18 , 1118 ’ I' r ' a " d b J 
methods are in use for the heads and feet of the posts of larep and boIts - &c - Various 

details which pertain more to the professional bridge 1 builder 8 P ’ b 1 We cannot her e treat upor 
This mode of trussing is also well adapted to long door beams- 
web members; as well as in long stretches of chonrifrom one point of support to^nother 8 ° bllqU * 






























































STRENGTH OF MATERIALS, 


MS 


Continuous beams. When a single beam, as a b, Fig 40, is supported not 
nly at its two ends, but at one or more ihterraediate points, it is said to be con- 
inuous. It is stronger than if it were cut into two parts, a c, b c, each supported 

at both ends; because the tensile 
strength of the particles at o (lower 
Fig) assists in counteracting the bendg 
or breakg tendency of loads on the in¬ 
termediate parts om, on, of the lower 
Fig. These particles at o must be torn 
asunder before the beam (if properly 
proportioned) can fail. Such a beam, 
m v, if very long and flexible, will, 
under its own wt, assume the shape of 
the reversed curve msosn; or if it be 
stiff, and heavily loaded, the same 
ffect will follow. The points s s, at which the curves reverse, are called the 
mints of contrary flexure; and the spans are virtually reduced from mo 
nd no, to ms and ns. When the beam is supported at only 3 points, as in the Fig, 
nd uniformly loaded, the point of contrary flexure is dist from the central support 
£ of the span; so that each span, om, on, becomes virtually reduced about % part; 



nd the defs will be but about jq^ as great as if there were two separate beams. The 
ections of the beam at s and s will then experience no hor strain ; but merely the 
ert one arising from half the wt between m and s, and n and s. The position 
»f the point of contrary flexure varies with the number of interme- 
iate supports, and with the manner of loading; and in bridges, &c, where the load 
aoves along the beam, it changes its place during the transit, so as to bring the points 
s considerably nearer to the central support o; thus reducing materially the ad- 
antage commonly supposed to arise from connecting together the ends of adjacent 
ridge-trusses; if indeed there is any advantage in so doing, which is doubtful. The 
rinciple, however, becomes very useful in the case of long rafters or girders, stretch- 
ng over several points of support, especially when uniformly loaded. Each interval, 
xcept the two end ones, will have two points of contrary flexure; and will then have 
iearly twice as much strength, under an equally distributed load, as a single beam 
to longer than said interval. 











516 


STRENGTH OF MATERIALS. 


Art. 33. 


Strength 

Beams. 


of hollow 



w 

Wf0JM 

1 



liillll 


Fig - . 13. 


Fig. 13. 


During the preliminary investigations 
relative to the construction of the Menai 
tubular bridge, a few experiments were 
made on the strength of hollow cast-iron 
beams of circular, oval, square, and rectan¬ 
gular cross-sections, supported at both 
ends, and loaded at the center. The clear 
span between the supports was in every 
case 6 ft, the thickness of metal in each 
beam, % inch; area of solid cross-section 

of each, 4.12 sq ins. The mean depth o o. Fig 12, of the circular tube, 3ins ; < 
the square one, Fig 13, 2 %; of the oval, 4% ; breadth, 2 %; and of the rectanguli 
one, mean depth, 3%; breadth, 1.833 ins. From these experiments Mr. Edwin Clar 
assistant engineer in charge, deduced the following constants,-and rules for cent 
breakg loads: 

Const for circ t ubes, .95 ; oval, 1; square, 1.14 ; rectangle, .91. Then, firs, 
finding the area of the solid part of the cross-section in sq ins, 

. . . Area of solid v Mean depth, o o, .. Corresponding 

Center breaking = iu sq in s x in ins x constant. 


load in tons 


Clear span in feet. 

Ex. Circular beam, mean depth o o, 3*4 ins; area of solid ring, 4.12 sq ins : deal 
span, 6 ft. Here, 

Area. Mean depth. Const. 

412 X 35 x -5 !£_= 2.28 tons, or 5107 lbs, breakg load. 


6 (length.) 

The thickness of the cylinder or tube is about 


and as a meai 


jL of the diam ;_ 

of 3 trials it broke with a center load of 2.287 tons, or 5122 lbs; span 6 ft. Henci 
we derive for similar tubes, the constant 530, to be used in the rule, Art 12; that is cen 
ter breakg load in lbs, of circular cast-iron tubes with a thickness of one-tenth of tin 
Cubeot outer diam (in ins)X 530 

outer diam = Clear gpaQ in feefc -; supposing Mr. Clark’s iron to have beet 

of average quality. 

The average breakg load of 3 square beams was 2.152 tons, or 4820 lbs ; of the rec 
^angular ones, 2.3 tons, or 5152 lbs; and of the 6 elliptic ones, 3.207 tons,or 7183 lbs 
See Art 9 f ° reg0,ng extraneoU8 load8 must be added half the wt of the beam itseli 

P and ride > P 488, give breakg loads about one-thin 
greater than Mr. Clark s results, except for the oval beam, where they agree closely 
ihe discrepancy is probably due to difference of quality of material/ b i 

«..nw < f!.* OW *u e r a *Vi S wro,, » h t iron were experimented on at thi 

same time ; and for these Mr. Clark deduced the following constants, to be used witl 
his foregoing rule for cast-iron ones : - 

Constants for thin riveted tubes, circular, 1.74 ; oval, 1.85; rectangular, 1.96. 

welded tubes, “ 1.09; “ 1.27; “ 1.51. 

Art. 34. The following experiments on riveted slieet-iron cvlindri 

5“* i , 7f? l Vi IS .i. are by / airba ^‘ lst * C - vlindcr 18 ft b.ng; 1 ft outer diam ; deal 
span 17 ft, thickness of iron .03/, or ^ of an inch; wt of tube 107 lbs. 



















STRENGTH OF MATERIALS. 


517 


Center load. 

Def. 

Center load. 

Def. 

Lbs. 

Ins. 

Lbs. 

Ins. 

1300 . . 


2368 . 


1920 . 

.41 

2480 . 


2114 . 


2592 . 


2256 . 

.60 

2704 . 



After bearing 2704 lbs. for 1 % minutes, failed by crushing at top. 

2<I. Cyl 16 ft 10 ins long; 12.4 ins outer diam; clear span 15 ft 7V< ins; thickness of 
ron .116, or full ^ inch; wt of tube 392 lbs. 


Center load. 

Deb 

Center load. 

Der. 

Lbs. 

Ins. 

Lbs. 

Ins. 

2000 .... 


8000 . 


4000 .... 

.34 

10000 . 


6000 .... 


11440 . 


With 11440 broke by the tearing of the bottom across the shackle-hole from which 

;he load was suspended. 



3d. Cyl 25 ft long; 17.68 ins outer diam 

; clear span 23 ft 5 ins; thickness .0631, or 

Tull inch; 

weight of tube 346 lbs. 



Center load. 

Def. 

Center load. 

Def. 

Lbs. 

Ins. 

Lbs. 

Ins. 

1000 .... 

.12 

5000 . 

.48 

2000 .... 

.21 

5280 . 

.51 

3000 .... 

.30 

5840 . 


4000 .... 

.40 

6120... 

.71 

With 6400 broke at bottom ; 25 ins from center, by tearing through the rivet-holes. 

4th. Cyl 25 ft long; 18.18 ins outer diam ; clear span 23 ft 5 ins; 

thickness .119, or 

leant % inch; 

wt of tube 777 lbs. 



Center load. 

Def. 

Center load. 

Def. 

Lbs. 

Ins. 

Lbs. 

Ins. 

2000 .... 

.15 

10000 . 


4000 ... 

.30 

12000 . 

.95 

6000 .... 


13000 . 

.1.04 

8000 .... 


14240 . 



Broke through the rivet-holes 3 ft 3 ins from center, after sustaining the load for 
lalf a min. 



The tubes were composed of sheets about 2]/^ ft wide; and so 
long that a single sheet sufficed to form the entire circumf of the 
tube. They were united by double-riveted lap-joints. The loads 
were placed on a platform, supported by a rod r, Fig 14, which 
passed through a hole h in the bottom of the tube s. This rod was 
attached at its upper end to a block of wood w, rounded at its 
lower surface, so as to fit the tube. 

Circular blocks of wood were fitted into the ends of the tubes, to 
prevent them from crushing at those parts under their loads; and 
the ends rested upon blocks hollowed out to correspond with their 
cylindrical shape, to a depth equal to about % part of their diam. 








































518 


STRENGTH OF MATERIALS. 



. ^ ie l 'PP e ^ u — 3 ins X 1 inch = 3 sq ins area; bottom rib b = \% ins 

X 1- ms — 18 sq ins area; total depth oo, 15 ins; clear span, 20 ft. Here 
18 X 15 X 2.166 584.82 

= 29.241 tons, the reqd load, including % the beam. 


20 20 _ . _ 

Now to find the wt of half the beam, we may proceed thus: Mult 

thp rnh 18 cr ' ,ss ^ ction u ' n sq ins, by the clear span‘in ins. This gives us 

the cub ms of iron contained in the beam; and these div by 8600, give the wt of tho 

beam in tnns • horanm fiOOA ;_. • . . . J 1,10 


iris Tib 1 t0 ^’, s b Tr a ;r 86 °? CU K ins of Cast iron weigh about 1 ton ; or near 4 cubic 
tain 2L the i “ 8 ? «S® the CM- 


*,.; n 01 __ .. . A *- °h lucu omot; me iwo nangey con¬ 

tain 21, the entile section is 33 sq ins; and the span being 240 ins, we have 33 X 240 

= 7920 cub ins of iron. And = .92 of a ton, the wt of the beam. One-half 


8600 


of this, or .46 ton, taken from the breakg load 29.241 tons, leaves 28.78 tons as the neat 
bieakg load , showing that in such cases as this it is scarcely worth while in practice 

toTSo Yse e e Sl6‘u T , r - be T 8 are not al ™y« ™de of the s.ame^tion 
throughout, (see tig 16,)but diminish toward the ends; this method is therefore not 

always strictly correct, but no great accuracy is needed in such cases. 

a *?® n< <he size of a IIodg-kiiiHon beam, reqd fo break under 
" center load, having the depth. Mult the given load in tons 

by the*clear span in feet. Mult tho constant 2.166 by the total depth oo in ins 

tf !f. lttst; the 9«ot will be the area of the bottom rib in sq ins' 
1 his d v by 6, will be the area of the top rib. The bottom rib is usually made from 
yo times as wide as it is thick; and the top one from 3 to 6 times The thickness 

toiM from T U ! Ua y all 5 tle / reater ^ bottom than at top; the average thickness 
bung from to ^ of the depth of the beam. See Rom. next page. 

O save iron, the width of the bottom flange, and of the top one also if thought 
-■ proper, may be reduced by curves 


ft 


"D—— 







Flgr16, 



to about % as great at each end 
of the beam as at its center; as 
shown by the middle sketch of 
Fig 16, of w hich the upper sketch 
is a side view. Or, leaving the 
dimensions of those flanges un¬ 
altered, the depth of the vertical 
rib may be reduced toward tho 
ends, as shown by the lowest 
sketch. The theoretical curve is 
here an ellipse. When the width 


is reduced, the very ends may, for stability, be widened 

The vert rib Is generally strengthened by casting brackets 


* In practice % is much better and safer than Y & . 












































STRENGTH OF MATERIALS. 


519 


on each side of it, as in the upper sketch. These should not extend entirely to tne 
upper rib, as they then expose the beam to crack as it cools. To prevent this tend¬ 
ency, they may be attached alternately to the top and bottom ribs. The upper ones, 
however, are rarely needed. 

In designing these beams, as well as in all other castings, it is important to avoid 
sudden transitions from thin to thick parts; and to keep all parts as nearly as possi¬ 
ble of the same thickness. Otherwise the castings are apt to warp and crack in 
cooling. Also, bear in mind that the resistance or strength per sq inch is considera¬ 
bly less in thick castings than in thin ones. 

Item. The above rule for breakg loads is safe when the load is equally disposed 
on top, or on each side of the vert web; and when said web and the flanges are pro¬ 
portioned to each other about the same as those used in Mr. llodgkinson’s experi¬ 
ments. But subsequent investigators have found his beams to break with but little 
more than half the loads given by the rule, when applied to only one side, as bo , or 
j uo, Fig 15, of the topor bottom flange. W. II. Barlow, C. E., London, experimenting 
| since Ilodgkinson, finds that when a cast-iron beam is liable to be loaded on only one 
side of ttie flange, the top flange should have an area equal to % that of the entire 
cross-section of the beam; and for beams so proportioned, he gives the following: 


■ntor / Area bot flauge\ , /Half area of Wf^ufer ("c'enter'oAx 

, y In ^ X Wand bot flange'r 


, Constant 
2.333 


Center 
bre 

*• load, .__ 

in tons Clear span in feet. 

Other experimenters recommend that even for loads pressing vertically through the 
upright rib, tho lower flange should have but about 3 instead of 6 times the area of 
f he upper one. Cast beams should always be tested. 

The average ultimate resistance of steel to compression being about twice that to 
ixtension, a Hodgkinson beam of that metal should have its lower flange of twice 
;he area of its upper one. jMuch uncertainty exists in the whole matter. 

Art. 36. For the purpose of ready reference, we give a few ex- 
Derimental results with cast-iron loams of various shapes: being the actual center 
oreakg loads in tons of sound beams. Some beams of Sterling’s toughened cast 
;ron gave results full 14 higher than those of common iron. 

Actual center breakg loads in tons, of cast-iron beams. Clear 
ipans in feet. Breadths and depths in inches. 


1.5*.5 

a.Sx.5 


to Span \\4 ft. 

Br load 2 tons. 


a 



The above inverted. 

Br load 2.3 to 2.9 tons. 


^1^.0 Span 4 % ft. 

*? Br load .125 ton. 


1‘1x\ 


.32 


2. 27x. 5 


fil 


The above inverted. 

Br. load .4 ton. 

to 

$2 Span V/ 2 ft. 

• f3.7 tons 

Br load ■< to 


( 4.2. 



Span 11% ft f 
Br load 20 tons. 


8x1* 



8 Jxl 


Span 18 ft. 

Br load 22 to 28 tons. 



Span 27^ ft.J 
Br load 29]^ tons. 


* As shown by dd. Fig 15. .. . . . . . „ 

t “After bearing 17 tons, the beam was unloaded, and its elasticity appeared to be but little if at all 
njured.” Def under 4^ tons, .15 inch ; tons, .3 inch; 17 tons, 1.1 inches. . 

I About two hundred of these beams were tested by ceuter loads of 12 tous. Def % to % inch. 


37 






























520 


STRENGTH OF MATERIALS. 


€• 5x1 



Span 15 ft. 

Br load 12% tons. 


1.75 x.42 A 

ITS 



Span 4% ft. 

Br loan 3 tons. 


6.67 ,_ a__ 

X.66 


Span 4% ft. 

Br load 10.5 to 
11.6 tons. 


H-x it 



Span 19 ft. 

Br load 50 to 54 tons. 


By formula, p 488, 
it should have been 
but 40 tons. 


4 w 


6 


15 x 2^: 



Span 30% ft. 

Br load 58 tons.* 


In describing such beams, it is better to give the entire depth of the beam; foi 
when the depth of the web is given, doubts arise whether it is meant to include the 
thicknesses of the two flanges, or not. Every writer, almost, that we have Been 
leaves us in this doubt. 

Rem.. In beams either larg-er or smaller than these, but whosp 
eross-sections are proportioned exactly as these are, and whose spans are the sami 
that these have, the center breakg loads will be as the cubes of their cross-sectioi 
lines. Thus, in a beam which is , % , % , 2, 3,. or 10 times as large ever; 


way, except in span, 




the breakg load will be T~5oo ’ ~i7 ’ y's > 8,27, or 1000 times as great. 

If the spans also differ, first find the load as above, as if they were th 
same; then say, as the new span, is to the span given in our Figs, so is the break 
load thus found, to the actual breakg load for the new beam. Thus, suppose we wis 
to make a cast-iron beam, 4 times as large every way as the dimensions given in th 
first of these Figs; except its span, which is to be, say 10 ft, instead of 4% ft. Iler li 
the first breakg load is found tobe4X4Xl = b4 times as great; or 2 tons X e 
«= 128 tons. Next, 

New Span. Span in Fig. First load. Actual load. 

10 : 4.5 :; 128 : 57.6 tons. 

In such cases we must, however, have regard to Rem 1, Art 11, p 494. The fore 
going process applies equally to beams of any other shapes, such as the followin 
ones; or whether solid or hollow, &c; and of any other materials; so that if w 
have all the dimensions, and the breakg load of any beam whatever, we may fin 
that tor another one of the same material, and of the same proportions of cross-se< 
tion. It may become advisable in important cases, to even make one or more mod* 
beams of some hitherto untried form ; and to break them in order to find the breal 
weight of the actual beam of the same material. In doing this, the defs should al 
be ineasd, in order to see whether those of the actual beam may not be too great. Sc 
A rt 26, &c. 


* This iron was “ Sterling's toughened, 4 having about 16 per cent of wrought scrap melted in i 
Each of the 89 beams was tested by a center load of 20 tons, which produced defs of from % to 
inch. Entire length, 34>$ ft. 






























STRENGTH OF MATERIALS. 


521 



Fig 18. 



Q 


Fig 20. 


1 beams, fig. 18; Channels, fig. IS; and Deck beams, fig 


20 . 


Art. 37. 

are made by _(_ 

Wr- 2C ' s ' ■“ h St-.ri.nad.lpM. 

16th and Market Sts., Philadelphia ) ’ ' J ‘ ( re P reseDte d by Morris, Wheeler & Co., 

SSSfi*i®D co & , Sh’a^“S *•. 

0l c unts on large orders. mag, etc., address as above. Special dig- 

,m°aX» the ,„ rgMt and 

)f web for each pattern. Any desired thicknp« V and minimum thicknesses 

tarnished Th.dll of flange Increases eouX tllwh "S" ext ™“« «M be 

heruc<™., s of flange, at root and at base, for an/ given pat tem^are"^ but 

ill thicknesses of web. ’ Pattern, are the same for 

IVglnU^j’ h"“Lo,^?o ‘ > 208 Fratk ! iin ,a /rN S 2 v T 334 fts P« 74 

t th St, Philadelphia. ’ rankhn St - New York J Esberick & Co, 263 S. 


In any bar^bean, itc, of wrought iron, of uniform cross section, 

area of cross section, 

in square inches X 10 


Weight per lineal 
yard, in pounds 


Area of cross section. weieht nf 

in square inches = —~—-_ am per lineal yard, in pounds 

10 ~ 

iSSSRSfe 3S%£ factSXS u^’S^S^SoSS^ 

fitly without°?nVreas C e h o S f°load^^The'teb 8 ™ sf ppose^to*^ 6 ?,!^. ? ield inde £ 
nr figs; and the beam supported at ifoth eldS^Sdstaved SSbfJv’iiM-*“ 

«" the a JaLge. e Then*^ ° f th ® “ de sup P orts llot exceeding 20 times the widuf 


Center ultimate load 

in lbs, including half wt of == 
clear span of beam 


Tabular coefficient for een ult 
load* for the given size 


clear span in ft 

Xxtr&neous cen ult load = cen ult load so found — ha,f wt of c,ear 

span of beam 

Distributed ultimate load = twice cen ult load 

Safe cen or distd load _ ll ^ cen or d »std load 

required factor of safety 
1 7’® : of safety should in no case he less than 3. When the lo>.d is 
» J from4 r e6 f P 43 . 5 > 0 , r . v |brat,ion, or is liable to be suddenly apmied 

• ic factor should be further increased when the length of beam 


■The coefficients are the center ultimate loads in pounds (including half tl.« 
i2ht of the clear span of the beam) for beams of one foot clear span, taking 42 000 
p r square inch as the modulus of rupture (p. 485.) * ® * ,uuu 



































522 


STRENGTH OF MATERIALS. 


■between lateral supports exceeds 20 times the^ width of a^eiiJth == 20 

readies 70 times said width, the factor should be double that for a length 

* Caution. With verv short, spans, the safe loads thus found, although safe 
as regards a ruptvre of the beam itself, may be so great, as to endanger a cjws - 
ing of the ends of the beam, or of the wall etc under them, unless the beam has 
at its ends a greater length of bearing than would otherwise be needed. &ee 

P *To find wliat beam Is required to sustain safely a given extraneous 
load in lbs, whether uniformly distributed or applied at the center. If uniformly 
distributed, divide it by 2. The quot is the equivalent center load. Then in either 
case add half the wt of the clear span of the beam itself. Mult together the sum 
the required factor of safety, and the clear span of the beam in ft. Use any desired 
beam whose coefficient for ult cen load is not less than the prod. 

for the deflection in ins at the center of the length of the beam, under any 
load less than half the ultimate load : 

Load in lbs X Cube of clear sp an in ft 

De ?n C i!i i0n = Wt of beam, v Square of depth D v Constant, 
in 1 in lbs per yd A of beam, in ins ^ given below. 

Constants for the above formula: 

For I beam, loaded at center.11200 

“ “ uniformly loaded.18000 

See also formulas, p. 505 a. 

Tlie strength of a beam or channel when used as a column, 

may be found by the formula, p 439, using the least radius of gyration, as given in 
the following tables. 

For separators for I beams, see pp. 523 b and 523 d. 


For channel beam, loaded at center, 10000 
“ “ uniformly loaded, 16000 



Kig. SI 

Fire-proof floors of I beams and brick-arches, Fig. 581. 

The arches are usually “four-inch ’’—or “half a brick” deep; span, s, from 4 to 
6 feet; rise about one twelfth to one sixteenth of the span. Tie-rods T, 3-4 inch to 
1 inch diameter, from 4 to 6 or 8 feet apart, and anchored into each wall with a stout 
washer, W. At each wall an angle iron, a, or a tee iron (see pp. 525, etc.) ia 
generally used instead of a beam. The spandrels are 
enclosing wooden strips m m. about 1 inch X 2 inches, 
these strips the flooriug is nailed. 

The weight of a 4 inch arch, with its concrete filling and wooden flooring, bu 
exclusive of the beams, Is about 70 lbs. per square foot of floor. 

A dense crowd of persons will hardly weigh more than 80 lbs. per square foot 
See foot notes pp. 606 and 623. 


iron (see pp. tyzb, etc.) is 
leveled up with concrete, 
two over each arch. To 



Fig. 22 shows the u°ual manner of centering the arches. Each center, 0, h 
fastened to it at each end a bent iron strap h, forming a hook by which the cen 
is suspended from the lower flanges of the beams. 

































































STRENGTH OF MATERIALS 


523 



Pen coy <1 Channels.— See page 521. 



«. J 

£3 

SS-a 

0) tH 

bg, 

Coefficient 

for ultcen load, 

in lbs. 

Least radius of 

cvration. in ins 

Mom 

Inc 

About 

XY. 

ents of 
:rtia. 

About 
W Z. 

S 204.5 

1,040,400 

1.10 

557.44 

24.74 

1 148.0 

842,700 

1.13 

451.51 

19.05 

J 160.0 

626,400 

.90 

268.51 

12.96 

1 88.5 

426,300 

.92 

182.71 

7.42 

j 101.5 

404,700 

.72 

173.51 

5.26 

) 60.0 

274,200 

.74 

123.71 

3.22 

j 106.0 

366,900 

.82 

131.04 

7.13 

1 60.0 

259,500 

.84 

92.71 

4.29 

j 86.5 

294,300 

.67 

105.16 

3.88 

1 49.0 

. 206,700 

.09 

73.91 

2.33 

j 93.0 

282,300 

.67 

90.66 

4.17 

j 54.0 

200,100 

.68 

64.34 

2.47 

j 61.0 

186,300 

.58 

59.85 

2.05 

| 37.0 

135,600 

.59 

43.65 

1.31 

j 80.5 

210,000 

.71 

60.00 

4.28 

| 43.0 

139,800 

.71 

40.00 

2.17 

1 54.0 

143,400 

.60 

41.03 

1.94 

i 30.0 

98,700 

.60 

28.23 

1.06 

( 73.0 

170,100 

.65 

42.57 

3.08 

41.0 

117,900 

.65 

29.51 

1.71 

49.0 

111,300 

.58 

27.86 

1.65 

26.0 

73,800 

.58 

18.46 

.90 

55.4 

126,300 

.63 

25.12 

2.56 

32.9 

85,500 

.07 

18.37 

1.46 

54.5 

124,200 

.01 

24.35 

2.03 

32.0 

72,900 

.00 

17.60 

1.15 

39.6 

90,200 

.52 

16.73 

1.07 

22.7 

54,300 

.51 

11.67 

.59 

46.0 

87,400 

.58 

14.20 

1.55 

27.3 

57,600 

.50 

10.29 

.86 

32.9 

62,500 

.47 

12.54 

.73 

18.8 

37,200 

.45 

6.67 

.37 

31.5 

47,800 

.52 

6.49 

.85 

21.5 

36,300 

.50 

5.16 

.54 

23.7 

36,000 

.49 

4.97 

.57 

17.5 

28,800 

.48 

4.14 

.41 

18.9 

21,600 

.47 

2.31 

.42 

15.2 

18,900 

.40 

2.03 

.32 

11.3 

10,000 

.43 

.80 

.21 

10.0 

7,600 

.32 

.88 

.10 

8.75 

6,700 

.31 

.48 

.08 

3.5 

2,300 

.17 

.38 

.01 





















































































523 a 


STRENGTH OF MATERIALS 


Pencoytl I Beams.— See page 521. 


X 

a 

3 

S5 

u. 

3 

JO 

U 


10 

11 

12 

14 

16 

18 


19 


20 


21 

22 


■M 


by 2 



10 } 

ioh , 

, 10U| 4J_ 
ilOU 4H 
■) 101 ? 


?* ill M 


4% 

4 3 4 

4% 

4§i 

4% 

4gi 

4^8 

*41 

4 

4 ^ 

& 

3 

2% 

3 

2% 


8 


IB 

A' 


234 


TB 
7 i 

f* 

| 

H 

! 7 
2 



00 

0) 

CO 



n 

a 

a 

licknes 

f fiang' 

iu ius. 




o 



At 

At 



root 

edge 


/ 

f 

He 

% 

I 

( 

i 

1 % 

% 

3 

} 

134 

11 

j 

I 

i 

f 

1 

1! 

3 

i 

f 

V/s 

§2 


( 

( 

11 

4* 


J. 

( 

n 

TB 


( 

f 


/4 


i 

f 

1 

t 7 b 


\ 

!i 

14 


l 

§2 

% 


i 

r 

n 

32 


3 

3 A 

% 


l 

ii 

11 


t 

i 

§1 

t b b 


\ 

34 

% 


L 

f 


3% 


3 

% 

3 7 2 


3 

TB 

14 


3 


4f 



a 

j= "" 
*>•© 
0) *- 

t* £ 


233 

200 

201 

145 


120 

161 

134 

134 

108 

109 

89 
137 
112 
106 

90 
122 

90 

88 

70 

109 

81 

75 

65 

88 

51 

63 

40 

40 

30 

38 

28 

21.5 

18.5 

28.6 
23. 
21.7 
17. 


Coefficient 

for lilt cen load, 

in tbs. 

Least radius of 

gyration,in ins. 

Moments of 
Inertia. 

About 

X V. 

About 
W Z. 

1.388.700 

1.17 

743.60 

31.89 

1,273,200 

1.20 

682.08 

28.50 

1,168,800 

1.05 

626.57 

22.16 

972,900 

1.08 

521.19 

16 91 

940,800 

1.16 

403.48 

26.10 

867,900 

1.17 

371.98 

23.19 

757,200 

.99 

324.61 

16.02 

636,600 

1.01 

272.86 

12.22 

708,600 

1.17 

265.74 

22.20 

644,300 

1.19 

241.63 

19.00 

585,400 

1.05 

219.53 

14.74 

521,100 

1.07 

195.42 

12.45 

480,900 

.94 

180.34 

9.59 

432,700 

.97 

162.26 

8.34 

544,200 

.96 

194.41 

12.63 

486,000 

.98 

173.58 

10.64 

452.400 

.93 

161.33 

9.17 

415,200 

.95 

148.31 

8.46 

436,800 

.94 

143.70 

10.78 

369,600 

.96 

118.81 

8.44 

331,500 

.87 

106.78 

6.66 

293,700 

.89 

94.44 

^.59 

345,900 

.93 

98.59 

9.43 

293,700 

.94 

83.93 

7.23 

260,700 

.86 

74.50 

5.55 

242,100 

.88 

69.17 

5.02 

232,800 

.80 

58.60 

5.63 

172,200 

.82 

43.08 

3.43 

144,600 

.64 

30.77 

2.58 

112,200 

.66 

24.10 

1.80 

81,600 

.58 

14.69 

1.35 

69,900 

.60 

12.50 

1.09 

63,000 

.62 

9.02 

1.46 

53.700 

.63 

7.69 

1.17 

39,300 

.50 

5.55 

.54 

36,000 

.51 

5.14 

.49 

34.500 

.58 

3.99 

.96 

30,600 

.59 

3.29 

.77 

28,200 

.52 

3.01 

.59 

24,600 

.53 

2.66 

.48 



* 

Steel 

20 

20 

1 % 

0 % 

6 

I'A 

H 

h 

144 

.16 

. 

1 h-JwHXO' 

| h-ttoh)00\| 

272 

200 

5.25 

2,310,000 

1,732,500 

4,320 

1.31 

1.15 

.36 

1,650.30 

1,238.00 

.135 

46.50 

26.62 

.0685 


These last three patterns are from the list of the New Jersey 
Steel and Iron Co., Trenton, N. J. The 20 inch are the 
largest, and the \% inch is the smallest, now made in America. (But 
see also p. 523 c). 































































K,Ho«»Ma9.mM K Ha«oo^ae, 0ll ^ M19 M Chart number 


SEPARATORS FOR I BEAMS 


5236 


STANDARD SEPARATORS FOR PENCOYD ROLLED 

IRON I BEAMS. 

Price per lb. of separators and bolts is about the same as that of the beams, 

see p. 521. 




Beam. 

Separator. 

Bolts. 

Depth, inches. 

Weight lbs. 
per yard. 

-1 

Minimum width, 
inches, out to out 
of the two flanges.* 

Weight, fos. 
of one sep’r. 

Number used. 

Diameter, inches. 

Weight, lbs. of 
1 bolt and nut 

For mini¬ 
mum width.* 

Additional 
for each ad¬ 
ditional iuch 
of width. 

For mini¬ 

mum length* 

Additional 

for each ad¬ 

ditional inch 
of length. 

15 

in. heavy 

186 to 233 

12 

22 

3.84 

2 

4 

1.75 

.123 

15 

in. light 

145 to 201 

11 

21 

3.13 

2 

4 

1.62 

.123 

12 

in. heavy 

168 to 194 

UK 

16 

2.76 

2 

% 

1.69 

.123 

12 

in. light 

120 to 163 

io T | 

144 

2.95 

2 

4 

1.58 

.123 

10>^in. heavy 

134 to 161 

11 

114 

2.10 

1 

4 

1.64 

.123 

10*4in. med. 

108 to 135 

10*4 

11 

2.06 

1 

4 

1.28 

.123 

10*4 

in. light 

89 to 109 

9% 

11 

2.03 

1 

4 

1.53 

.123 

10 

in. heavy 

112 to 137 

9 4 

10 

1.93 

1 

4 

1.56 

.123 

10 

in. light 

90 to 106 

9 

10 

1.93 

1 

X4 

1.52 

.123 

9 

in. heavy 

90 to 122 

9*4 

9*4 

1.63 

1 

4 

1.52 

.123 

9 

in. light 

70 to 88 

m 

9 

1.63 

1 

% 

1.48 

.123 

8 

in. heavy 

81 to 109 

91 4 

64 

1 36 

1 

4 

1.50 

.123 

8 

in. light 

65 to 75 

84 

6 4 

1.49 

1 

4 

1.46 

.123 

7 

in. heavy 

65 to 88 

84 

4 

1.26 

1 

% 

0.96 

.085 

7 

in. light 

51 to 88 

8*4 

4 

1.26 

1 

% 

0.91 

.085 

6 

in. heavy 

50 to 63 

6% 

3 

1.24 

1 

% 

0.90 

.085 

6 

in. light 

40 to 63 

6% 

3 

1.24 

1 

% 

0.87 

.085 

5 

in. heavy 

34 to 40 

6 

2% 

1.10 

1 

K 

0.43 

.055 

5 

in. light 

30 to 40 

6 

2 4 

1.10 

1 

K 

0.42 

.055 

4 

in. heavy 

28 to 38 

6 

2 

0.85 

1 

K 

0.42 

.055 

4 

in. light 

18.5 to 21.5 

4% 

2 

0.85 

1 

K 

0.39 

.055 

3 

in. heavy 

23 to 28 6 

6% 

i K 

0.69 

1 

k 

0.38 

.055 

3 

in. light 

17 to 21.7 

4% 

iK 

0.69 

l 

K 

0.31 

.055 


♦The flanges of the two beams in contact. 


































































523 o 


STRENGTH OF MATERIALS 


Steel beams. Carnegie, Bros. & Co., Limited, Pittsburgh, Pa., 
roll steel beams as described in table below. Owing to the gi eater 
strength of the material, they are rolled somewhat lighter (chiefly 
in the web) than iron beams of the same depth. 

The prices per pound are the same as for iron beams. See p. 521. 

To flncl tlie strength in any given case, use the formulas 
on p. 521, which apply equally to steel beams, except that the 
coefficients are of course to be taken from the following table. Said 
coefficients are calculated upon ail assumed ultimate longitudinal or 
« fiber ” strain of 50,000 pounds per square inch. Experiment shows 
this to be a safe assumption. 

Deflection. In the absence of full experimental data, we may 
assume that the modulus of elasticity for the steel used in beams is 
about one-tenth greater than that of iron, and use the formula tor 
deflection, p. 522, as follows: 

Deflection Load in lbs. X Cube of clear span in feet 

in inches — Weight of beam, v Square of depth D 77 Constant, 
iu lbs. per yard x of beam, in inches A given below. 

If the beam is supported at both ends and loaded at the center ; constant = 12000. 
i: « « “ “ “ “ uniformly; “ =19500. 

See also formulae for deflection, p. 505 a. 



CARNEGIE STEEL. I BEAMS. 


P4 

2 

00 

® 

A 

o 

.2 

a 

•H 

ft 

a 

bi[) 

a 

a • 

« $ 

o o 

a 

& 

<D 

* . 

° & 

CO D 

00 

® a 

Thickness 
of flange 
in inches. 

t per yard * in 
pounds. 

?fficient lor 
:e center load 
a pouhds. 

1 

cj 

Si 

of 

2 -S 

a 

n 

a ~ 

Moment of 

Inertia. 

V- 

o 

o 

a. 

CD 

ft 

+-> 

p 

S? 

a 

M 

o 

• H 

3 

H 

at 

root. 

at 

edge. 

rS 

to 

'3 

£ 

o S'" 

OS 

+3 

"5 

Least l 
tion 

About 

XY 

About 

WZ 

301 b 

20 

7.00 

0 60 

1.14 

0.66 

240 

2415000 

1.39 

1449.0 

45.6 

301 a 

20 

6.25 

0.50 

0.93 

0.55 

192 

1910000 

1.20 

1146.0 

27.3 

302 c 

15 

6.16 

0.785 

1.25 

0.875 

240 

1723800 

1.29 

775.7 

38.8 

302 b 

15 

5.75 

0.45 

0.95 

0 55 

150 

1177<X)0 

1.20 

529.7 

21.0 

302 a 

15 

5.50 

0.40 

0.78 

0.40 

123 

912500 

1.08 

424.1 

14.0 

303 b 

12 

5.50 

0.39 

0.88 

0.50 

120 

781400 

1.20 

281.3 

16.8 

303 a 

12 

5.25 

0.35 

0.72 

0.35 

96 

617500 

1.04 

222.3 

10.3 

304 b 

10 

5.00 

0.37 

0.82 

0.47 

99 

537500 

1.10 

161.3 

11.8 

304 a 

10 

4.75 

0.32 

0.65 

0.32 

76.5 

412200 

0.99 

123.7 

7.32 

305 b 

9 

4.75 

0.31 

0.75 

0.42 

81 

409700 

1.07 

110.6 

9.10 

305 a 

9 

4.50 

0.27 

0.60 

0.28 

63 

312300 

0.95 

84.3 

5.56 

300 b 

8 

4.50 

0.27 

0.67 

0.35 

66 

299400 

1.01 

71.9 

6.62 

306 a 

8 

4.25 

0.25 

0 56 

0.26 

51 

240600 

0.91 

57.8 

4.35 

307 b 

7 

4.25 

0.27 

0.65 

0.35 

60 

236500 

0.97 

49.7 

5.52 

307 a 

7 

4.00 

0.23 

0.53 

0.25 

46.5 

183700 

0.87 

38.6 

3.47 

308 b 
308 a 

6 

6 

3.625 

3.50 

0.26 

0.23 

0.59 

0.50 

0.34 

0.25 

48 

39 

159000 

130500 

0.83 

0.77 

28.6 

23.5 

3.24 

2.27 

309 b 

5 

3.13 

0.26 

0.54 

0.33 

39 

104700 

0.72 

15.7 

1.99 

309 a 

5 

3.00 

0.22 

0.44 

0.23 

30 

82600 

0.661 

12.4 

1.29 

310 b 

4 

2.75 

0.24 

0.49 

0.30 

30 

64400 

0.645 

7.73 

1.23 

310 a 

4 

2.625 

0.20 

0.38 

0.20 

22.5 

49000 

0.584 

5.90 

0.752 


* Messrs. Carnegie state the weights of their beams in pounds per foot. We give 
them in pounds per yard for the sake of uniformity with our other tables. 













































SEPARATORS FOR I BEAMS 


523 d 


STANDARD SEPARATORS FOR CARNEGIE ROLLED 

STEEL I BEAMS. 

Price per ft. of separators and bolts is about the same as that of iron or steel 

beams, see p. 521. 




Beam. 


Separator. 


Ja 

•a s 

C -3 

s CO 

'jr 

© 


a> 

P3 

o 

□ 


a. 

<0 


301b 

301 a 

302 c 
302 b 

302 a 

303 b 
303 a 
303 b 

303 a 

304 b 

304 a 

305 b 

305 a 

306 b 

306 a 

307 b 
307 a 

V 308 b 
1 308 a 
309 b 

309 a 

310 b 
310 a 


20 

20 

15 

15 

15 

12 

12 

12 

12 

10 

10 

9 

9 

8 

8 

7 

7 

6 

6 

5 

5 

4 

4 


u 

CD 

P. 


J 2 "*T 
a 

rO 

bt> 

'5 


240 

192 

240 

150 

123 

120 

96 

120 

96 

99 

76.5 
81 
63 
66 
54 
60 

46.5 
48 
39 
39 
30 
30 

22.5 


'a 


* 

jz o • 

^ GO 

- a> 
3 fcdO 
> O a 
^ q 32 

a +* S3 
a 

-p § © 

s o * 

• G GO ® 

^ a rC 


Weight, lbs. 


a 

'I 

o 

pH 


* 

+2 


14% 

13% 

12% 

12*4 

11% 

H% 

11 % 

11 % 

H% 

10 % 

10 % 

10*4 

9% 

9% 

9% 

9 

8% 

7% 

7% 

6% 

6 % 

6 

5% 


18% 

17% 

11 

12 % 

9% 

9% 

9% 

7 

7% 

6 % 

6 

5% 

4 5 H 

8 

2 % 

& 

1$ 


I H 

— "O 'O 

a3 <S C 
c 

o £2 

StfS'i 

S * § ' 

4-< T3 


2.43 

2.46 
1.72 
1.81 
1 85 

1.43 

1.47 
1.43 
1.47 
1.18 
1.22 
1.07 
1.09 
0.95 
0 97 
0.82 
0.84 
0.52 
0.54 
0.42 
0.44 
0.33 
0.35 


Bolts. 


f 

GO 

CD 

GS 

0 

G 

•rH 

lT 

CD 

Weight, lbs. of 

1 bolt and nut. 

T5 

0 ) 

-4 Q0 

M © 

* 

1 -G 

•r-t -+-» 

C bp 

Tell" • 
S 54 c-P 
5 ^ 

r © 

? .© 

3 a 

° p 

S 

<9 

s 

'5 ,2 

0 a 

6 p 

a 

0 — _ tc 
5 u cs a 

•-it 3 2 .® 

S " g- 
° 

4-1 T3 

2 

% 

1.75 

.165 

2 

V* 

175 

.165 

2 

% 

175 

.125 

2 

% 

1.625 

.125 

2 


1.5 

.125 

2 

% 

1.6 

.125 

*2 

1.5 

.125 

1 

3 ? 

1.5 

.12 

1 


1.5 

.12 

1 

% 

1.5 

.12 

1 


1.5 

.12 

1 

% 

1.5 

.12 

1 

3Z 

1.25 

.12 

1 

% 

1.25 

.12 

1 

% 

1.25 

.12 

1 

% 

1.25 

.12 

1 

% 

1.25 

.12 

1 


1 

.12 

1 


1 

.12 

1 

% 

.75 

.12 

1 

% 

75 

.12 

1 

% 

.75 

.12 

1 

% 

.75 

.12 


Separators for beams 7 inch and larger are made % inch thick. Separators for 
mis smaller than 7 inch are made % inch thick. 


* The flanges of the two beams about half an inch apart. 

f Messrs. Carnegie state the weights of their beams in pounds per foot. We give 
>em in pounds per yard for the salte of uniformity with our other tables. 







































































524 


BRIDGES OF I BEAMS. 


Rolled iron I beams for railroad bridges of sliort span. 

(See pp. 521, etc.) A single 7-inch beam, 88 lbs per yard, under each rail, will suffice 
for 3 or 4 ft span ; one of 15 inch, 200 lbs, for 8 or 10 ft; two of 12 inch, 194 lbs, side 
by side under each rail, as in Figs 50 A and 50 B, for 12 to 14 ft; and two of 15 inch, 
145 lbs, under each rail, for 15 ft. By employing a greater number of beams, or by 
introducing a truss-rod, r r, Fig 52, p 514, the spans may be increased, or lighter 
beams be used. Care must be taken to insure lateral stability, by means of hor ties 
and struts. 




INCHES FEET 




Figs 50 A and 50 B, for which we are indebted to the courtesy of Mr. Jos. M. 
Wilson, C E, Engr of Bridges and Buildings, Penna R R, illustrate a standard 
form of rolled beam girder in use on that road. For spans of 14 ft, each beam is 12 
inch 180 lbs per yd; for 15 ft, 15 inch, 150 lbs. The two beams, B B, forming a 
girder, are held at the proper dist apart from each other by separators, S, each of 
which consists of a short piece of channel irou placed vertically, and having its 
flanges riveted to the webs of the beams. On some roads, cast-iron separators, and 
bolts which pass through them and through the webs of the beams, are used in 
stead. The longitudinal dist, R R, Fig 50 A, between the separators, is about 5 time* 
the depth of the beam. 

At the same points, R R, &c, of the span, are placed transverse tie-struts, T, Fi 
50 B, each composed of two 4-inch 21 lb channel bars placed back to back (so as t 
form an I) but % inch apart. Between the two channels are riveted angle-plates 
A A, % inch thick, the flanges of which are fastened to the webs of the inner beamt 
of the girders by the same rivets which hold the separators. At their centers, th< 
two channel-bars are separated by a piece of bar-iron, % inch thick X 4 ins square 
riveted between them, as shown. The cross-ties, notched to the girders as shown 
give additional lateral support. 

At the ends of the span, the lower flanges of the beams are riveted to rectangula 
bolster-plates,” P, which rest upon slightly larger wall>plates, W 
The rivets are counter-sunk under the plate, P. Both plates are of rolled-iron,^ 
iuch thick. They are held in place on the abuts by bolts passing through both platei 
as shown. At one end of the span, the bolt holes in the bolster plate are slightly 
elongated, to allow for contraction and expansion. 







































































ANGLE AND T IRON. 


525 


Angle and T Iron. 



t*-- 


Fig 1. 







w 

n 

*r 


X' m 

1 


r 



[ 



S 

i 

r 

i 



i 

i 

K 

• 

r> 



i? 



• 

: 

B 

i 

i 

i 

i 

• 


i 

t 

B 


! x 

Y 


• 

1 





Li ^ 

4 

_/ 


! X 



\X- 


if 4 



A i. < 

_ 


r r 

o r 

... 1 


C— \ 

*— 

« 

j. . 

k f.~ 


r\ 


Fig 2. 


Fig 3. 


-. ± — 

Fig 4. 


B 


The sizes A and B are, in all cases, measured from out to out. as 

indicated in the Figs. ’ 

Ang le iron, of any given dimensions, A and B, in the tables, can he rolled 
to any desired thickness between the maximum and minimum thick¬ 
nesses given for that size. The dimensions A and B vary slightly with the thickness 
Those given are the dimensions corresponding to the minimum thickness. In order- 
mg angle iron of thicknesses between the max and min, give either the thickness 
in ins, or the wt in lbs per yard, wanted ; but not both. 

The thickness of each size of T iron is fixed as given in the table 
and cannot be varied except by changing the shape of the rolls, or making new 
ones. This is always expensive. It is important, in designing,or iu ordering rolled 
iron of any kind, to bear this in mind, and not to introduce sizes that have to be 
specially made. 

The area iu square ins of cross section of any bar of rolled 

iron of uniform dimensions throughout, is = Us "eigh t m lbs per yd 

The ultimate or crippling center load, for a beam consisting 
of a single bar of angle or T iron, supported at both ends, may be found by the 
formula, p 488, using the moment of inertia as given in the following tables* Of 
approximately, and much more simply, 


Fit cen load 

in lbs 


} 


2800 X Area of ^ ross " v De P tl1 of 

^ section in sg ins x Beam in ins 

Clear span in ft. 


The ultimate distributed load is twice the ult center load. 

For the safe center or distributed load, divide the ultimate center 
■ or distributed load (as the case may be) by the required factor of safety, which 
should in no case be less than 3, and should generally he from 4 to 6, according to 
circumstances. ° 

Under any load less than half the ultimate load; 


]>eflection in ins ~| Load in lbs X Cube of clear span in ft 
with load at centerJ 68000 X A ^. e ^' n X Squa -® ?! depth 

iub in ids. 


Deflection in ins 'l Load in lbs X Cube of clear span in ft 

with distributed load j ~~ 108000 X i n X Squa F e ? f de P th 

bij[ ins in ins* 


The breaking load, for a column consisting of a single bar of angle 
or T iron, with flat ends, firmly fixed, may be found by the formula, p 439, using the 
least radius of gyration as given in the following tables. 





































526 


ANGLES AND T IRON, 


Angles with equal legs. Fig. 2. 


< £ 

.a 
» *** 

|.s 

2« 

o a 

5* 

Thickness, ins. 

13 

t- 00 

g 

fl 

U .H 

p£ 

Moment of In¬ 

ertia about X Y 

Least Rad of 
Gyration in 
ins. 

Dist d from 
base toneutr’l 
axis. ins. 

Dimensions, A 

and B, in ins. 

Thickness, ins. 

>% 

. 03 

03 S3 

-4J 

* 

Moment of In¬ 

ertia about X Y 

Least Rad of 

Gyration in 

ins. 

Dist d from 

base to neutr'l 

axis, ins. 

1 

X 

1 

34 

2.3 

.02 

.20 

.30 

2^X2^ 

X A 

22.5 

1.23 

.49 

.81 


it 


<z 

4.4 

.04 

.20 

.35 


k 

13.1 

.95 

.55 

.78 

134 

X 

134 

{/ 

3.0 

.05 

.26 

.36 

it 

A 

25.0 

1.67 

.54 

.87 

it 


5.6 

.08 

.26 

.40 

CO 

X 

CO 


14.4 

1.24 

.60 

.84 

134 

X 

134 

A, 

5.3 

.11 

.31 

.44 

ii 

ff 

33.6 

2.62 

.59 

.98 

it 

s 

9.8 

.19 

.31 

.51 

3^X334 

24.8 

2.87 

.70 

1.01 


X 


A 

6.2 

.18 

.36 

.51 

ii 

% 

39.8 

4.33 

.69 

1.10 

tt 

% 

11.7 

.31 

.35 

.57 

4 X4 

% 

28.6 

4.36 

.81 

1.14 

2 

X 

2 

Yo 

7.1 

.27 

.40 

.57 

ti 

k 

54.4 

7.67 

.80 

1.27 


i* 


% 

13.6 

.50 

.39 

.64 

5 X5 

TB 

41.8 

10.02 

1.00 

1.41 

2 34 

X 

234 


10.6 

.50 

.45 

.65 

ii 

i 

90.0 

19.64 

.98 

1.61 

ii 

7 

17.8 

.79 

.44 

.72 

6 X6 

x 7 e 

50.6 

17.68 

1.19 

1.66 

234 

X 

234 

s 

11.9 

.70 

.50 

.72 

ii 

i 

110.0 

35.46 

1.17 

1.86 


Angles w ith unequal legs. Fig. 3. 


Dimensions, B 
and A, in ins. 

00 

.2 

Cfl 

91 

0> 

a 

.2 

jS 

CD 

£ 

.2 

U 

0) 

o* 

£ 

Moment of 
Inertia. 

Least Rad of Gy¬ 
ration, in ins. 

Dist in ins 
from base 
to neutral 
axis. 

Dimensions, B 
and A, in ins. 

00 

00 

00 

Cj 

a 

.2 

S 

E- 

Wt per yd, in lbs. 

Moment of 
Inertia. 

Least Rad of Gy¬ 

ration, in ins. 

Dist in ins 
from base 
to neutral 
axis. 

-*3 

ol* 

About 

W Z. 

a 

s 

About 

X Y. 

About 

W Z. 

a 

* 

3 X2 

34 

11.9 

1.09 

.39 

.46 

.99 

.49 

5 

X3 


54.4 

13.15 

3.51 

.69 

1.84 

.84 

ti 

A 

22.5 

1.92 

.67 

.46 

1.08 

.58 

5 

x&A 

% 

30.5 

7.78 

3.23 

.80 

1.61 

.86 

3 x$A 

T5 

16.2 

1.42 

.90 

.54 

.93 

.68 


“ 

k 

58.1 

13.92 

5.55 

.79 

1.75 

1.00 

tt 

U 

25.0 

2.08 

1.30 

.54 

1.00 

.75 

5 

X4 

% 

32.3 

8.14 

4.66 

.87 

1.53 

1.03 

3^X2^ 

A 

17.8 

2.19 

.94 

.56 

1.14 

.64 


ti 

i 

80.0 18.17 

10.17 

.86 

1.75 

1.25 

ii 

U 

27.5 

3.24 

1.36 

.56 

1.20 

.70 

514X314 

% 

32.3 

10.12 

3.27 

.81 

1.82 

.82 

3^X3 

tt 

21.2 

2.53 

1.72 

.64 

1.07 

.82 


it 

a 

52.3 

15.73 

4.96 

.80 

1.91 

.91 

it 

k 

36.7 

4.11 

2.81 

.64 

1.17 

.92 

6 

X3*4 

re 

39.6 

14.76 

3.81 

.82 

2.06 

.81 

4 X3 

A 

24.8 

3.96 

1.92 

.67 

1.28 

.78 


«t 

i 

85.0 29.24 

7.21 

.81 

2.26, 

1.01 

ti 

/& 

39.8 

6.03 

2.87 

.65 

1.37 

.87 

6 

X4 

ie 

41.8 15.46 

5.60 

.92 

1.96 

.96 

4 X3K 

A 

26.7 

4.17 

2.99 

.74 

1.20 

.95 


tt 

i 

90.0 30.75 

10.75 

.91 

2.17 

1.17 

ii 

% 

43.0 

6.37 

4.52 

.73 

1.29 

1.04 

6^X4 

x 7 e 

44.0 19.29 

5.72 

.94 

2.18 

.93 

4^X3 

% 

26.7 

5.50 

1.98 

.69 

1.49 

.74 


it 

i 

95.0 38.66 

11.00 

.93 

2.38 

1.13 

ti 

% 

43.0 

8.44 

2.98 

.68 

1.58 

.83 

7 

X&A 

% 

61.7 30.25 

5.28 

.85 

2.57 

.82 

5 X3 

% 

28.6 

7.37 

2.04 

.70 

1.70 

.70 


ii 

i 

95.0|45.37 

7.53 

.84 

2.71 

.96 


T iron with equal legs. Fig 4. 


Dimen¬ 
sions, ins 

A and B. 

Thicknesses, ins. 

Wt per yd, 
lbs. 

Moment of 
Inertia about 

X Y. 

Least Rad 
of Gyr. 

Dist d, from 
base to neutral 
axis, ins. 

Stem, B. 

Base, A. 

At 

root. 

At 

edge. 

At 

root. 

At 

edge. 

1 xi 

34 

35 

34 


3. 

.03 

.26 

.30 

i % x 134 

ii 

te 

ii 

1 % 

4.5 

.07 

.27 

.37 

i>4xi34 

tt 

ii 

ti 

it 

6. 

.13 

.32 

.45 

'Axi-% 

ii 

ti 

it 

tt 

7.1 

.21 

.37 

.50 

2 X 2 


34 


34 

10.5 

.38 

.43 

.60 

2*4 X 2*4 

k 

ti 

k 

it 

11.75 

.52 

.50 

.61 


t 5 s 

ii 

Te 

tt 

12. 

.54 

.47 

.65 

234 x 234 

tt 

tt 

tt 

tt 

17.52 

.97 

.53 

.75 


tt 

tt 

tt 

tt 

19.5 

1.12 

.55 

.75 

3 X 3 

A 

xe 

t% 

ti 

26.0 

2.10 

.62 

.90 

314 x 334 

%i 

ii 

34 

7 

1 e 

31.0 

3.47 

.74 

1.00 

4 X 4 

it 

tt 


ii 

36.5 

5.26 

.84 

1.14 










































































































ANGLES AND T IRON, 


527 


1 iron witli unequal leg's. (Using the lettering of Fig. 4.) 


Dimensions 

Thicknesses, ins. 

•V3 

-*3 

O o 


- ^ 

Eis . 

in ins. 

Stem B. 

Base A. 

gi 

'S • 

a 83 

03 * 

« K 

P P K 

C *~ 


B. 

At 

At 

At 

1 At 

Q-H 

S~X 

O fc. 

V. w 

03 

<D O 

2 0.2 
t! g 

A. 

root. 

edge. 

root. 

edge. 

ts 

S c 
►—< 


c s s 

2 

9 

is 

R 

IS 

fs 


14 

5.88 

.01 

.13 

.17 


i 

9 

35 

/4 

44 

44 

7. 

.05 

.26 

.27 


^A 

5 

IS 

U 

TSB 

(4 

8.75 

.16 

.43 

.43 


A 

4i 

TB 

14 

(4 

6.5 

.01 

.12 

.18 


Vyi 


/i 

4» 

44 

9.1 

.10 

.33 

.32 


J /4 


% 

35 

i 5 b 

18.75 

.56 

.55 

.66 


2 


a 

44 

44 

21. 

.83 

.55 

.75 

3 

Si 

M 

k 

14 

J4 

11.2 

.19 

.41 

.37 



k 

/2 

j 7 b 

% 

23.8 

1.38 

.63 

.82 


A ‘A 

k 

IB 

k 

15 

28.25 

3.12 

.61 

1.10 

4 

2 

A 

% 

t 5 b 

IB 

20.4 

.68 

.58 

.54 


3 

k 

A 

44 

4. 

25.25 

2.09 

.82 

.84 




% 

IB 

% 

25.9 

1.94 

.86 

.77 


’ 6 Vi 

% 

i 9 b 

f4 

IS 

41.8 

4.65 

.88 

1.09 

^A 



1b 


• 4 

44.5 

5.27 

.91 

1.16 

0 

2 H 

u 

IB 

% 

% 

30.7 

1.61 

.72 

.67 



•* 

44 

7s 


33. 

1.63 

.70 

.64 


&A 

% 

H 

k 

k 

48.44 

5.37 

1.04 

1.05 


4 

hi 

k 

44 

44 

44.1 

6.24 

1.09 

1.08 









































528 


BEAMS WITH THIN WEBS. 


RESISTANCE OF OPEN BEAMS. 

Roams, Ac, in which all of the longitudinal resistance is 
regarded as being exerted by the flanges. 

Art. 1. On page 480 we saw that the strength of “ closed ” beams, or beams of 
solid cross-section, is proportional to the squares of their depths; because the resist- 
ing moment of each fibre is proportional to the square of its distance from the neu- 
tral axis. 

Art. 2. But, as in an open beam, or truss, ixea Fig 6 (or in a closed beam with 

a thin web ir, Fig 22, p 537) we may place as 
many of the fibres as possible in the chords , ix 
and e a Fig 6 (a and b Fig 22) or as far as possi¬ 
ble from the neutral axis « Fig G, so that they 
may exert their maximum resistance. The 
chords are thus made to bear most of the hori¬ 
zontal or longitudinal strain. For conveni¬ 
ence’s sake we assume that they bear it all. 
For the resistance of the web, see Art. 5, p 529. 

It is plain that the breaking moment of the 
load is the same as in the closed beam Fig 2, p 
479, = (Fig 6) the load X its leverage ( ^ ea or 
ix) about the neutral axis ». But the resisting 
moment of the beam now consists of 

/longitudinal tensile its leverage in*\ /longitudinal compres- its leverage \ 

I resistance of the X about the neu-) + I sive resistance of the X ne*about the \ 

' upper chord i x tral axis n J \ lower chord e a neutral axis n) 



Ii=l 


_ sum of the longitudinal re- 

sistances of the two chords ^ * ia ^ depth - of beam 


But we may express this more simply by regarding the neutral axis or fulcrum as 
being at the inner end i or e of one of the chords ; and by writing 


Moment of resist¬ 
ance of beam 


_ longitudinal resistance of whole depth 
either one chord ix or ea ^ ie* of beam 


‘ o onf p the beam sustains its load, the longitudinal resistances of the two chords 
are equal, although the longitudinal strengths of one or of both mav be much trreater 
than this resistance. If the strength of either chord becomes less than 


Resistance = of rupture 

depth* of beam 


then that chord fails, and the beam gives way. 
Hence the 


maximum or ulti¬ 
mate moment of 
resistance of beam 


ultimate longitudinal 
strength of the weaker X . ° p depth* 
chord ix or ea i e of beam 


In ‘ closed beams,” as explained on p 485, the longitudinal tensile and com¬ 
pressive resistances exerted by the fibres against rupture, are aided to an imnortant 
*,!« mutual adhesion of the m„i which resist's their slidiug uZ each 
other, for without such sliding, rupture cannot occur. But in “open beams.” 6ucli as 
we are now treating of, the thickness of the flanges or chords is so slight compared 
with the depth of the beam or truss, that the sliding between their fibrw teSSS* 
insignificant. The resistance due to the adhesion of the fibres to each other is 

‘ch»ncor“Sp“ S i“«JI?ns *" 08 *“'* l°agi!udi„al 

let the concentrated load E at its outer end be 1 ton. This load tends to milYthe 
beam into the dotted position by stretching or tearing apart the fibresof e 
upper chord at i. Now with how great a moment of rupture dcles it lend to o 
this, and how strong must the fibres of the chord be at /in order that their mo 
ment of resistance mav oppose it safely ? We shall here ipa V p tPp „fV ln K mo 

itself out of consideration. When requiredto t^ nch ded see ?a s r G p 48*2 ^ 

Regard the lines i e and e a as the t wo arms of a bent lever resting on its f„l 
This lever is plainly acted upon and balanced by Uvo enual momenta 
one &t each end a and x\ namely at a v,he moment of runturp r*f rv 1 <. * 

(1 ton X 6 ft leverage a e) = 6 ft tons ; and at™the resistinemoLJn t0 

equal to the lior pull or strain on the fibres at the chord i y I - ft m e Kam ’ 
But we do not yet know what amount of Imrpullby thel brS at f is re'qffi to 

-,!.L he dei>ths aud halfde P ths arc me^r^rom^b7^7 s of gravity ot cross section of^he 
























BEAMS WITH THIN WEBS. 


529 


balance the moment of the load. It is however very easily found by merely divid¬ 
ing the 6 ft tons moment of the load by the 1.5 ft leverage of the fibres, that is, by 
the depth of the beam. Thus we get (6-f- 1.5) ~ 4 tons pull at t; and we then liave 
the 6 X 1 = 6 ft tons moment of the load, balanced by the 1.5 X 4 = 6 ft tons mo¬ 
ment of the fibres. Therefore in order just to balance the moment of the load, 
the chord at i must be strong enough to bear a hor pull of 4 tons; or for a safety 
of 3, 4 or 6, Ac, strong enough to bear a pull of 12, 16, or 24, Ac, tons. The web 
members of course carry this 4 ton hor pulling strain from the upper chord to the 
lower one upon which it acts as an equal hor compressing one. 

In shape of a formula the above stands thus. 

Hor strain at any Moment of load Load X its leverage 
point in a hor flange _ at that point _ at that point 
of an open cantilever Dept h of beam ~ Depth of beam 

Hence if we know the size and of course the ultimate longitudinal tensile and 
compressive strength of the flange or chord, we have by transposition the ulti¬ 
mate or breaking load of the hor open beam, thus, 

Breaking- load at any _ Ultimate strength of flange X De pth of beam. 
point of a hor open cantilever Leverage of load at that point. 

And for a safety of 3, 4 or 6, Ac, we have 

Safe load = ^ % or &c ’ the uIt strength of flange X Depth of beam. 

Leverage of load at that point. 

Art. 4. Also in a hor open beam or truss supported at both 
ends, after having found the moment of the load at any point (by “moments,” p 
479, &c ) the strain on the beam as also its load in lbs or tons are found in the same 
way or by the same formulas.* 

Rein. 1. The longitudinal strains on the flanges of hor closed beams 
with thin webs such as common rolled I beams, as well as their loads, are also fre¬ 
quently computed in this same ready way, instead of the more troublesome one, p 488. 
The webs are left entirely out of consideration as regards the hor strains. Although 
not strictly correct, it is sufficiently so for ordinary practice, and is safe. With these 
assumptions the dimensions or sectional areas of the top and bottom flanges are 
proportioned to the safe unit strains of the material. Thus Hodgkinson having 
I found that the ultimate compressive strength of cast-iron averaged about 6 times 
as great as its tensile one, gave his upper flange only one-sixth the area of the 
* lower one, in order that both should be equally strong. In wrought-iron the ten¬ 
sile strength is somewhat the greatest, which would lead to making the lower 
flange the smallest, but here this consideration is outweighed by the practical 
i ones of greater ease of manufacture and of handling or placing which require 
i equal flanges. 

Rem. 2. If the flanges are not horizontal, although the beam or 
truss itself may be so, the longitudinal strains on the flanges will be increased; 
and the transverse or shearing strains on the webs will also be changed as stated 
t. in Art 6. If the beams are inclinefl, modifications arise which we shall 
| not treat of. Strangely, most of our standard authorities on bridge building do 
( not even allude to them. 

Item. 3. The principle of the bent lever in open beams explains why the 
'strength of a truss is as its depth, (the length of its vert lever-arm) 
instead of as the square of its depth as in closed beams. The strength however is 
inversely as the length in both kinds. 

Art. 5. The web members of an open beam or common truss like Figs 
i 10 and 11, p 558, uniformly loaded, carry the vert or shearing forces of the load 
and beam from the center each way, up and down alternately from one chord to 
| the other, until finally the end ones deposit it as load on the supports or abut¬ 
ments. For each member receives and carries its share of the shearing force in 
the shape of an end load, thus changing the shearing tendency into an alternately 
milling and compressing one according as the alternate members are ties or struts. 
In doing this any web member that is oblique is (on account of its obliquity) 
strained to an extent that exceeds its load in the same proportion that the oblique 
length of the member exceeds the length it would have had if it had been vert, as 
explained in Art 11, p 557, Ac. This excess of strain over the load on the obliques 









530 


BEAMS WITH THIN WEBS. 


exhibits itself at their ends as lior pull along one chord, and hor compression along 
the other; and these hor strains on the chords are the same as those found by mo¬ 
ments. Thus it is seen in Figs p 558 that the hor strain at the center of each chord 
(as there found by tracing up the diff loads or shearing forces in their journey along 
the obliques) is set down at 16 tons. Now the whole distributed load on one truss 
of 64 ft span, and 16 ft depth, is 32 tons; and by Case 10, p 483, we find that at the 
center the moment of this load is 16 X 16 = 256 tons; and this divided by the depth 
of our open truss or 16 ft gives 16 tons for the hor strain at center as before; and 
so at other points. The oblique web members are plainly the only ones that can 
convey their loads laterally , that is in directions tending toward or from the abut¬ 
ments. Vertical members merely convey their loads vert up or down from one 
chord to the other, at which last they transfer them to oblique members which 
can convey them laterally. If both a pull and a push act at once in opposite 
directions on a web member, their ditt is the actual strain. 


Rem. As a mat ter of economy in small spans it is often better not to 
proportion the sizes of the individual members to the strains they have to bear; 
but to give to the flanges throughout their entire length the same dimensions as 
are required at their most strained part, namely, at the center; and to make all 
the web members as strong as the most strained or end ones. 1 his avoids the 
extra trouble and expense of getting out and fitting together many pieces of 
various sizes. 

Art. 6. Oblique or curved flanges. We have hitherto supposed 
the beams and their flanges to be horizontal; but a beam may be hor, and yet 
have one or both of its flanges oblique or curved as 
at A and B. In such cases the longitudinal strains 
along the flanges become greater; and the vert or 
shearing strains across the web in most cases less. 

See Rem at end of Art. It is plain that such flanges 
must as it were intercept to some extent (depending 
on their inclination) the vert force at any point, 
and convert it into an oblique one along the flanges, 
somewhat as the oblique web members of an open 
beam do. 

To find these new strains at any point o, 

Figs A, B, of either an upper or lower oblique or 
curved flange, first ascertain by “Moments,” the 
hor strain at that point for a beam with the depth 
o e; and by “Shearing,” the shear also. 

Then from that point o draw a lio'r line A equal by 
scale to the hor strain; and from its end draw v 



tU 111 V llv/ 1 Ollulll , ** UU 

vert and ending either at the flange (produced if necessary) if straight as in A; 
or at a tangent l from o if the flange is curved as at B. Then will l in either fig 
give by the same scale the longitudinal strain along the flange at o; and A and v 
are the components of that strain. As a formula, the Rule reads thus, o being the 
angle formed by A and l at o. 

Wtniiit . „ mom of rup 

oblique flange = hor strain cosine of o = - 


cosine of o. 


I 





depth of beam 

Here v shows by the scale how much of the vert or shearing force has been eo 
verted into a longitudinal one; and if it be taken from the total shearing force 
before found, the remainder will show how much of said force still operates on tht 
iceb at o. For exceptions, see Rem. The foregoing applies also to oblique flanks 
of open hor beams. 

In the lior triangular flanged beam D with a concentrated load a 

its free end, draw a o vert and equal by scale to the load, 
and draw o c hor. Now here the whole load rests upon the 
upper end a of the oblique flange a n , which therefore sus¬ 
tains all of it as an end load, which it deposits as vert press¬ 
ure at n, and thus entirely prevents it from exerting any 
shearing force whatever upon any part of the beam. The 
shaded web is therefore of no use here. The line a c meas¬ 
ures the strain along the oblique flange; a o the vert pressure 
at n; and o c the lior pull of the load all along the upper 
flange a e. Also a o and o c are the components of a c. 

So also in Fig E, with a concentrated load l suspended by a string from 
The string carries the load up to the two oblique flanges c a, cb, which conver 
its shearing tendency into two oblique pulls along themselves. A 







































































BEAMS WITH THIN WEBS. 


531 


the abutments or supports these pulls along c a and c b become converted into ver¬ 
tical pressures, together equal to the load l; and into hor pressures compressing ab. 

Here also the shaded web is unnecessary; as would 
likewise be the case if the load were transferred to e, 
and a single vert post (shown by the dark line) provided 
to carry it dow n to c, as the string before carried it up 
to c. If there is no such post the web acts, and the 
strain on either oblique flange is found as for A and B. 
But it is only in a few similar eases that the oblique 
flange entirely supplants the continuous web. 

Humber gives for finding' the strain at 
any point of an oblique or curved web as follows. 
First find the shear as before as for a horizontal 

lange. Then 

Jf the coin pressed flange is inclined down to the nearest support, or 
Jf the stretched flange is inclined down from “ “ 

;ake the diff between the vert component v and the shearing force. But 
Jf the com p ressed flange is inclined down from the nearest support, or 
If the stretched flange is inclined down to “ “ “ f 

;ake the sum of the vert component v and the shearing force. 



Rem. Hence in these last two cases (which do not include any of our above 
igs) the vertical force on the web is increased. 

As Humber remarks, in girders or beams with curved or oblique flanges the 
greatest strain in the web is not always where the greatest shearing strain is 
produced. 




38 





532 


VERTICAL STRAINS IN BEAMS, ETC, 


VERTICAL STRAINS IN BEAMS, ETC. 


VERTICAL OR SHEARING STRAINS IN BEAMS, ETC. 
Art. 1. When a loaded beam, a v. Fig 1, rests upon two supports, Z and r, the 


weight of the beam and load between the supports, and the upward reactions ot the 

two supports, tend not only to bend the beam, as in Fig 
3 , p 550, but also to cut ot shear it across vertically, as 
in Fig 1. 

In practice, beams rarely fail by shearing. 

and then only when heavily loaded close to their points j 
of support, as in Fig 1. 

But the vert forces which tend to pro¬ 
duce shearing exist in all beams and 

In the latter, they cause all the strain on the verts and a part of that 



pOOQQof 

a 


mm, 


mv/. 




trusses 

in the obliques (whether web members or flanges); and in beams with thin webs 
and considerable area of flange, such as box and plate-girders, ps 537, Ac, it is usual 
and sufficiently (although not strictly) correct, to assume, for convenience, that al 
of the vert strains are borne by the web , while the flanges are regarded as resisting j 
only the longitudinal strains. The following instructions are therefore given foi j 
finding the amount of vert or shearing strain in the different parts of a beam nndei i 
different conditions of loading. For the sake of simplicity, we neglect the wt of j 
the beam itself, unless otherwise stated. 

Art. 2. Imagine tJie beam. av,Fi" 2 (supported at both ends, ant 
loaded at its center with 3 tons), to be divided into a number of slices 

?t s t, Ac, by the vert planes whose edges ar< 
shown in the fig. If we take any two adjoin 
ing slices, as m and n, to the right of the load 
it is plain that we may regard the left-hum 
slice, ?)/, as being pressed downward by tha ) 
portion (1*4 tons) of the load which goes b ] 
the right-hand support,? - ,while its neighbor j 
ing slice, n, on the right, is upheld by th j 
equal upward reaction of the right-hand sup 
port. r. There is, therefore, a vert strain \ 
equal to 1}4 tons, or to one of these forces ) 
tending to shear the beam across on the ver a 
plane separating the two slices; and this ten ( 
deucy must Iks resisted by the cohesive fore t 
of the beam in that vert plane.* Since the downward pressure exerted upon th t 
right-hand support undergoes (in this case) no increase or diminution between tli 
load and the support (there being no intermediate load or supporti, and since th t 
upward reaction of r is also exerted, unchanged, at every point between r and c, i 
follows that tlie shearing strain is equal at all those points. jVnd since the load i 
at the center of the beam, the upward reaction of the left abutment, l, is equal to tin 
of r, and there must, therefore, he an equal shearing strain of 1% tons at each poii, 
in the Ze/?-hand half of the beam. In other words, a load, conceit truted a 
tbe middle of a beam supported at both ends, exerts a nnifori 
shearing strain, equal to half of said load, throughout the beam. 

Art. Ji. But while, to the right of the load, each slice tended to slide downwar 
past its neighbor on the right; the reverse is the case to the left of the load; eac 
slice there tending to slide downward past its neighbor on the left. In other word: 
to the right of the load the vert downward strain on each section comes from th 
left, and vice versa. 

Art. 4. At Die vert section Immediately under or over a conceit 
tented center load there is, strictly speaking, no shearing tendency betwee « 
the two slices to the right and left of that section, because they evidently have n 
tendency to slide past each other. But there is, in the two slices, a combined crust 
ing strain equal to the entire load; because each of them sustains half the load, an 
is pressed upward by the equal reaction of the abut. If we suppose a v to be a trus 


* So long as the joint between m and n remains intact, n is of course also pressed downward 1 
the half load, and m is also upheld by the support. But this does not affect the shearing strain ; 
the joint, because, in order that n may receive the downward pres of tbe half load, said pres must I 
transmitted to it from m throuah the joint in aucstion : and so with tho nm.rH 
from n to m. 


: through the joint in question ; and so with the upward reaction transmittt 
It is the transmission of the original action and reaction through any given ioint th 
causes the shearing strain in it. 1 6 J 




























533 


VERTICAL STRAINS IN BEAMS, ETC. 

and the vert line between 7c and in to represent a vert post, then said post will have 
to hear the combined crushing strain, which, in the beam, comes upon slices k and 
in (= the entire load of 3 tons), and its pin at c in the lower chord will sustain a vert 
shearing strain of 3 tons in addition to the hor shearing strains from the chord. 

Art. 5. If, in Fig 2, we make al and vr each equal, by 6cale, to the upward 
reaction of its abutment, or (in this case) equal to half the load; and draw Ir hor; 
then vert lines (of uniform length in this case) drawn between av and Ir will give 
the shearing strain at each point in the beam, except of course at the cen, c, as ex¬ 
plained in Art 4. 

Art. 6. Figs Sand 4 show the application of the foregoing’ to 
common forms of trusses. In each tig, a load of 3 tons is supposed to be 
sustained entirely by only one truss of the span. The members sustaining tension 
, are shown by light lines; and those under compression , by heavy lines. It is plain 


C 



that each vert member, in either fig, is strained 1 ]4. tons.* Each oblique sustains a 
total strain greater than 1^ tons; but the vert comp of each is only 1^ tons. Also, 
v the doivnuHird strain on each web member to the right of the load comes from the 
r. left, and vice versa. We see, also, that while the two central obliques have no ten- 
i deucy to slide past each other, their combined vert strain (compression in Fig 3; ten¬ 
sion in Fig 4) is equal to the entire load; and the pin at c in either fig sustains a 
vert shear equal to the entire load. And the cen vert rod in Fig 3 sustains a pull 
equal to the entire load, or 3 tons. Drawing a? and v reach equal to the upward 
reaction of its abut; Ir hor; and the vert dotted lines; the latter give the vert 
strains (uniform in this case) at the several panel points, except of course at the cen, 
is just explained. 

The arrows pointing downward represent the downward pressures caused by the 
bad; while those pointing upward represent the upward reactions of the abuts. In 
Fig 3 the doivnward pres at each section is applied at its foot, and the upward pres 
it its head; and the vert members are therefore in tension. In Fig 4 the reverse 
of all this is the case. The directions of the arrows should be carefully noted, as 
they have an important bearing upon what follows. The lengths of the arrows indi¬ 
rate the amounts of the forces which they represent; and their positions show 
whether those forces are applied at the upper or lower chord, and whether the force 
comes to the section from the right or from the left. 

Thus, in Fig 3, the arrow pointing downward, immediately under the load, and at 
the bottom of the diagram, shows that the force represented by it is applied imme- 
i Jiately at the joint (coming neither from the right nor from the left) and that said 
wjoint is in the haver chord. 

i‘ This force is equal to the entire load, or 3 tons. The two equal arrows imme- 
idiately to the right and left of the cen line, and at the top of the diagram, are made 
mch half as long as the central arrow just referred to, in order to show that the 
rf force represented by each is half as great as that represented by the central arrow. 
:lThese two arrows represent the upward reactions of the two abuts, coming from 
1 the right and left, respectively, and meeting at c in the upper chord ; 

Rem. A single concentrated load produces its greatest 
shearing strain when placed at one end of the span, immediately over the 
I'ede-e of an abut; at \yhich point the shear is then equal to the load; but there is 
ithen no shearing strain in any other part of the beam. As the load moves along the 
qbeam, the shear in front of it increases uniformly, and that behind it decreases uni- 
ijforml’y until the load reaches the other end Of the span. At whatever point 
the load may he, the shear at any instant is uniform 
^throughout either one of the two segments into which the load 
[divides the span. See Figs 5 and 6. 

* Except that the center vertical member, in Fig 3, belongs, as it were, to both 
\ halves of the truss, and therefore performs the same duty as two side verts, by 
, supporting the entire load of 3 tons = twice the half load, as explained in 
Art! 4. 



























534 


VERTICAL STRAINS IN BEAMS, ETC, 


Art. 7. In Figs 5 and 6, the concentrated load Is not at the 

cen of the beam. Therefore the upward reactions of the abuts are unequal, 
as are also the shearing strains in the two portions of the length of the beam. 

Hor diet from cen of 
1 he reac- load X grav of load to th®, 
lion of = other abut 

either abut - 






r 




1 

t t 

i t 

1 t 

iti 



J 

Fig.o 

1 

L - 




_i_4- 





; in 



h 

i 

i 

it 

t 1 

t l\ | ijt I'ft 


Fi 

g.6 



3 


Ill 


_jU— ii 


w 


Fig. 7 


3 


.3 


Span. 

Art. 8. In Fig 7 we have two concentrated 
loads of 3 tons each, and each placed at a dist 
from an abut equal to one-third of the span. 
Here it might be supposed that the shear at 
any point might be found by simply adding 
together its shear by Fig 5 and that by Fig 6; j 
but this is not the case, except for those points 
between an abut and the load nearest to it. At 
such points the vert forces in Fig 5 and those 
in Fig 6 act in the same directions, and thus 
assist each other when combined as in Fig 7 ; 
while at the sections between Vie two loads, 
they act in opposite directions, and consequently 
counteract each other. 

Thus, if we compare section c in Figs 5 and 
6, we will see that in Fig 5 slice m tends to slide 
downward past k; while in Fig 6 the reverse 
is the case; so that in Fig 7, which may be: 
regarded as a combination of Figs 5 and 6, these 
two equal ami opposite shearing 
tendencies counteract each other, 
and there is consequently no shear at c. 

Art. 9. Similarly, in Fig 10, the shear in 
W ]) is equal to the difference (1 ton) between 
those (l ton and 2 tons, respectively) in W D in 
Figs 8 and 9; while that in a W is equal to the 
sum (5 tons) of those (1 and 4) in a W in Figs 8 
and 9; and that in Du is equal to the sum (4) 
of those (2 and 2) in Du in Figs 8 and 9. 

A rt. 10. The following general rules 
are illustrated by the foregoing: 

R tile 1. The shearing or vert strain at 
any point of any beam, fixed or supported at 
one or at both ends, and loaded in any manner, 
is equal to the diff between the upward vert 
reaction of either abutment, and any load or 
part load on the beam between that abut and 
said point. To find the upward reaction of 
either abut, see Art 7, above. 

Rule 2. Let all that part of the load tt 
the right of the given section be called R; anc. j 
that to the left of it, L.. Then the shearing 
strain at that section will be equal to the diff | 
between that portion of R that goes to the left- 
hand support, and that portion of L. that goes 
to the n^/if-liand support. 

Rem. 1. In applying either of these rules 
to a section immediately under or 
over a concentrated load, as at W, 
Fig 9, or concentrated portion of a load, as atj 
W, Fig 10, we must, theoretically , consider the! 
section as being the dividing line between the 
two portions of said concentrated load or part 
load which goto the two abuts respectively; 
and must regard said portions as forming parts 
of the loads on the two portions into which 
the given section divides the beam. 

For instance, in Fig 10, at any point, P. be¬ 
tween a and W, Rule I gives shearing strain = 
upward reaction of abut at a — load between a and P = 5 — 0 = 5 tons ; or, = load 
between u and P — upward reaction at u = 9 — 4 = 5 tons; and Rule 2 give’s shear -\ 


a 


T~ 

D 

T 


V 

1 

j 

L JL 

A 


7 



Fig. 6 

J L_J_ 




a 


p \V 

i l 


) 



j - 


1 

jj 


i 

LL 


V 

V 




i 


-I 


Fig.lO 





































































. 


VERTICAL STRAINS IN BEAMS, ETC. 


535 


in? strain — portion of R going to a — portion of L going to v--= 5 — 0 = 5 tons. 
1 nt, at IV, the wt of 6 tons, resting there, divides; two-thirds of it, or 4 tons* 
a, ( ld one-third, or 2 tons, to v. Therefore “ L ” (or load to the left of 
nr \v in Q tons ; a |? d “R” (or load to the right of W) ='the remaining 2 tons 
q „ i V u l° nS ’ ~ 5 to ? s ' Here * Rule 1 gives shear at W = upward reaction 
at a —load between a and W = 5 — 4 = 1 ton ; or, shear at W = load between v 

u re rd react,on at tt = 5 — 4 = 1 ton ; and Rule 2 gives shear at W = 
one-third of D, or one ton of R going to the left. abut, at a. b 

t ( ,^, 1 n t i ,, , , ti >, aC ! iCe w it safer t° ,,e « lect such refinement, and 

o?. o 1 e * Ct °? V y, as having a shear equal to the greater of the two shears 

or 5 tons 8 '** 6 ° f ° r ’ m tFus case ’ e 0 ual t0 tbe shear at any point between a and W, 
The following are applications of Rules 1 and 2: 

Rem. 2. The shearing force at any cross section of a cantilever (a project- 
in^ beam fixed at one end and free at the other), no matter how the load is disposed 

is equal to the wt of that part of the beam and its load which is between said section 
and the iree end. 

.. .liVL.V. 1 . ". ‘i"",I,*’?. 1 ‘ supported at both ends, is 

uniformly loaded throughout, the shearing strain is greatest at the 
abuts: at each of which it is equal to half the load. From 
each support it diminishes uniformly to the center, where it 
1 is zero. Therefore, if we make a l and v r each equal, by scale a 
to half the load, and from l and r draw straight lines, l c and 
r c, to the center, e, of the span ; then a vert line, as o s, drawn 
from any point, o, in the beam, to either line, as l c, gives the 
^hearing force at said point, o. A ITig.ll 

Art. 12. When a beam, a v. Fig 12, supported at 
both ends, is nniformly loaded from one of its supports, as r, 
part way across, say to n. as when a train, as long as the span, or longer, 
mines part way upon a bridge, the greatest 
shear is at the abut, r, and is equal to the 
jortion of the load borne by that abut. From 
hat point it decreases uniformly to zero at a 
sero point, z, w hich is always within the load 

tself. To find the dist, n z, of the * N 'U { 

Eero point, z, from n, when the load . Vi 

jxtends from v any given dist, as vn, toward ITig.l^J 


]X C 

H 


x 


V 


7 


NP 


la 




Twice . Length, vn, of bridge . . Said covered 
the span* occupied by load ** length, vn 


n z, 


or nz — 


XI 

vn 2 
2 a v' 


The zero point of shear is also the section of greatest moment of rupture. 

From the zero point, the shear again increases uniformly to the end,n, of the 
oad. and at the same rate of increase as from z to v. At n, and from n to the 
ither abut, a, the shearing strain is equal to the portion of the load sustained 
>y said abut, a. 

Art. 13. Fig 13 shows the shearing strains which take place 




'v.„ 

: ***•—.* 


7 6 5 4 3 2 1 


O 

P 


lTig.13 




--- 


nccessively at a given point, 6, in a bridge, 0-8, while a train, as long 
s the bridge, comes upon it, passes across it, and leaves it, all in the same direction. 


























536 


VERTICAL STRAINS IN BEAMS, ETC. 


It also shows the sticoes^i ve shears at eaeli abut during the passage of 
the train. The train is supposed to move in the direction of the arrow, or from 
right to left, and, for convenience, is supposed to be of uniform wt per ft run through¬ 
out. The several shearing strains are found by Art 12. The vertical distances, 
8-e and 8 -g, are made each equal, by scale, to half the wt of the train, which is’ 
equal to the shear at 0 or at 8 when the head of the train is at 8; and when, con¬ 
sequently, the train just covers the span, as in Fig 11. (It must be borne in mind 
that we neglect the wt of the bridge itself.) 

The vert lines, 6 -d, 11-/, Ac, show, by the same scale, the amounts of shearing 
strain at 8 when the head of the train comes, respectively, to 6,11, Ac. Similar vert 
lines, drawn from points in the hor line, 16-0, to the lower curve, 16-^-0 would show 
the corresponding vert strains atO; and the heavy vert lines, 4-?n, d-d 8 -h 10-t 
Ac, from 16-0 to the heavy curve, 16-14-d-0, show the successive shearing strains at 
the point 6, as the head of the train reaches 4, 6, 8, 10, Ac. 

It will be noticed that at each abut the shear, just before the train touches 
the bridge at 0, is zero; and that it increases until the train just covers the bridge 
vhen it is equal to half the wt of the train, as in Fig 11. It then decreases, reaching 
zero again when the rear of the train leaves the bridge at 8* Hut at anv interme* 
diate point, as 6, the shear increases from zero until the head of the train comes to 
sam point. During this time the shears at the point are the same as those at the 
ab . llt 8 beyond it (see n, Fig 12), and consist solely of shearing strain passing through 
it to that abut. But now the point, 6, begins to pass strains to both abuts, and con- 
tinues to do so until the rear of the train has passed it. These opposing strains 
partly neutralize each other (see Art 8); and the resultant, or remaining, shear at 6 
diminishes until the head of the train reaches such a point, z, that 6 becomes the 
zero point. It then again increases until the head of the train reaches 14. The rear 
J ?, now at 6 - Fi ;om now on all the shear going to 0 (and no other) passes 
through b. Consequently, the shear at 6 is, from now on, the same as that at 0, and, 
like it, decreases, and becomes zero as the rear of the train leaves the bridge at 8. 

I he greatest shear that can occur at any given point is when the 

{STwEtoe rtol e.’P‘ £ ,£££ ,0 ,hat po ‘“- “ is sreater P** 

* But, as indicated by the two curves, O-e-16 and 0-O-16, the shearing strains at 

Ttrat a SrriA and f°s ( }° n0t increase an<1 decrease uniformly or equally 
Thus. at the left abutment 8 (see upper curve O-e-16), the increase of shear as the 

train comes upon the bridge, is at first slow, and afterward more rapid as the head 

bf at flrstTl approache . s 8 ' s » n ‘ilarly, the decrease at 8, as the train leaves the bridge 
is at first slow, and afterward more rapid as the rear of the train approaches 8 At 
the ? ight abutment O, the exact reverse of this is the case. The sheaiino- strains at 
the two abutments are not equal at any given moment except when the train cover* 
the entire bridge, and when there is no train on the bridge; 




RIVETED GIRDERS. 


I>oT 


RIVETED GIRDERS. 

Art. 1. Plate girders. Figs 21,22, and 23; and box-girders. Figs 24 
and 25; with hor flanges, and supported at both ends. In these, it is usual, and suf- 




WD 

4= 

•r- 

cO 


▼■lei 




< > 




4 


6A 

'i- 


Fig. 35. 


ficiently correct, to assume for convenience that the flanges sustain all of the hor 
strains, and no other; and that the web sustains only the vert strains. 

On this assumption, 


The total lior compressive] Moment of rupture, in ft-lbs, at that point, 

or tensile strain, in lbs. in I_ as found by pp 481, &c _ 

eilber Mange, at any point in { The yert dist in tt between the centers of 

the span, is J gravity of cross-section of the two flanges 

at the same point. 


If the moment of rupture is in inch-tons, the vert dist must be in ins, and the 
flange strain will be in tons, &c, &c. 


Flange strain per square inch 
of cross-section of flange 



Total strain in one flange, found as above 


Area of cross-section of one flange in 
sq ins. 


Art 2. The total strains on the two flanges at any given point in the span, are 
plainly equal to each other ; but that on the upper flange is of course compressive; 
and that on the lower one, tensile. 

Inasmuch as the vert dist between the cens of grav of cross-section of the two 
flanges is approximately the same throughout the span, the strain on either flange 
at any point may be taken as proportional to the moment of rupture of the g r-der 
at that point. Therefore, if we draw a diagram of these moments, as dnected for 
various methods of loading, in pp 482, Ac, and let any one of the ord nates equal 
by scale the flange strain at that point; then the other ords will give, by the same 
scale, the flange strains at their respective points. 

Art. 3. Aireas of cross-section of flanges. 


Minimum allowable area 

in sq ins of effective cross- 
section of lower flange at 
any point in the span. 


Tensile strain in lbs on the flange at 
the given poi nt, as found by Art 1 

Sale tensile strength of the metal in 
lbs per sq inch 






































538 


RIVETED GIRDERS. 


The effective cross-section of the lower flailsre is = the cross 
section of the flange plate ,plus that of both legs of both the angles that fasten it to the 
web. minus the rivet holes. When the rivets are staggered, as in Fig 25 A, it is 

usual to consider the effective section as equal to a 
net section taken on a staggered line, X A B C D Y, 
between the rivet holes, the oblique lines, A Band 
(' T>, being taken at only three-fourths of their actual 
lengths. But if this should give a less area than 
would be obtained by deducting the area lost in all 
of the holes, from that of the full cross-section on 
the straight liue, X Y, then this last is taken as the 
effective area. Fig 25 A is a plan of part of the 
lower flange of a plate girder. F F are the pro¬ 
jecting edges of the flange plate. II II are the 
horizontal legs, and V V the vertical legs, of the 
angles by which the horizontal flange plate, F F. is 
joined to the vertical web, W, of the girder. The 
rivets joining the vertical legs V of the angles to the web plate W, are omitted 
in the figure to avoid confusion. 

For girders supporting quiescent loads, such as walls, floors, &c, the safe ten¬ 
sile strength is usually taken at about 10000 lbs per sq inch for wrought-iroa 
Buch as is used in girders; but, for railroad bridges, from G000 to 7000 lbs per sq inch. 

Having decided upon the shape of the upper flange, its area of cross-section 
may be found by means of the rules for iron pillars, p 439 etc. 



IPigi 35 


If the flange is of the usual T shape: 


Square of least radius ) _ (Approx) square of width of flange 
of gyration J in ins h- 22.5. 


The effective cross-section of the upper flange is equal to its entire sec¬ 
tion, because tlie loss of metal occasioned by punching the rivet holes is compen¬ 
sated by the rivets themselves, which also resist compression. 

Art. 4. The width of the tipper flange is governed by its length of 

the greatest longitudinal distance between those supports which prevent the girder 
from yielding sideways. Thus, in railroad girders, these supports consist of the 
transverse bracing; and, in order that the flange may contribute sufficient lateral 
stiffness to the girder, it is usual, in single-web girders, to make its width not less 
than about one-twelfth of the greatest longitudinal distance between the points of 
attachment of that bracing. In buildings, where the load is quiescent, one-twentieth 
is the usual minimum. In these, however, there is frequently no sideways support 
between the abutments, so that the proper width of flange becomes one-twentieth 
of the span. 

Since the loiver flange is in tension, its width is a matter of minor importance (pro¬ 
vided sufficient area is given), and is generally fixed by considerations of practical 
convenience. 


Art. 5. As the moments of rupture increase as we proceed 
from the ends of the span toward its center, the total flange strains, 
and the required area of cross-section of flange to withstand them, increase in the 
same proportion. The width of each flange is usually made uniform throughout 
the span, and the required increase of area is given by increasing 
their thickness. This is done by increasing (toward the center) the 
number of plates of which each flange is composed. The plates and 
angles composing a flange are riveted together throughout their 
length. When a flange-plate, or flange-angle, is too long to be 
made in one piece, the two lengths which compose it must be 
joined, where they abut together, by special splices or “covers” 
-»— 1 (see butt-joints in “Riveting,” p 469). In thus splicing the angles , 
h h, Fig 25 B, rolled 44 angle-covers,” c c, about two feet long, 
are used. 







Fig. 25. B. 



















RIVETED GIRDERS. 


539 


Art. 6. Rivets. The vert rivets in the flanges, which join together the 
flange-plates and the hor legs of the flange-angles, are generally so spaced that their 
w pitch,” or dist apart from cen to cen in a line parallel with the length of the 
girder, is from to 6 ins. Where the flange is made up of two or more plates, as 
in Art 5, the added plates are made so much longer than the length required by the 
moments of rupture, as to give room, outside of said required length, for a suflicient 
number of rivets to transmit to the plate its share of the longitudinal flange strains. 

The hor rivets joining the vert legs of the flange-angles to the web-plate; and 
those used in splicing together the several lengths of the web-plate; may be spaced 
by the following rules: 


Greatest allowable strain "I toq 0 q v Crippling area of 
on each rivet, in lbs j ' X oue j n 8( j j nSi 

The crippling' area of a rivet, in sq ins, is = its diam, in ins X the thickness 
of the web-plate, in ins. 

In buildings, &c, where the load is stationary, 1C000 may be used instead of 13000. 

Number of rivets in the Total vert or shearing strain 
depth of the girder between / at the joint, in lbs, 

flange-rivet lines; or in a — --—-— -——-— : - 

length of flange equal to said ( Greatest allowable strain 
depth J on each rivet, in lbs 

The vert or shearing' strain may be found by pp 532, &c. 

If this, in any case, makes the pitch of the rivets less than about 2% ins, the thick 
ness of the web-plate should be increased. 


Art. 7. The web has to resist the vert forces acting upon the girder, and to 
transmit them t-o the abuts. In doing so, it is regarded as acting like the web mem¬ 
bers of the Pratt truss, Fig 31, p 595. In that truss the compressive strains are re¬ 
sisted by the vert posts; and the tensile, strains by the main obliques, c c c. In the 
plate girder, vert stiffeners of angle or T iron, riveted to the web, take the place of 
the vert posts, and resist the crushing strains; while the web itself, in the panels 
between the stiffeners, takes the place of the obliques, and resists the diag tensile 
strains. For the vert strain at any point in the span, see pp 532, &c. 
The diag tensile strain on the web will be = 

i Said vert strain X length of diagonal drawn across a panel 


depth of girder. 

This last strain, however, need not be specially considered ; because, if the web is 
made strong enough to resist the crippling tendency of the rivets, it will also be 
strong enough to resist the diag tensile strain. 

Art. 8. The longitudinal hor dist between two stiffeners is usually 
made about equal to the depth of the girder; except that it is seldom less than about 
3 ft, or more than 5 ft, whatever the depth of the girder may be. The stiffeners 
are generally placed somewhat nearer together toward the ends of the span than at 
its center# 

In such railroad bridges as Nos 4 and 6, in our list, p 545, in which the road is car¬ 
ried by transverse floor girders; a heavy stiffener is placed at the end of each floor 
girder. The stiffeners are thus about 8 ft apart; and, as this exceeds considerably 
the usual limit, a lighter stiffener is placed between each two of the principal ones. 
See foot-note fl, p 545. 

Art. 9. Dimensions of stiffeners. Find by pp 532, &c, the vert strain 
on the girder at the point where the stiffener is to be placed. Then, having decided 
upon the shape of the stiffener, find its area of cross-section by means of the lules for 
iron pillars, pp 439 etc. 








540 


RIVETED GIRDERS. 


If the stiffener, as usual, is of angle iron, or T iron, riveted to the opposite sides 

of the web plate; _ . . 

Square of width of stiffener in ins 

Sauare of least ) _ t 4 \ measured transversely of the girder 

radius of gyration / ~ approx; -- 

In order to have as great a radius of gyration as possible, the narrower leg of an 
angle stiffener (if the legs are unequal) is riveted to the web, leaving the wider one 

projecting. ... 

The area of hor cross section of the small portion of the web between the two 
angles or T’s of the stiffener, is included in that of the stiffener; but that of any 
packing pieces (see foot-note §, p 545) is not,because the latter have no firm hearing 
upon the flanges of the girder, and therefore give comparatively little support to 
the stiffener. 

Art. 10. As already stated, if the web is proportioned as in Art 6 , so as to l>e 
of sufficient thickness to be safe against crippling by the rivets, it will also be strong 
enough to resist safely the tensile strain. 

Art. 11. The web may also he regarded as resisting the shearing and 
buckling tendencies of the vert strains in the girder. The amount of the vert 
strain, at any point in the span, may be found by pp 532, &c. 

The average ult shearing strength of rolled bridge plate is about 45000 lbs per sq 
inch; and its safe strength say 9000 lbs; but, owing to the considerable depth of the 
web as compared with its thickness, it is more liable to fail through buckling. In 
resisting the buckling tendency, it acts like a flat column ; and its ult load may be 
found approx by the following formula, which is similar, in principle, to that used 
for columns; 


Tit buckling load) 

in lbs per sq inch of vert V 
cross-section of web ) 


30000 


Id- 


depth 2 , in ins 


1000 X thickness 2 , in ins 


in which 30000 is taken as the ult crushing load in lbs per sq inch, of wrought iron 
in short blocks. 

Ult bucklin g load _ 

Sale buckling load = Factor of safety,say from 3 to 6 according to 

circumstances 

Art. 12. In tlie box girder. Figs 24 and 25, if we could be certain that 
the strains were equally divided between the two webs, that on either one would 
of course be half that as found for the whole girder; but in practice one web is likely, 
through unavoidable inaccuracies in riveting, &c, to receive more than its share of 
the strain ; and considerable margin should be allowed, according to circumstances, 
to cover this uncertainty. 

For this reason, plate girders are more economical than box girders. 
They have, also, the advantage of being more readily accessible for inspection, re¬ 
pairs, painting, &c. On the other hand, box girders have greater lateral 
stability. 

Art. 13. Formulas for the ultimate crippling strength of well made 

plate and box girders, so proportioned as to be secure against buckling and against 
yielding sideways, and loaded with a quiescent weight; taking the breaking tensile 
strength of wrought-iron at 44800 lbs, and its elastic limit at half this, or 22400 lbs 
= 10 tons, per sq inch. 

Area of cross Depth in ins between 

section of the cens of grav of .. itkqq 

= lower flange ^ cross section of the 
in sq ins flanges, at cen of span 


Quiescent cen 
crippling load, 
including % 
the wt of clear 
span of girder J 




Quiescent extraneous crippling load in lbs 


Span in feet. 

The load so found is that which would cripple the girder by bending it beyond 
recovery. Calling it W ; then 

half the wt of the 
W — clear span of the 
girder in lbs. 

Quiescent distributed crippling load in lbs, including ) _ . . w 
the wt of the clear span of the beam j 

Quiescent extraneous distributed crippling load, in lbs = 

/* • w\ weight of entire clear span 
(twice W)— nf Hr.W in 11 .* * 


of girder in lbs. 

In railroad bridges, a factor of safety of 3 is used with the above loads. 


















RIVETED GIRDERS, 


541 


Art. 14. Formulae for deflections within the limit of elasticity: 


Reflection in ins at center of t 
span under a uniformly dis- > 
trilmted quiescent load j 


Load in tons of 2240 lbs X Span 3 , ft 


220 X Depth 2 , ins X 


Area of cross section 
of one flange in sq ins 


Reflection in ins at center of 1 
span under a quiescent con- > 
centrated center load J 


Load in tons of 2240 lbs X Span 3 , ft 


137 X Depth 2 , ins X 


Area of cross section 
of one flange in sq ins 


It is very important that the rivet holes in the web should agree exactly in size 
and position with the corresponding holes in the vertical legs of the flange-angles; 
as otherwise considerable deflection may take place while the rivets are coming to 
their bearings, and undue strains be brought upon the web. 


Art. 15. The following are the results of an experiment with a box 
beam like Fig 25 made by Trenton (N J) Iron Co. Channels 6 ins X 2% ins X 
y 2 inch. Sides inch thick, 18 ins deep. Weight 207 lbs per yard. Length 20 ft 
3 ins; .dear span 19 ft 5 ins. The ends steadied sideways, but otherwise unconfiued. 


Cen load. lbs. 

Def. Ins. 

0 

Vs 

12990 

3 

T(T 

19920 


24230 

K 

28744 

7 

32284 


37844 

9 

42387 

Ps 

46923 

H 

51460 

5? 

55985 

13 

60553 

1 5 

T'ft 

65089 


69954 

iU 


Cen load. lbs. 


Def. Ins. 


76862 

Interval of 26 days. 
81342 

85524 at once a 

crackling noise 
commenced. In 
10 min, 




i 11 

,1 

, 1 5 

its 


2.5. 
1 6 


In 1 hour, 2_7^ 

90302 3 t ^ 

With a side defln of 1% 

This increased 
until the side plates gave way 
at their bottom edges, in an 
hour. 


Art. 16. 


Weight of 

entire girder of 
uniform cross 
section, in lbs 



Area of vert 
cross section of Length 
plates and an- X of girder X 10 
gles alone,in sq in yards 
ins 


Allowance Allowance 
, for heads , for vert 
alone of stiffeners, 
rivets Ac. 


The weight of the two heads of a rivet, after driving, may be 
roughly taken as averaging about two-thirds of that of the entire rivet. More 
exactly: 

If the dlam of rivet is % % %i, nch > 

the weight of its 2 heads is .167 .2 .25 .4 lbs. 


Riveted girders, erected, cost, per pound, about twice as much as the plates. 
See p 402. 


Art 17. The plates are usually from to % inch thick, and from 1 
ft wide up to 20 ft long, to 6 ft wide up to 15 ft long. The angles (see pp 52o Ac) 
are from 21X X 2U X% to 6 X 6 X 1 inch. The rivets (see pp 469, Ac) are 
from 5 X to ins diam, usually % inch; and are spaced from 2J4 to 6 ins apart 
from center to center. This dist from cen to cen of rivets is called their pitch. 
















542 


RIVETED GIRDERS. 


Art. 18. Figs 26, 27, aD(l 28 illustrate methods of attaching the vert stiffeners 
to the webs of the girders, formerly, they were sometimes bent, both at top and 
at bottom, as at i i. Fig 28, in order to pass the hor angle irons of the flanges. Now, 
however, their upper and lower ends generally abut squarely against the 
lior flanges of those angles, as in Figs 28 A, 28 B, 28 C, and 28 D; and should be 



Fig. 28. 

trimmed to fit them closely. Different methods are employed to enable the 
stifle no rs to pass the i^ert flanges of the angles. Sometimes, as in 
Figs 28 A and 28 B, “packing pieces” (flat bars of rolled-iron) are placed between 
each stiffener and the web; sometimes the upper and lower ends of the flange of the 
stiffener are cut away, as at a a, Fig 28 C; and sometimes the stiffeners are crimped 
or bent slightly, as in Fig 28 D. 



Art. 19. In cases where a girder is placed directly under each rail, the dist 
apart of the two girders of a single track bridge is about 5 ft. Frequently, 
however, and especially in long spans in order to give the bridge sufficient lateral 
stability, the main girders are placed further apart; and the cross-ties, T, Fig 28 B, 
on which the rails rest, are then carried by longitudinal stringers, S, of iron or tim¬ 
ber, which rest upon transverse floor girders, F; and these, finally, rest upon the 
main girders, G. In such cases the latter are generally about 12 ft apart for single 
track, or where 3 girders are used for double track. Where only two girders are 
used tor double track they are placed about 16 ft apart. 

The transverse floor girders are generally about 8 ft apart. They have a vert stiff¬ 
ener under each rail, and frequently others. 

Art. 20. Where transverse floor girders are used, they, with light hor diag ten¬ 
sion rods, give sufficient lateral bracing; but where these are wanting, as in 
Fig 28 A, special transverse strut-ties, T, tic, are used. They are generally made of 
angle or T iron. 




































































rA r\ r\ 


RIVETED GIRDERS 


543 



g-a £ 

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544 


RIVETED GIRDERS. 


Fig 28 B shows one of the side girders, G, and parts of the transverse girder, F, Ac, 
of bridge No 6. As in No 4, the lower flange ot the track stringer, S, consists only 
of the hor legs of the two angle-bars, and has no flange-plate proper. It rests upon 
plates, ins, through which pass the rivets which fasten it to th# 

flange of the transverse girder. . 

The track stringers are stayed by bent, plates, B, which are riveted to them and 
to the upper flange of each cross girder. A is a transverse brace of 2 y 2 X 2*4 X M 
inch angle-iron. One of these is riveted, by means of a connecting-plate, to the 
upper flange of each track-stringer, at its joint with the next one. The plate, 0, ot 
y inch iron, is placed between, and riveted to, the two inner stiffener-angles, K 
(only one shown), and two angles, L, 3 X 3 X an( i t w0 others, M, 2)4 X 3 X /b* 



On p 545 arc given the principal dimensions, weights, Ac, 

for different spans. The numbers (1, 2, 3, 4, 5, C) are our own, and are used merely 
for convenience of reference. Where the girders are 5 ft apart (Nos 1, 2, 3, and 5) 
the bridge is for single track, the cross-ties rest directly upon, and arc notched to, 
the upper chords, and one rail is placed directly over each girder. No 6 is for double 
track, and the roadway is arranged as shown in Fig 28 B. No 4 is also for double 
track, with roadway as in Fig 28 B, except that the track stringers, S, rest upon the 
lower chords of the transverse girders. 

The lower flange of each main girder is riveted, at eacli end, to a rectangular 
rolled-iron “ bolster plate.” which rests upon a slightly larger “-wall 
plate.” The two plates are held to the abut by two bolts which pass through 
both of them. At one end of the span, the bolt holes in the bolster plate are slightly 
elongated in the direction of the length of the bridge, so that the bolster plate may 
slide on the wall plate when the girder expands and contracts under tlu 1 
influence of lieat and cold. 


































































RIVETED GIRDERS 


545 


Standard Plate Girders, Pennsylvania Railroad. 

For loads, see page 546. 


Girders. 

Length. 

Span *. 

Dist apart, cen tocen. 
Approx gross wt,f lbs 
One girder, alone .. 
Two girders, with 
transverse brac’g 
for single track .. 
Middle girder, alone 
One side girder, 

alone. 

Three girders, with 
transverse brac'g, 
for double track.. 
Upper flange,! 

Width. 

Thickness, 

At cen of span... 
At ends “ 

Lower flange,! 

Width. 

Thickness, 

At cen of span... 
At ends “ 

Web-plate. 

Depth. 

Thickness. 

Angle stifir- 

eners. 

Size§ at cen of span.. 

“ at end9 *• 

Dist apart 

At cen of span. 

Near ends II of span 

Trans\erse 

girders. 

Dist apart cen to cen. 

Flanges. 

Web. 


Track 

stringers. 

Upper flange. 

Lower “ . 

Web. 


No. 1. 

No. 2. 

No. 3. 

No. 4. 

No. 5. 

No. 6. 

33 ft. 

49 ft. 

59 ft 6 ins 

61 ft 6% ins 

61 ft 8 ins 

70 ft % ins 

25 to 30 ft 

40 to 45 ft 

50 to 55 ft 

56 ft. 

55 to 60 ft 

65 ft 6 ins 

5 ft. 

5 ft. 

5 ft. 

12 ft 2 ins 

5 ft. 

12 ft 2 ins 

6000 

12000 

17500 


21500 


1 (000 

27700 

39300 

' 

49300 





29500 


3ROOO 




20000 


95000 




115000 


741^00 

10 ins .... 

12 ins.... 

12 ins.... 

14 ins.... 

16 ins.... 

14 ins 

1 y s ins ... 

2 3-16 ins. 

2 9-16 ins. 

4 ins. 

2% ins... 

4 % ins 

15-16 in .. 

1 in. 

1 in. 

1 in. 

1 iu. 


10 ins.... 

12 ins.... 

12 ins.... 

20 ins.... 

16 ins.... 

20 ins 

1% ins... 

1% ins.... 

2% ins ... 

3% ins... 

ins... 

3 7-16 in* 

15-16 in .. 

1 iu. 

1 in. 

1 in. 


1 in 

36 ins .... 

48 ins ... 

58 ins.... 

60 ins.... 

64 ins.... 

72 ins 

Ye in. 

% in. 

Vs in. 

Ye in. 

Ye in. 

% in 

2JSX3X?* 

3X4X%.. 

3X4X%.. 

IT 

3X3^xy 8 

f 

3X3>*X>S 

3X4X%.. 

3X4 X%.. 

IT 

33^X3>^X% 

IT 

3 ft 6 ins.. 

4 ft 3 ins.. 

4 ft 10 ins 

4 ft 2 ins. 

4 ft 11 ins 

4 ft 2 ins 

2 ft 4 ins.. 

3 ft 6 ins.. 

3 ft 11 ins 

4 ft 2 ins. 

* 

3 ft 11 ins 

4 ft 2 ins 






8 ft 3% in* 




9"V?j"... 


9"X %" 




19 "X%" ■■ 


I9”y , %" 




8"XH 16” 


B”X%" 




84§”X V4". 


8%"X7-16" 




12 WXYe” 


12>*"X%" 






* By “ gpnn ” here is meant the clear distance between the abuts, taken just below the coping. 
The span to be used in ascertaining moments of rupture, <fec, is measured between the centers of the 
wall-plates on which the girder rests; and is generally from 1 to 2 ft less than the length of the girder. 

t These weights include the weights of the entire rivets (shanks and heads), and that of the plate* 
and angles as ordered from the mill, and before any subsequent reduction by trimming, or by punch¬ 
ing for rivet holes. Ac. 

! Under “ flange ” we include not only the bor flang e-plates, but also the hor legs of the two angle- 
bars by which said flange-plates are fastened to the vert web. The thickness of these angles is in¬ 
cluded in the flange thickness. Together they are generally narrower than the flange-plates. 
By “ width of flange” we mean its greatest width, or the width of the widest flange-plate. 

^ Each vert stiffener has, between it and the web, a “packing” consisting of a flat bar of 
rolled iron as widens that leg of the angle stiffener which is fastened to the web of the girder, or 
wider, and as thick as the angles of the upper and lower flanges. As shown in both figs, the stiffeners 
extend between the hor flanges of the upper and lower angles ; but the packing pieces only between 
the edges of their vert flanges. The packing pieces, by keeping the stiffeners away from the web, 
render it unnccessarv to bend them, as at i i, Fig 28, or as in Fig 28 D, or to cut away part of their 
flanges, as in Fig 28 0, in order to enable them to pass the flange-angles. • 

II At each end of the span, over the abuts, two or more vert stiffeners are placed on each side 
of the beam and quite near each other, in order to withstand the severe strains at those points. One 
wide packing piece is placed under these on each side of the girder. Across each end of the girder a 
“cover-plsite ” is riveted to the end stiffeners. It is about Ye inch thick, as wide as the flanges 
(tapering when these are of unequal width), and as high as the extreme end depth of the girder. 

IT In Nos 4 and 6, the stiffeners, at the points where the transverse floor-girders are attached, 
Consist each of two angle-bars placed together with a plate between them, so as to foim a T. The 
angles are about 3Y X 5 X H inch, and the plates between them are about K X 8 inch. The inter¬ 
mediate stiffeners are single angle-bars with packing pieces, as in the smaller spans. See Art 8. 











































































































546 


RIVETED GIRDERS. 


The bridges in the table are required to carry safely either one of the three fol¬ 
lowing moving loads, A, B, or C. If the bridge is double track, it must carry such 
a load on each track at the same time, the two loads headed in the same direction. 


“Typical ” 
consolidation 
locomotive. 


Tender. 


“ Typical ” 
consolidation 
locomotive. 


Tender. 


Train. 


.00000 

or, O O O O 0> 
^OOOOO 

<N <M (M <M 


OOOO 
OOOO 
OOOO 
CO CO CO CO 


o OOOO o 


OOOOO 

00000 

OOOOO 
<M r* -r -p -p 

Cl ^ !N <N 


O 

O 

O 

CO 


O o 
o o 

8 8 



000 


oO QQQ 00 o 


M * S-> 


ft. 

7.5 4.5 4.5 4.5 

10.5 

5.0 5.5 5.0 

8.0 

7.5 4.5 4.5 4.5 

10.5 

5.0 5.5 5.0 

3.0 






B 





“ Typical ” 
passenger 
locomotive. 


Tender. 


“ Typical ” 
passenger 
locomotive. 


Tender 

Train, 

00 

OOOO 

8 8 8 8 
OOOO 

o 

o 

o 

o 

8 § 8' 
o o o 

O O O 


OOOO 

OOOO 

OOOO 

CD CD O O 

t — 

O 

CD 

O O o 

8 8 8 

CD CD CD 

"1 3000 ) 

per ft > 
1 run ) 


o o O O 

o 

o o o 


o o 66 

O 

o o o 

ft. 

5.5 9.0 8.0 9.5 

5.0 5.5 5.0 

8.0 

5.5 9.0 8.0 9.5 

5.0 5.5 5.0 

3.0 


Shifting 
locomotive 
‘‘class M'* 
Penna R R. 


Tender. 


Train. 


000 
000 
000 
-too 
CO CO (Cl 


o 

o 

o 


o 

o 

o 


o 

o 

o 


000 00 


o o 



ft. 6.0 4.7 13.3 4.8 3.7 4.8 5.0 


The above “ typical ” engines were de¬ 
signed (in skeleton) by Mr Wilson, for 
use in planning bridges for the Penna 
It It. They were purposely made some¬ 
what heavier than the engines actually 
in use on the road, in order to provide 
for the constantly increasing dimensions 
and weights of the latter, which, how¬ 
ever, are fast approaching these “typ¬ 
ical ” figures. Indeed, the shifting en¬ 
gine 11, here given (“Class M ”), which 
is in actual use, produces, with certain 


jnuuutCB, wiiu veil (till 

lengths of span and panel, greater strains in a truss than either of the two typical 
engines. In calculating the strains on web members, the cross-girder load under the 
foremost pair of drivers is to be considered as the head of the train; any load upon 
the preceding cross-girder being neglected. 

Equivalent uniform loads. Owing to the great diversity in the design 
of locomotives and in the distribution of the load upon their several pairs of wheels 
the method of specifying the actual or assumed wheel-loads, as above necessitates 
much laborious calculation of strains by bridge-builders. To obviate this Mr Geo 
II. Pegram, C. E., suggests* that the strains in plate girders and in the chords 
(i a and m x, Fig 13 f, p 564) of trusses, be calculated from an assumed total load of 
/ o w > » , 60000 lbs f \ 

{■ Vm ms + ~ 8pan in ft j Xspan in ft, uniformly distributed over the entire span as 


1 


span in ft 

in Fig 41, p483; and that the strains in th etoeb members (a x, cr. etc. Fig 13 f) of trusses 
be calculated from an assumed load consisting of a train weighing 3000 tbs per ft run 
and of a single concentrated load of30000 lbs. In order to find the strains on the several 
web members in turn, we suppose the train to extend from one end, as m Fig 13 f of 
the span, first to o, then top, etc, and thus to each panel point in succession: the con¬ 
centrated load being supposed to be placed always at the panel point « o, etc next 
behind the head o, p, etc of the train. The first half panel load is to be neglected in 
the calculations. Thus, if the train he supposed to extend from m to a Fig 
13 f, the concentrated load would be assumed to be at p, making the original panel 
load at n and at o each — 3000 tb X length of one panel in feet; that at « 8000 

lbs X Panel length imp in ft + 30000 lbs; and the half panel load wq would be ne¬ 
glected Mr Pegram finds that by using 2900 lbs and 25000 lbs respectivelv instead 
°f the above 3000 lbs and 30000 lbs, we obtain strains practically equal to the greatest 

aLin'm 6 Ca V^ J;?4 iny ° f th ® above arr angements of wheel loads; and that by using 
3500 lbs and 35000.tbs respectively (as would seem to be advisable in view of the rapid 
increase in the weight of rolling stock), we should add but about 10 per cent to the 
weights of bridges designed for the above wheel loads. 




* Transactions, American Society of Civil Engineers June 1886 
f ith 00000 lbs uniformly distributed, the breaking moment at any point is 
same as would be caused by 30000 lbs concentrated at that point. J P 

























TRUSSES. 


547 


TKUSSES. 


A rt. 1. When the span of a bridge, roof, Ac, becomes so rreat that single solid 
{5ME25? r ?“«> compound beams. 



ifi_^ Vi ; , . .’ — v oi mmorianL aeiau, tnat. 

like the building of locomouves, cars, Ac, they have become a specialty, or a dis¬ 
tinct branch of business, to which persons coniine themselves to the exclusion more 
or less of other departments; and thus attain a degree of skill beyond the reach of 
the general engineer.- Ihe latter, however, should possess a knowledge of the sub¬ 
ject sufficient at least to enable him to form a well-grounded opinion of the general 
merits of a design; and to guard him against the adoption of one involving serious 
imperfections. In a volume like this we can aim at nothing more than an attempt 
to illustrate some few general principles. We shall confine ourselves to such trusses 
as are in common use; showing first the effects of uniform stationary loads as in 
the case of roofs; and then those of moving loads, such as an engine and train on a 
bridge. 


Art. 2. Most of the bridge trusses in common use have two long, straight par¬ 
allel upper and lower members 1t, a p ; and l t, ap, Figs 10, 11, called the 

chords; or in England, the booms. Vertical pieces placed between, and con¬ 
necting the upper and lower chords, are called posts, when they sustain compres¬ 
sion ; and vertical ties, or suspension rods, Ac, when they sustain tension 
or pull. The oblique pieces seen in these figs are called braces, §(ru(-K>racc,s 
main-braces, Ac, when resisting pres or thrust; or tie-braces, tension- 
braces, main oblique ties, oblique suspension-rods, Ac, when 
resisting pulls. Sometimes the same piece is adapted to bear both tension and com¬ 
pression alternately; and may then he called a tie-strut ora strut-tie. The 
oblique members alluded to are sometimes called main-braces, whether they are 
struts or ties; to distinguish them from counter-braces, or counters. These 
last are not shown in Figs 10 and 11, but are seen in Figs 28 and 31, crossing the 
main braces diagonally. These posts, braces, counters, ties, Ac, serve not only to 
keep the two chords asunder, and to prevent them from bending; but to transform 
he transverse strains produced by the wt of the truss and its load, into other strains, 

| "ting longitudinally, or lengthwise, along the diff members; and to conduct said 
i trains along the truss, to the firm supports of the abuts. A load placed at any one 
i f these members is, of course, partly supported by each abut; one part of it tra\els 
P and down alternately between the chords, and along the successive members, 
mtil it reaches one abut; and the other part, in like manner, goes to the other abut! 
^hese members, therefore, perform the duty of the vert web of the Hodgkinson 
’’ earn ; or of the I rolled beams, or of the tubular girder; and on this account are col- 
ictively called the web members, in contradistinction from the chords. Each 
j oortion of any load, while being transferred by the web members, from the spot at 
which it is placed on the truss, to its final point of support on the abut, produces a 
strain equal to itself upon every vert web member along which it travels bet ween the 
parallel chords ; while upon each oblique member encountered on its way, it produces 
i strain greater than itself, in the same proportion that the oblique member is longer 
than a vert one. 

Whether the web members are strained by compression, or by tension; or, in 
>tlier words, whether they act as struts, or as ties, the amount of strain will be the 
■ame. In either case the straining agent is the same identical force, namely the wt, 
jr vert force of gravity of the truss itself, and of its load; and (Art 25, of Force in 
Rigid Bodies) whether this force exhibits itself as a push , or as a pull, neither its 
imount, nor its direction undergoes any change. So far, therefore, as regards the 
jroad principle involved in the duty performed by the web members, they might bo 
livided simply into verticals, and obliques. We 6hall frequently so desig- 
late them. 

Whatever amount of strain the upper end of an oblique produces in one direction 
jigainst the upper chord, that same amount will its lower end produce against the 
arallel lower chord ; but in the opposite direction.- That is, if the top or head of 
ay oblique, pushes the upper chord toward the right hand ; its foot will pull the 
ower chord to the same extent toward the left hand. This, however, is not^re- 


* The first writer to whom we are indebted for a knowledge of correct principles on this subject is 
i. Wurrri.E, C. E , the first edition of whose book (beyond all doubt the pioneer one) bears date, 
'tica, X. York, 1847. He was followed by Bow, of England, and Hanpt, of this country, both in 1851. 
fhe Murphy-tVbipple bridge (of which Mr. John W. Murphy, C. E., has built several of the best) 
iwes its name tG these two gentlemen. oq 







548 


TRUSSES. 


rise ly correct, inasmuch as when the oblique is a strut, the pres at its foot is som< j 
what greater than at its head, because the foot supports also the wt ot the strrj 
itself; or if the oblique is a tie, with its head attached to the uppet chord, then tli 
btraiu is a little greater at the head than at the foot; because then the head uphoh 


the wt of the oblique, and the foot sustains none of it. This remark applies, c 
course, to verts also. Another exception is. when the ends of two obliques me< 
each other: as those at the center of the trusses, in Figs 10 and 11. If, in such case 
the ends of the obliques abut against each other, instead of being separately attach 
to the chord, they will at that point exert their strains against each other, instead 
against the chord. 

In any oblique, as c d, Fig 1, the vert dist a c between its ends; and the hor di 
a d between the same, are called its vert and hor spreads, or stretches, 

reaches. 

Art. 3. There is a great diff in principle between two classes of trusses ^ 
common use. In some of them, two chords are absolutely essential, as in tl 
Howe truss, p 594; the Pratt, p 595; the Lattice, p 596; the W arren, p>">69; ai 
their various modifications, known as the Murpliy-W hippie, the Linville, the Lat 
&c, &c, which differ only in certain unessential details. In the Ilowe and Pi 
trusses there is no diff whatever of broad principle, the distinction between tl 
consisting chiefly in the fact that in Howe’s the verts are ties, or suspension ic 
and the obliques, struts; while in Pratt's, the verts are posts; and the obliques, t 
In all these the strains on the verts and main obliques (not on the counters) are lea 
at the center of the truss; and increase gradually toward the end of it; while tho 
on the chords (as in an ordinary wooden beam) are greatest at the center, and lea 
at the ends. Hence, also, such are called bcuin trusses. The strains on t 
counters are also greatest at the center. 

Hut there is another class, called suspension trusses, of which the Fin 
Figs 46 and 47; and the Bollman, Figs 44,45, are the principal representativ 
In these but one chord is essential for a perfect truss. From this chord the w 
members are suspended; and to it alone do they all transfer their strains; a 
the strain on this hor chord is uniform from end to end. Figs 45 and 47 sin 
perfect bridges, with but one chord each. In Figs 44 and 46, n n, n n, appear 
be chords; but strictly speaking they are not; they are merely longitudinal piec 
for upholding the cross-beams of the flooring, when the roadway is placed at t 
bottom of the truss. They have not to resist tension, as in beam trusses. 

In all the forementioned trusse* the roadway may be placed on either the top or the bottom cho |,!,)i 
constituting in the first case a top road, or a deek bridge: and in the second, a bolt 
road, or a through bridge. 


Si 


.ha 


if) 


It 


si 

f* 


Art. 4. That part of a truss, such as Figs 10, 28, 31, Ac, that is comprised 
tween two adjacent verts, is called a panel; thus, in Fig 10, e ij d,djk c , Ac; a 
in Fig 31, of Pratt, tynw, is a panel. The Triangular or Warren truss, Fig 11 U 
558 has no verts, as essential parts of it; and its subdivisions are called sin 1 L 
triangles; and a panel is a length of truss equal to the width of a triangle. V* , 
are sometimes added to it when the spaces a b,b c,c d, Fig 11, become too long 
safely supporting the roadway without them; thus dividing the truss into li 
panels. It is not a matter of practical importance as regards strength, whet, 
the number of panels in a truss be odd or even; but it is usually even, with a v 
at the center of a truss. 

A panel-point, as a, b, d, c, o, or n, Fig 1, is one at which web-memb 
meet a chord, or a rafter in a bridge or roof, as Fig 14, b, c, k, Ac, 

The length of a panel is its horizontal measurement. T ' 1 
best inclination of obliques, as regards economy of material in 

web, is when their least angle (t p o, or s o n, Ac, Fig 10) with the chord is - 
This applies also to the admirable Warren truss; in which the triangles are, howe 1 

usually made equilateral. When the spar 
great, and the height of truss correspondingly 
if the panels be made square, or nearly so, v «r 
a view to secure this inclination of about 45 c 
the obliques, the verts (as to, sn, &c, Fig 1 
will become so far apart, that the stretcher 
dists p <>, o n, Ac, become too long to be safe 
upholding their loads of engines, cars, &c, w 
out additional precautions. When, therel 
the expense or inconvenience resulting from i > 
would be too great, the verts may, as in Fig ] 


mil 



placed so near together as to make half pa 


4 

*5. 


'ill 



















I 


TRUSSES. 


549 


r dJitted^counter^Mhen^nnf a ’ Ul th ° ° ,,H( l Ues (both the main ones > aild the 

"I I Ear! IrT 8 one vert, as in the Fig: or across two/if neces- 
: f , U1< '* arren girder, tlie expedient is to introduce verts • or else a sernml 

bridges, the main obliques, instead of being each in one piece arc usually undo nf 
. „ r more pHral !, 1 pi«** dtaporcd l„ „tch a manner E to letS 
ietween them diagonally, without mutual interference. Each lower chord tuidVe 


In European iron bridges, the chords are generally attached to the web mom- 
,K rs by riveting; but in America this is done by means of cylindrical iron or 
steel pins as shown for a lower chord in FiV ni n fii 9 y a 1Lai lron or 


I i ° - — a o uuur uy lilt: 

i >teel pins as shown for a lower chord in Fig 61 p 612 

, Th ® members in the web, like the lower "chord, generally consist of 

U'ound, flat or square iron bars, with eves Fi<r ♦ of 


fa 0 ge n of1he°pinr are ^ barS * With ’^ es <•* Fi & 61 > at thefr endsXrThe pa°^ 

rd, and the struts or compression members in the web, were at 
ade of cast iron; but they are now almost universally of rolled 
d give them sufficient lateral stability as pillars without an n- 
tue expenditure of material, they are made hollow. Many different, shapes are 
Jsed That shown in Figs 13 and 14 of “ Trestles”, p 757 . is a common one - fre- 
uentlv 'n ith channel bars (p »21) instead of the side plates and angles of Fig 

lJio T ^ P i X « 0, »* nn <P 449 ) is also very largelv used for this pur- 

lie pasla h ge on S he f pfn e s StrUtS ’ tb ° Se ° f the ,les ’ are furnislied with eyes for 

When the web of an iron or steel bridge truss consists of inclined and verti- 
al members they are so arranged that the vertical, or shorter, members shall bear 
lie compresst vets trams, and the inclined , or longer , members the tensile strains 
s in Fig 31, p 59o;* because (see lines 11 to 18, p 457) a short .pillar is stronger 
han a longer one of the same material and cross section. In great spans 
here the truss is necessarily very deep, the obliques are often made to cross’ 
vo panels, as in Fig 1, thus intersecting each vertical post at its center In 
tch cases the oblique is sometimes fastened to the post by a pin at, the point 
intersection, in order to further strengthen the post and to prevent the Ion"’ 
r clique from sagging. ° 

In 'J° od £P bridges, the verts are generally ties, and the obliques, posts, as in Fig 
> P 008 . The Howe, F tg 28, p 594, has oblique wooden struts, and vert iron ties, 
bln long spans provision must be made for the expansion and contrac- 
< on of the truss under the effects of heat and cold. See p 614. 


If in Fig 10 we imagine lines crossing the panels diag, as the main obliques shown 

I the Fig do, but in the opposite direction, as shown in Figs 28 and 31, they will rep- 
•jsent counter-braces, or counters. These, like the main obliques, are in 
’me cases struts,and in others ties. Although important members, they are’less so 
ia an the main obliques. They are unnecessary when the load is uniform and station- 

II y, as is usually assumed to be the case in roofs; and are required only when the 
ft id is unequal, or a moving one, as in a train crossing a bridge. In this last case they 

t chiefly while the span is but partially loaded. If the train at any moment covers 
)(e entire span, and is of uniform wt, their action ceases for that time. Their office 
solely to counteract the deranging tendency of the unequal loading of diff parts 
I, the truss, as shown in Figs 9^. In Fig 9 5, an excess of load along a o 

(»uld tend to derange the main braces ho and ta \ and this would be counteracted 
4 counters across co and ts. The same thing may be effected by arranging the main 
f tees, ho, ta, so as to bear tension as well as compression. The bad effects of une- 
n al loads must plainly become greater in proportion as the load is heavier than the 
f iss itself; and when the bridge becomes very heavy, so that the load must extend 
r :r several panels before its effects become serious, but little counterbracing is 
j jded; and that at and near the center only; whereas, in a very light bridge, the 
inters should extend from the center, where they are most strained; to near the 
; Is, where the strain upon them is least. Inasmuch as we shall first speak of uni- 
••mly loaded trusses, we shall not here say more respecting counters. See Kemark 
; t 10 . * ’ 

• t would at first sight appear that the several parts of a bridge truss must he most 
t lined when covered from end to end with its maximum load; but this is true only 
i the chords; and of the main obliques and verts, as la, tp, Fig 10, at the ends of 
i truss. The other web members are more strained by a part of the load as it passes 
ng the truss; so that if they be correctly proportioned for a full load, they will 


: Except that, when the roadivay is on the lower chord, the two web members 
ting upon the abutments are generally inclined struts. 






550 


TRUSSES. 


»,e too weak for a partial one. If all be made as (strong as the end ones, tboy will il 

. . .. « - J . il.in »’..nl/l vaoniro on OYI1PIKP (it niJltfiriAI LllHil 


De too weaK tor a paruai uue. xi <*“ f . * • , flia 

is true, be safe for a passing load; but this would require an expense of nmtenal tha 
would be justified only in the case of moderate spans, especially of wood , in wfiicr 
the additional trouble and expense of getting out and fitting together pieces o! 
many diff sizes, may more than counterbalance the saving in material. 

Art. 5. Trusses with moving losuis require calculation diff fron , 
that for uniform loads. W e shall first treat of the latter only ; and in so doing shal 
not employ the shortest methods, but such as will render the general principles ch a 
to any one acquainted with the simple elements of “ Composition and Resolution o 
Forces.” The strains on trusses may be found with all the accuracy needed for prat 
tical purposes, by means of diagrams drawn to a scale, lhe °^* e “J*!' 

that answers for a foot of length, may also represent a ton, 1000 lbs oi any otlu 
convenient wt, load, or strain, and may thus be used for measuring the lines wine. 

The chords, verts, and obliques heretofore mentioned, constitute ali the essentu 
elements of a complete truss; but other pieces are necessary for a complet 
bridge; such as roof and floor beams; transverse bracing for connecting two pai 
allel trusses with one another, so as better to resist lateral or sidewise motion froi 
winds or lurchings of trains; bars for tying the truss to the piers and abutments i 
some cases, Ac. The same may be said of the extension frequently made at the end 
of either an upper or a lower chord of a bridge as shown at v v in the bottom choi 
of Fig ill Here the trusses are perfect without the extensions; but the. brut, 

requires them, to allow the load to reach and to leave it. They may be needed 1 < 
the same purpose in an upper chord of a top-road bridge; or for extending a rot 
over an entire span, Ac. The end vert posts pu, of the same Fig are not parts < 
the truss, but supports for upholding it; also, the posts p and c/, Jug 28, are not t 
sential to the truss . * 

Rem. Besides the forms of truss already mentioned, there are many others in some of Whi «a 
arches are introduced either as principal members, or merely as auxiliaries; as JL own & Liftttl( 
Fig 33 ; the Bow and String, Fig 35; and the Burr, Fig 36, all of much mem. The Latt 
and the Burr have both fallen into uudeserved disrepute, trom the fact that being the nr&t trus. 
that were extensively introduced upoD the railroads in this country, they were built too weaa tor t 
heavy engines and trains of the present day, and consequently failed. ^ 

Art. 6. Chords. When a beam a b, Fig 3, supported at both ends, brea 

either und^r its own wt, or under t 
action of a load placed on top of it, a 
suspended from it below, it does so 1 *. 
cause the lower fibres, near its centj^ 
l, arc pulle.it asunder; and its upp^ 
ones at u, crushed together to such 
extent as to offer no effective rosi 

a soi 



I 


S' 


ance. The fig shows this in 


what exaggerated manner. The 
treme upper particles at u, and the '< 
treine lower ones at being the m 


iris 

T 


ine 


aej 


IV.MV Cl Cf 11CO Cm t * 

strained, give way first; and the strength of the beam being thereby diminished, i 
adjacent ones give way in rapid succession. The compressed particles of the be J 
are all above a certain point n ; while the extended ones are below it If we imag 
an infinitely fine needle to be held perp to this page, and in that position to be sti 
through the point w, passing entirely through the beam, or page, then the infinit 
fine hole thus made will pass along what is called the neutral axis of the lies 
It is so named because the fibres situated in that line, and which were cut in twe 
the needle, are neither compressed nor extended, until the strain becomes so gi 
that on its removal the beam will not entirely recover itself; or, in other wo 
until the strain exceeds the elastic limit of the beam. Within the limits of elastic 
the neutral axis may be assumed to pass through the cen of gravof the cross-secl 

of the beam. Thus, if the cross-section lx 
any of the forms shown in Fig 4, then so 1 
as the beam is safe, or the load within 
elastic limits, the line na will pass alonj 
cen of grav; which is at the same tiim 
neutral axis. But the chords of a t 
differ essentially in condition from 
fibres of the beam, as will be seen by comparing pp 485 and 486 with Art. 2, p 
where it is shown that while the resistance of a closed beam is in proportion tc 


% 

lit 


an, 


w 


Hi 





\v licit; lb lo onuwii iiicvb wmic biic icoiotctiii/c; ui a. twocw ucam 10 111 u 

square of its depth, that of a truss, or open beam, is proportional simply to its dep 
The same quantity of material that composes the beam ab Fig 3, will preseu 
more resistance to bending or breaking if it be cut in two lengthwise along the 1 

















TRUSSES. 


551 



Fig. 43^. 


a J:,vwK° nVert f d int ° tor,and bottom chords of a truss; because the 
of the cliont- , 1 t lG re if 8tance ac . ts is thus g r ‘‘ atl y increased. Besides, the depths 

the r fibre mn JI f° sma11 compared with their distance from the neutral axis, that 
fibres in th ' * <! aSs ' lin< ; d to act unitedly and equally. Hence, practically, all the 
.„ *!? t : Q u PP er chord must be crushed, or all those in the lower chord pulled 
iA Fi’tr 1 thT f %mtant ' beture [ he ^ubs can give way ; whereas, in the solid beam, 

pittsr or or flbr ' s Rm • " ,cn «*«» ne “ ,u 

a A - r ** 7 * n , tB,e designing of trusses, especially such as mav have to 
l( Jear unequal loads at different parts, as in a bridge the ^ 

:t Joint chiefly to be aimed at is to dispose its various 
:1 jarts so as to form a series of properly connected t ra¬ 
mbles, because in that shape they present more 
resistance to derangement of form, than in figs of a 
tj;ieater number ot sides. Thus, in the three beams at 
(, 4igs 4/4, with a bolt at each junction or joint, the 
i! riangular form evidently cannot be changed by any but a force sufficient to either 
n >end or break either the beams or the bolts. But in the 4-sided fig 0 the form mav 
a eadily be changed to that at c, by a force at n entirely too smalf to injure either 
; r beams or the bolts. In a the bolts assist to prevent change of form - but in b 
5 hey are merely pivots, around which great changes may easily take place 

i su-n, propor - 

nnlar trusses already built. This bccome/the more necessary asthe truss Ssefitfit^To^ha 
s own wt becomes greater in proportion to that of the load. The table n fitvi f! *’ ° tha f 

,1- t. for bridge trusses; and p 580 will aid in the case of roofs In ven^mall fplns eledal'.Tof 

S? Sect^d emlreir 7 8 ° “ UOh ^ ^ Wt ° f the truss - that the latter’mi^ht almost 

‘,1 ^ or fi Q ^ing the strains on a paneled truss by means of a drawing it 

best to represent each member by a single line, as in Figs 1,10,14, 23, Ac. Such 
', calle , d . u S ** eleto , n rawing or diagram of the truss. Each of the parts 
ito which the panel-points divide either chord, or a rafter, is to be regarded as 
11 separate member. 6 

\ As wil J bfe shown farther on, a load consisting of some portion of the wt of the 
uss and its load, ts assumed to be supported at each panel-point. All the forces 
Inch meet at any panel-point (namely, the aforesaid partial load, and the forces 
t :ting lengthwise of the members which meet there) hold each other in eouili- 
; "mm. 1 

! T, V e il for P e f «pon a truss (omitting wind) are the downward 

ae of the wt of itself and load; and the upward one of the reaction of the abut- 
i ent , s : and these two forces are equal. They produce all the strains along the 
embers. 6 

K 

I 


It consists of two 


Fig- 5 Is the most simple form of a roof truss. 

ial rafters o a, o l; 

•d » bor tie-beam a b. 
tbre, as iu roofs gen- 
illy, the entire weight 
the truss, and of its 
id of roof - covering, 

)w, wind, foe, may bo 
iriumed to be uniformly 
tributed across the 
* ole span. A roof con. 
as of several trusses, 
ced usually from 8 to 
li ft apart; but some- 
tes much less, and at 
ers much more. The 
sses rest on longitudi- 
1 timbers, p, p, called 
ll-plates, stretching 
ing the top of the wall; 

1 serving to distribu te 
wt of the truss and 
load over a greater 

a. On the rafters, and at intervals of a few ft, are fixed pieces of timber called purlius, of 


V 



y 

"p 

w 

p- 

v i 



Fig. 5. 


ci 























552 


TRUSSES. 


small scantling, running across from truss to truss ; to which the laths or hoards are nailed whi^h 
support the shingles, tiu, or slate, &c, which forms the roof covering. . ,, g 

A truss plainly supports all the purlius, roof-covering, snow, &c, <fec. which occupy the space half¬ 
way ou each side of it to the next truss. Thus, suppose a span ot 30 It, and each rafter to he lo.H ft 
long - and that the trusses are say 12 ft apart from center to center, and assume (as it is generally 
well to do ) that the wt of the truss, covering, snow, &c, may amount to 40 ths per sq ft of area of ' 
roof Then each truss has to sustain 33.6 X 12 X 40 = 16128 lbs, including its own weight. Strictly ,,, 
the wt of the tie-beam should he omitted; because in Fig 5 no part of it is upheld by the rafters 
It is very trifling however in comparison with the load. 

To iind the strains upon -the different parts of a truss. ;; 

Fiji - 5. First calculate in the manner just shown, the entire wt in lbs of a trust 
and its load Through the center U of either rafter draw a vert line II r. From o draw a hor lint * 
oH Join 11 a. Now on the vert line H r, lay off H I by any convenient scale to represent tl* etotir - 
uniformly distributed wt of one rafter and its load ; aud draw the hor line I E. Then will I K giv 
by the same scale the bor force at the head of the rafter; aud H i£ the amount and direction of th 
oblique force which presses the foot of the rafter. The hor force at the toot of the rafter will be equa 
to that at its head ;* and equal also to the hor pull along the whole length of the tie-beam.t 

Or, consider the force of gravity, GR, Fig 5% ( = HI) as resolved into two com- ni 
ponents; one, LR, in the direction of the length of the rafter ; and the other, GL j;. 



at right angles to it. The latter is the force which tends to break the rafter tram 
versely, or like a beam. .Since the rafter is uniformly loaded, the cen of grav G is a •- 
the center of its length. Hence, by the principle of the lever, Art. .54, p 3:49, op, — } 

G L, is exerted against the top of the other rafter; and the other half a q at the aim 
a. At the top, op causes, or is resolved into, two forces; first., the horizontal prei “ 
sure o b ( = I E) against the other rafter; and, second, a longitudinal thrust o z alon 


o a 


+ This thrust i oz) is uniform from o to a. Hut at each point between o and a 


ilk 


is added to by a jiortion of the other longitudinal thrust L II which arises directly fro | u 
the pres of the load. Since the load is uniformly distributed, this last thrust i 1 
creases uniformly from nothing at the top o. to its full amount L It at the foot, a. t 
the top therefore, the total longitudinal thrust is o z. At G it is o z f half L It. At 
it is a k = oz + L It; and combines with the transverse pres aq there, to form tl 
resultant pressure a v of the foot of the rafter, which is of course the same as H E, tl 
resultant formed by the load II T and the horizontal pres I E of the other rafter, 
will l>e noticed that the horizontal components, n q and s k, of a q and a k, are in 
site directions. Their difference (= t v) is the pull on the tie beam, and is = 1 E = o 
But their vert components, an and as, are both downward; and their sum ( — a t) 

= HI = the load on«« = the upward reaction of the abut a. 

Tlie sizes in I ig 5 may be found in ilie following' mannei 

Take, for example, a truss of white pine, of SO ft span, and 7)4 ft rise. The wt of the entire ro 
snow, &o, &c, 40 lbs per sq ft of roof area. Trusses 12 ft apart from center to center: so that ea 
truss will have to sustain a total load (including its own wt,) of 33.6 X 40 x 12 = 16128 lbs, which 
may call 16000, or each rafter 8000 tbs. We will calculate each part with a safety of 3; which 


* The foot of each rafter tends to slide or push outward^iorlzontally in the direction of the am 
{ and v ; each with a force equal to I E. But the tie-beam prevents them from so doing, and thus c 
verts their pushes into pulls against each other; and thereby into a pulling strain along the wl 
tie-beam itself, to an amount equal to one of the foroes ; as two men pulliug against each other at 
two ends of a rope, each with a force of 10 lbs, only strain the rope 10 lbs. In other words, it requi 
two equal opposing forces of 10 lbs each, to produce one strain of 10 lbs. 


>»,: 

% 


*• 


T And there is a hor strain to the same degree generated at every point along the length of e , t 
rafter. 

t It is Immaterial whether we thus resolve o p directly into o b and o z, (as thougli the head of 
rafter rested against a vert wall at o); or whether we first resolve it between the two rafters, into 
and o r. For in the latter case we must add to o c a thrust (=or = ci) produced in o a by the tra T 
verse pres (similar to op) of the head of the other rafter; ami the sum of these two (o c and o r ' 


«h 
















TRUSSES, 


553 


hink is abundantly sufficient, with the assumption of 40 ibs per sq ft. First prepare a diagram 
f the truss, on a scale of say % inch to a ft. This diagram will consist of but three lines. We will 
ise the same scale of % inch to represent 1000 ffis of either wt or strain. Make HI by scale equal to 
inch ; that is to the 8000 lbs uniformly distributed wtof one rafter and its load. Also draw I E, and 
nensure it. It will be equal in this case (accidentally) to H I, or 8000 Ibs; and this is the amount of 
mil along the lie-beam.*Now we see by table, page 463, that average w hite pine breaks under a pull 

10000 

f 10000 Ibs per sq inch; so that for a safety of 3, we must not subject it to more than—-— =3333 

>s pull per sq inch. The weakest part of the tie-beam is where it is cut into, near the ends, for foot- 
ag the rafters; and even what is there left by the cut, is usually still farther reduced by the holes 
f the bolts or spikes driven into it through the feet of the rafters. Therefore, allowing for these 
hings, we must give to the tie-beam at that point a transverse section.of solid wood, equal at least to 
000 . 

— *.t 3< I las - This would no doubt be sufficient to resist the pull; but there are other cousidera- 

H 0,ls - such as danger of sagging or breakiug down if persons should get on it; or if a moderate load 
il muld chance to he laid upon it, &c, which cause the tie-beam (even when unloaded even by the wt of a 
lastered ceiling below,! as is here supposed to be the ease,) to be made about as large as a rafter. 
If, instead of a beam, we had used an iron rod to resist the 8000 Ibs pull, we should have reqd one 
ith a breaking strength of 8000 X 3 — 24000 lbs; aud by the table of bolts, page 409, we see that a 
iam of full i| inch would suffioe if upset; or of full 1.04 inch if not upset. See Rem p 408, 

Now, as to the ratters, each of them is an inclined beam, supported at both 
nds, and uniformly loaded with 8000 Ibs; which is equal to a center load of 4000 Ibs. But for a safety 
f 3 against 4000, we will Bud its dimensions for a breaking center load of 12000. •• Its 

20, p 497, we 
Therefore to be 



ife with 4000 lbs center load, each rafter must at its head be 5 ins broad, bv 9 ins deep.f 
We may take G L (= <200 Ins) instead of H I (= 8000 Ibj as the load ; but then o a (= 
lust be taken as the span instead of w a (— 15 ft). The result is the same in both cases. 


16.77 ft) 



train is concerned, the i-after might be of less cross-section at top than at foot; but in practice the 
xpense of cutting the timber to that shape would generally more than compensate for the slight dif- 
:rence of material, even assuming that it could be saved.* It is uncertain how the transverse and 
ingitudinul strains are distributed through the cross section; for the least saggitig of the rafter 
ould throw most of the lougitudinal compressive strain on those fibres ;on the upper side of the 
lfter) which are already under compression due to the transverse strain. For safely we will add to 
ie cross-section required by the rafter as a beam, a sufficient area to be safe in itself against the 
reatest lougitudinal compressive strain, or that (ak) at the foot, which we have just found to be 
- 10800 lbs; aud will make the area uniform throughout. Now we find by table p 436 that aver- 
a ge white pine or spruce crushes under a pressure of say 6000 Ibs per sq inch.' Therefore it will have 
safety of 3 under 2000 Ilia per sq inch ; so that we must provide for each rafter — say 5}^ sq 

is of area of cross-section in addition to the 5x9 ins already found. These 5H sq **is may be added 
ther in the breadth of the rafter, thus making it say 5.6 X 9; or to its depth, making it say 
X 10.2. 

In the next three trusses we shall not enter into this detail 
f ealculation; as we conceive that this example suffices to 
1 lucidate its principle. 

d Art. 8 . Next to Fig 5, in point of simplicity, is Fig 6; which represents a truss 
r either a bridge or a roof of mode- 
tc span. It bas two equal rafters, 
li id a hor tie-beam a 6 as before; 
ilfit with the addition of a kiug- 
ist, king-rod, or suspension-rod 
n. Either the tie-beam, or the 
" fters, or both, may be uniformly 
aded. It is immaterial whether 
e load on the former be equal to 
at on the latter or not. We shall 
■re consider the truss only as that 
a roof. Let y g be points half- 
ay between the king-rod and the 
(hutments. Then will the king-rod 
i stain all the weight of the portion 
jy of the tie-beam and its load, 
le portions of the tie-beam and its 
id between y, y, and the walls 
to, are sustained diiectlv by the 

alls. The entire wt of the truss aud Its load, it is plain, is sustained ultimately by the abuts, or 
ills x to ; but the wt of y y and its load does not reach the walls until after having, as it were, first 



* The pull I E along the tie beam will thus be equal to the uniformly distributed load on the rafter 
ilv when the rise o to is one fourth of the span. If o w exceeds oue fourth of the span, I E will be 
s* than H I, and vice versa. 

t The weight of an ordinary lathed ami plastered ceiling; is 

bout 10 Ibs per sq ft; and that of an ordinary floor of 1% inch boards, to- 

•tlier with the usual 3 hv 12 inch joists, 15 ins apart from center to center, is from 10 to 12 lbs per sq 
. | n preliminary calculations it is well to take the two together at 25 lbs per sq ft. 
j This is not a bad proportion of breadth to depth. If we bad assumed say 15 ins for the depth, we 
lould have got a rafter so thin as to be laterally weak. Frequently, two or three assumptions and 
actuations may have to be made before we hit upon a satisfactory proportion. 




















554 


TRUSSES. 


traveled tip the king-rod to o, and from there down the rafters to a and 5: or. indirectly, hr a cir¬ 
cuitous route. That the king-rod sustains all between y and y, will be evident when we reflect that 
a beam a b, when firmly suspended at its center n, may be regarded as two separate beams b, » a. 
One-half of the beam n b and its load would, in that case, manifestly be borne by the wall x, ana 
the other half by n; and so with n a. Therefore, n upholds one-balf of the beam u l aud its load, 
or, iu other words, all between y and y. The king-rod transfers the wt of and on y y, to the heads 
of the rafters at o. This wt may, therefore, be considered precisely iu the light of one resting upon 
o; aud we 
33. 

ami draw its hor diag m d.~ 'Then wilTo m, o d measure' the strains produced by said total weight 
only, along their respective rafters; aud cm, c d the pulling forces produced by the same wt only 

. V . . . . I • _ .__tt .. on O of thorn 



along the tie-beam a b ; causing strain all along it equal to one of them. 

o d must be added to the other longitudinal strains oz and L K (1* lg iA) 1D 


The strain om or i 
rafter, found as already explained. 

For n^Ikt railroad bridges, a truss like Fig 6, of 30 ft span and 10 ft 

high, may have a chord of 15" by IS''; rafters 10" by 10"; one rod of 2%" diam ; or 
two rods of \%” diam, aud several inches apart transversely of the bridge: which 
is far better than one. 


The pull on the tie-beam will be I E added to cm or cd. Find the 
safe area bv dividing their sum by 3333, whieh is the number of lbs per sq inch, giving a safety of 3. 
Then regarding half the length of the tie-beam supported at both ends, aud loaded at its center with 
only one-fourth of the wt of and on the entire tie-beam, find its safe dimensions by the rules, p497, 
or by table, page 499. The resulting area, added to the safe area for the pull just found, will be the 
entire section of the tie-beaiu, uuless some addition be made to the depth, to allow for what is cut 
away for the feet of the rafters.* See Rem, p 500, also Rem, p 355. 

As to the vertical king-rod, n o, it must be strong enongh to bear safely 

a pull equal to its own weight, added to the weight of and upon yy. If the rod is of good bar iron, 
it should have one square inch for a safety of 3, of cross-section for each 20000 lbs of said weight 
If of wood, it must, for a safety of 3, have at least one sq inch for about eact 
S333 5>s of said weight. A safety or 3 will be enongh if the bar is not liable to vibration. 

When the kiug-rod is of wood, it is improperly termed a king-post. Since a post is intended to sns 
tain a load on Us top. the term might lead to the inference that the upper ends of the rafters resiet 
upon, or were upheld by the king post; whereas, as we have seen, they actually uphold it. 


We add the calculated approximate dimensions lor a truss 

Fig 6, of 30 ft span; and 7U ft rise. Trusses 12 ft apart cen to cen. Wt of 

rafters and load on top of them, 40 Jb.-Tper sq ft of area of roof. Wt of and on the tie-beam, includ 
ing floor, ceiling, load, and momentum, 100 fia per sq ft. Timber white pine. Safety or each piece 3 
Rafters ins broad, by 11 deep. Tie-beam 8% broad, by 11 deep 

without any allowance for cutting at feet of rafters. King-rod 1 jf*g inch diam if upset; or full 1J4 i 
not upset.f 


With no floor or loading on the tie-beam, except its own wt, saj 

500 lbs, we have, approximately enough, rafters ins broad by 9 ins deep. 

Tie-beam, say same as rafter, or S}<; X 9. King rod, inch diam ; bu 

it would be expedient to make it rather more. Trusses 12 ft apart center to center 


Art. ». In Fig 7 we have a truss consisting of two rafters, a 6, a d; a tie-beam 
l d; n king rod. a e; and two struts or braces, ee, he. Either the rafters or the tie-beam, or both 
may be supposed to be uniformly loaded. 



Here, as in Fig 6 , the hing-rod a c, upholds the weight of the portion y 

of the tie-beam, and of any load < 45 
floor, ceiling, people, etc, that may 1 1 j 
placed upon that portion; togetht 
with its own weight. But it also su: 
tains, in addition to these, the weigl 
of the two struts e c. he; part or tl i 
weight of the portions z r, and x i 
of the rafters; and part of the weigl 
of the roof-covering, snow, &c, thi , 
may rest on said portions. That it u 
holds itself, y y, and the struts, is » 
most self-evident; but that it npoli 
part of z r, and xu,aDd their ioads, 
not at first sight so apparent. Su< 

struts are introduce 
into trusses when the rafte 

become so long as to be in danger 
bending too much, or of breaking u 
der their loads; or else requiring t 
use of inconveniently large timbers to make them of. They act like posts in atf irding partial suppr 
to the rafters. They carry a part of the strain upon the rafters down to the foot c, of the kmg-ro- 
and the king-rod carries it from there up to the tops, a. of the rafters. From a it passes down throui 
the entire length of the rafters to their feet. Thus, it is seen that the action of the struts consists 
relieving the rafters from a transverse, or cross-strain which would endanger their safety; and 


* In cases where no appearance of sagging would be admissible, it is not alwavs enough that t 
rafters and tie-beam be safe; for they mav be perfectly safe, and yet sag too much for some porpos 
When such is the case, refer to table, page 512. 

t We have known country road bridges. Fig 6, of 30 ft span, aDd 7^ ft rise, of two trusses 18 
apart, in which neither the timbers, nor the probable loads, were larger than in this example. 





















TRUSSES, 


555 


converting it into a longitudinal strain in the direction of their length, in which they can resist it 
with less danger. As we proceed with the subject of trusses for bridges as well as roofs, it will be 
! seen that this is the grand duty of such struts and obliques generally. Iu roofs they thus assist the 
j rafters ; aud iu bridges the chords. 

i Mi here each rafter is a solid unbroken piece, as in our figures, it is uncertain what portion of the 
load zr or xu is actually borne by the strut ec or h c. The introduction ot' the struts thus renders 
it impossible to calculate the strains with certainty by the above described method. For this reason, 
and lor safety, we adopt another method, in which we begin with an assumption which is not strictly 
correct, but which simplifies the problem and enables us to calculate with exactness for each mem¬ 
ber a strain which is sufficiently near to the probable true one for most practical purposes. This as¬ 
sumption is that the rafter is in two parts, aU and U b; that these parts are connected by a per- 
j fcctly flexible joint at U, so that all of the load of aud on zr rests upon the strut ec ; aud that the 
' load of and on xj and jz rests directly upon j. Draw e o aud h n vertically ; and make each of them, 

I: by any convenient scale, equal to the weight in lbs of either z r or x u, and its load. From o and n 
j draw the dotted lines, oi, nw, parallel to the struts; aud ok, nv, parallel to the rafters ; thus oom- 
pie ting the parallelograms of forces, ekoi, and hwnv. Draw the horizontal diagonals i k, and vw.* 
Theu by Composition aud Resolution of Forces, either e k or 7i v, measured by the same scale as be- 
: fore, will give the longitudinal strain in lbs upon each one of the struts. This strain presses the 
struts lengthwise from head to foot. They are also strained longitudinally and transversely by their 
owu weight, as the rafters in Fig 5 were strained by their own weight and that of the roof; but in 
,1 practice these strains iu the struts, due to their own weight, are so trilling compared with that from 
; the roof portious which they sustain, that they’ may be neglected. 

Therefore, each strut may be regarded as If a vert pillar, 
bearing a load equal to cli or hv. Now, the strain ek, along the strut 

ec, is compounded or composed of the vert strain es, (which is equal to half of eo, or oue-half of the 
wt of and on zr;) aud of the hor strain sk. And the strain h v along the strut Ac, is compounded 
ol the vert strain h l, (which is equal to half of h n, or oue-half of the wt of aud on ruj ;* and of the 

I hor strain tv. These two hor strains sk and tv neutralize or counteract each other, by pressing 
agaiust each other at the feet of the struts ; and therefore only the vert ones es aud h t pull upon the 
king-rod ; and they pull it to au extent equal to half the weights of and on zr and xu.* 

The Ring-rod, therefore, upholds in all, 1st, the weight of the two 

struts; 2d, the wtof and on yy: 3d, half the wt of and on zr and xu:* and 4th, its owu wt. It 

* must, therefore, have sufficient sectional area to safely sustain a pull equal to the sum of these four. 

* This area may be found by means of the table of bolts on p 409. 

Make a g by scale equal to the sum or these four wts, plus the weight of xj aud jz, which rests di- 
i, reetly upouy. Draw y in, gl parallel to the rafters; aud Irn hor. 

! Tor the dimensions of the rafters, ab, ad, commencing with what 

they require as beams, supported at the ends, bear in mind that the introduction of the struts ec,hc 
4 converts each rafter, as ab, into two shorter ones, a e, eb ; each of which sustains, in the present 
‘ case, only one-half the load of and on the whole rafter; or only % of it as a center load. Find the 
>! safe dimensions tor the short beam, with its smaller center load, by rules, p 497, or by table, p 499. 

The compressive strain on a U is am. That ou V b (or the greatest comp strain) is = am + e r. 

,( Divide this sum by 2000 (or by whatever other number of lbs may be considered the safe crushing 
strength of the timber). The quot is the safe area in sq ins reqd for that purpose. Add it to the area 
J previously found for the rafter as a beam. The sum is the entire area required. 

The tie-beam. The pull on the tie-beam is fm -f si. Divide it by 3333, the 

safe pull in lbs per sq inch. The quot will be the safe area reqd for that strain. Then consider one- 

II half of the tie-beam to be a uniformly loaded beam supported at each end ; and find the safe dimen- 
- sions by rules, p 497, or by table, p 499. To these dimensions add the area just found for the hor 

pull; the sum is the entire area reqd for the tie-beam, unless some addition be made to compensate 
for the cutting away at the feet of the rafters.f 

Below are the calculated dimensions for two trusses. Fig 7, 

of 4b ft span ; 10 ft rise; and 12 ft apart from center to center. In the first of 
. these the tie-beam with its floor, ceiling, and other load, are assumed at the rate of 100 lbs per sq ft 
i of floor; while, in the second, no specific load is assumed for that member, for reasons before given. 

In both, the wt of the rafters, with their roof-coverings and load of snow, and wind, is taken at 40 lbs 
s-iper sq ft of roof surface between the centers of two trusses. The safety of each separate part is taken 
at 3 ; except that the unloaded tie-beam is fixed by rule of thumb. Timbers white pine. The great 
west dimension in each case is the depth. Dimensions in inches. 1st. Kafters 8X10. 
^Tie-beam 8 X 15. Each strut 4X X 4 l W. King-rod 1 % diam if upset; or 2 ins if not upset. In praa- 
; : ;tice it is better to make the struts as broad as the rafters. 2*1. Ihifters 6X3. The 
1 'tie-beam requires, theoretically, only 16 sq ins area; we will make it 6 x ft, like the rafters. F.aeh 
strut IX X 4X; (the same as in the other.) King-rod X diam if upset; or scant 1 inch if not upset, 
lfa tie-rod were used instead of a tie-beam, its diam would be 1 inch if upset; or 1.6 if not. 

■t Art. lO. Fig 9 is a truss with a tie-beam a b ; two rafters w a, zb \ two queen- 
, rods,J or queens, wt. zt, and a hor straining beam d. It may represent a roof uniformly loaded 
' along the rafters and straining beam ; and having a uniform load along the tie-beam. Or only one 
.. of these loads may be supposed to exist, as in a bridge with a load along ab; or a roof with its load 
i along aw zb. The queen wt supports, besides its own weight, all the weight of aud on the part s y 

I ---------—-----— 

/ * Each strut will thus bear half of the wt of and on zr, or zu, only when, as in Fig 7. the incli- 

i nation of the strut is the same as that of the rafter. If the strut is steeper than the rafter, it will 
i bear more than half; but if it is less steep than the rafter, it will bear less than half; the remainder 
i being iu every case borne by the rafter. The parallelogram of forces will of course show all this. 

. When the inclinations of a rafter and strut are not equal, we cannot draw hor diags ik,vw; but 
i from the points i,k,v,w, we must draw hor lines to the vert diags e o, and hn. 

j t Wints a tijs-ukam is so Loivo that it must »K splicbd, allowance must be made for the weaken¬ 
ing effect of the splice. For Splices, see p 61 1 ; and for other joints, p 613. 
t The queens are frequently made of wood. 






556 


TRUSSES. 



of the tie-beam; and the other one s *, that of and on uy ; « 

only strains on the queens; so tliat \ 
their proper diams can be found by 1 
table of bolts, p 409. The parts of I 
the tie-beam from s and u to the 
abuts, or walls, as well as whatever 
loads those parts may bear, are sus¬ 
tained directly by the abuts. 

The queens transfer, as it were, j 
the weights of themselves and of s y 
and uy, with their loads,directly to 
w and z. To find the strains on the j 
various parts of the truss, first from 
the center U of a rafter aw, draw a 
vert line U II: and from tv draw a 
hor line tcH to meet it. Join H a. 
Make H I by scale equal to the wt of 
only one rafter and its uniformly 
distributed load. Also draw og vert, and equal, by the same scale, to the wt upheld by the queen-rod 
to t, added to one-half the wt'of the straining-beam d. and its load ; for it also presses vert at o. 
Draw g m hor, or parallel to the straining-beam ; aud y c parallel to the rafter; thus completing the 
parallelogram ocg m of forces. 

The strains on the straining'-beani d. The hor line IE and oc to¬ 
gether, give all the hor pres against the end to of the straining-beam d ; and it is plain that a similar 
p-ocess on the other side of the truss, would give an equal pres against the eud z. These two equal 
pressures reacting against each other, produce a strain, equal to one of them, throughout the entire 
length of the straining-beam ; and therefore, the beam must be regarded as a pillar with a load equal 
to this strain, on its top ; and the dimensions aud area of section, for safely supporting it, may be 
found by the rule, p 458 ; or table, p 459.* 

But beside this, the straining-beam, if loaded, must be regarded also as a beam supported at both 
ends; and the area necessary for this, as found by tables, page 499 or 512, must be added to that al¬ 
ready found. 


Tlie strains on the rafters. First, consider a rafter w a as an inclined 
loaded beam supported at both euds; aud find the proper dimensions and area, by the rules on page 
496; or hv the tables, p 499, or 512. , _ 

Second, add together om aud the other longitudinal strains oz and f, R (found as in Fig 5X> w the 
rafter, aud divide their sum by 2000 (or by whatever other number of lbs may be considered the safe 
crushing strength of the timber). The quol is the safe area in sq ins reqd for that purpose. Add it 
to that already found for the rafter as a beam. This last sum is the total area reqd. 

The tie-beam. The hor strain, or pnll on the tie-beam, will be equal to tho 

push on the straining-beam; and is represented bv 1 E and o c together. Find the safe area by table, 
page 463 : or by dividing the hor strain by 3333, which is the pull in lbs per sq iuch that ordinary 
building timber will bear with a safety of 3. 

Then, since in this truss the queens divide the tie-beam into three lengths, each of these must be con¬ 
sidered as a separate beam, (loaded or unloaded, as the case mav be.) supported at each end. Its safe 
dimensions being found, add the area just found for resisting the pull. Add, if reqd, an allowance 
for the cutting away at ti.e feet of the rafters. 


Relow are the calculated approximate dimensions for two 

trusses. Fly: 9, of sixty ft span : 15 ft rise; aud 12 ft apart from center to 

center. All the conditions the same as for the preceding example of Fig7. 1st. Rafters 12 ins broad, 
by 14 ins deep. Straining-beam 12 broad, by i2 deep. Tie-beam 12 broad, by 12 deep. Each queen 
rod lt£ ins diam if upset; 1% if nnt.t 

2d. Rafters 10 X 11% Straining-beam 10 X 11. Tie-beam, say 10 X 12. Each 

queen-rod ^ inch diam. Unloaded tie-rod, 1*^. 

The proper size for each piece, so that they shall all be suitable 

for the truss, cannot- be determined at ouce. We must find any dimensions that 
will answer for each piece by itself; and afterwards adjust them by recalculation, perhaps 3 or 4 
times. Great accuracy is not necessary in doing this. See Note, p 573. 


* A strut or tie cannot be strained along the direction of its length 

by a force acting at one end, unless there is at the other end an equal force acting in the same straight 

line but in the opposite direction, aud which may be either one 
single force, or the resultant of two or more forces neither of 
which acts in that direction. Hence if in 

Fig 9 we place a load at Z only, a parallelogram v e g n of forces 
will not give the hor strain v e along the beam Z W, because 
there is then no equal reacting hor force at the other end in the 
direction from H towards W. In that case a load at Z only, 
(represented by z c in Fig X) produces at z the two strains z n, 
ee ; which last pressing towards a tends to make z It revolve 
around b as a center, thus forcing z downwards, and the joint ui 
upwards, thereby causing the distortions seen in Figs 9V£. 9?£. 
The force z e therefore evidently tends to break the joint to: 
and with a moment equal to the force z e (in tons or lbs. &c) 
mult by its leverage w o perp to z a. If the moment of resistance of the joint can withstand this the 
truss will remain unchanged; but a simple strut from z to a would remove all danger, by sustaining 
the whole of the force z e effectively, aud thus relieving the joint ut entirely, 
t See Rem, p 408. 























TRUSSES. 


557 



iron bar by pulling it apart; therefore the truss "would have remained safe, and unchanged in figure; 
for the bar, while preventing cm from lengthening to ct, would, as a consequence, prevent en from 
shortening to e s. Or, omitting the irou bar at cm, suppose a stiff, unbending inclined post to 
be inserted between e and n. This also will divide the whole truss into triangles; and it is then 
plain that en could not be shortened to e s by any strain less than one sufficient to break the post by 
crushing it. Therefore, in this case also, the truss would have remained safe, and unchanged in 
figure ; for the post, while preventing e n from shortening to e *, would, as a consequence, prevent cm 
from lengthening to ct. Either the bar or the post would be a counterbrace against the effect of un¬ 
equal loading. With a uniform load it is not needed. Neither are additional counterbracing pieces 
needed in bridge trusses of the forms Figs 10,11,12, 13, provided each web member is so constructed 
as to bear alternately compression and extension. 

The next Fig 9 h, shows the bad effect produced in a £ t h l Z 

truss longer than Fig 934. when the web members are . / V \J V|\ / N 

not so constructed. In the Burr bridge, Fig 36, and in \ / / / , \ \ 'W 

some others, although the truss is divided into trian- ' - ‘ —* 

gles, yet the inclined braces, i c, &c, are often impro- Fig'. 9® 
perly adapted to bear compression only; their ends not 
being firmly attached to the chords. Consequently, 
with a heavy load at a, the derangement shown in Fig 
9 b (analogous to that in Fig 93-0 takes place. To pre¬ 
vent it. counterbracing must be resorted to, either bv 
inserting struts or ties along the dotted diagonals; or 
by making the braces capable of resisting tension as 
well as compression. The last method shows that counterbracing can be performed without the ad¬ 
dition of pieces specially called counterbraces, an 1 denoted by the dotted diagonals. All that is re¬ 
quired in the principle of counterbracing, is to so arrange and connect the several web members, 
that the strain produced by unequal loading at any point, as a, between the abuts ; or along any 
portion of that distance, shall be properly transferred by them to both abutments. 


0 


Fig-. 96 



Art. 11. The strains in such trusses as Figs lO and 11, 

may be found by three very simple processes when the truss and its load are uniform 
from the center each way.* When this is the case it is usual and safe to assume that 
the half load ep, Fig 10 or 11, on the right hand of the center e, rests on the right 
hand support p ; and that the half load e a on the left hand of the center e, rests on 
the left hand support «.f It is often assumed also, for simplifying the calculations, 
: that the entire weight of the truss and its load is distributed along one chord only. 
This is plainly incorrect; but inasmuch as the extraneous load (such as the covering 
of slate, snow, etc., on a roof, and the travelling load on a bridge) in many cases ac¬ 
tually does rest on one chord only, and is great in comparison with the weight of the 
truss alone, the error arising from the assumption in such cases is not of practical 
importance. 

But in bridges of great span the weight of the truss may bear a large proportion 
to that of its load; or there may be an upper and a lower roadway, one resting on 
each chord; and a roof truss may have to bear not only the covering, snow, &c., on 
its upper chord or rafters: but a floor with a plastered ceiling beneath it, and all 
the load incident to any ordinary room, on its lower chord. In such cases the entiro 
weight of the truss and load must be properly distributed along both chords before 
we can correctly find the strains. But this will in no way affect the principle of the 
three processes which we are about to explain, and as we proceed we shall give di¬ 
rections for both cases. 


* It is not necessary that the entire load should in itself be uniform ; but merely 
uniform each way from the center. Thus at e may be say 1 ton ; at d and m each say 
5 tons; at c and n each 2 tons, <fcc. 

fTIiis assumpt ion is untrue, and opposed to the unvarying law that 
avery individual portion of the entire weight rests partly on each support. Thus, 
ane portion of the load at o rests partly oh p and partly on a; and so with every 
ather portion ; and on this fact depends the difference in the methods of calculating 
the strains from uniform, and ununiform or moving loads. When, however, the 
weight of the truss and load is uniform each way from the center we obtain correct 
-esults, and more readily by adopting the erroneous assumption. 




















558 


TRUSSES. 





















































TRUSSES. 


559 


Beginning then with uniformly distributed weights of 
truss and load, and assuming all of said weights to rest on the long chord a p , 
prepare a correct skeleton diagram of the truss (or at least of one-half of it), such as 
1* igs 10 and 11, in which the height or depth e i, Fig 10, is the vert distance between 
the centers of the depths of the two horizontal chords. A scale of from % to of 
an inch to a foot will generally be large enough. 

ihen the first, process is the very easy one of ascertaining how much of the 
total uniform weight is to be considered as sustained at each point of support along 
cither the top or the bottom chord, as the case may be; remembering that one half 
of each end panel is sustained directly by the abut nearest to it, as in the preceding 
cases. . 

In order more fully to illustrate the following Articles, we shall assume each of 
the trusses, pp 558, 570, 571, 572, 574, 585 and 686, to be 04 ft long, and 16 ft high, 
and to % be divided into 8 equal panels. Total uniform wt of one truss and its load, 
32 tons; or 4 tons to a panel. Consequently there will be 9 points of support to each 
truss. Thus, in ligs 14, 15, and 16, in which the load is supposed to rest on top of 
the truss, and in Figs 10 and 11, in which it rests upon the bottom, the points of 
support are at a, b, c, d, e, in , », o, p. Some of these are not shown in the first three 
Figs. If both chords are loaded, there will be points of support in the short one 
also. Thus, in Fig 10 there will be 7, and in Fig 11 there will be 8 of them. Now, 
in Figs 10 and 11, w, x, y, etc., being midway between the points of support, it is 
plain that (assuming all the weight to be on the lower chord) the point a must sus¬ 
tain that portion of it comprised between w and x; wall between y and x; while 
the abut p sustains directly the portion from w top. The same principle applies to 
all the other trusses; and equally so whether the panels be of the same width or 
not; each point of support is assumed to sustain all the uniform wt of truss and load 
between itself and the two points midway to the adjacent points of support, how¬ 
ever unequal the two distances may be. In our Figs 10 to 16, the strong dotted lines 
of the web members represent ties; the full lines, struts. The dots intimate that 
chains may serve as ties. When the panels are of equal length, p o, o n, etc., the dis¬ 
tance from p to w will be but half a panel; so that each abut will directly sustain 
but half as much wt as each other point. Therefore, to find the amount of wt sus¬ 
tained at each of the nine points of support, we have only to div the total wt (32 

32 

tons) by a number less by 1 than the number of points. The quot — = 4 tons, will 

8 

be a full panel-load, to be at each point, except the two end ones, a and p, at the 
abuts; at each of which it will be but half of one of the full panel-loads, or two 
tons.* The amounts of these panel and half-panel loads should at once be figured on 
the sketch at their proper points, as is done in our Figs ; a 2 being placed at each end 
ot the truss; and a 4 at the other points. Each of these panel-loads of course causes 
a vert strain equal to itself where it rests. As the strains on one half of the truss 
are the same as those on the other half, the numbers need only be written on one of 
them; indeed, the sketch, as a general rule, need show but one half of the truss. 

If there is a load on the other chord also, it must be in the same 
way divided among the points of support of that chord, and be figured as before. 

The second process. All the panel-loads are of course eventually trans¬ 
mitted through the truss to the abuts; as is manifest from the tact that each abut 
sustains half the total load. But each panel-load, while travelling, as it were, up 
and down alternate web members from its original point of support, to the nearest 
abut, places, so to speak, an additional load, or more correctly produces an addi¬ 
tional vert strain equal to itself, at every intervening point of support in each 
chord.f Our second process consists in finding the amount of this additional vert 
strain at each point of support. 


* This of course is only when the end panels are of the same length as the others. 
When not so, the loads at the points of support and on the abutments will plainly 
vary from the above. 

f In trusses like Figs 10 and 11, with two horizontal chords, the panel loads are 
transferred directly from their points of support via the web members to the abuts. 
In such trusses the strains in the web members are least at the cen of the span, and 
greatest at its ends ; while those in the chords are greatest at the cen and least at the 
ends. This is indicated in the Figs by the diff thicknesses of the lines representing 
these members. But in Figs 14, 15 and 16, the panel loads divide at their respective 
panel points, as explained in Item p 573 ; a portion of each panel load going directly 
to the nearest abut via the sloping rafter; the remainder going first to the cen e via 
the web membei’S, whence it finally reaches the abuts via the rafters Figs 14 and 16, 








560 


TRUSSES. 


In Figs 10 and 11, with parallel horizontal upper and lower chords , the vert straius 
are very easily found, thus : Remembering that only half of the center panel-load 
strain at e goes to each abut, begin with the 4 tons at e. 

In Fig. 10 these 4 tons tirst go up the tie ei to i, where they produce a vert strain I 
of 4 tons, which figure as in the diagram. But at i these 4 tons separate ; 2 of them 
going to the abut p, and the other 2 to the abut a. The last 2 first pass down id to 
u, where also they produce a vert strain of 2 tons, which also figure, as must be done t 
with all that follow. At d these 2 tons unite with, or as it were take up and carry 
along with them the 4 tons already there ; and the entire 6 tons go up the tie d j to j 
where they produce a vert strain of 6 tons. From j these 6 tons go down the strut' 
j c to c, where they also produce a vert strain of 6 tons. At c these 6 tons take up 
the 4 tons already there, and the entire 10 tons go up clc to k; and thus the process 1 
continues until 14 tons find their way to the abut «, where they meet the 2 tons of !' 
load already there; thus making 1G tons, or one-half the wt of the truss and its load: i 
which is a proof that our work is correct so far. 

In F ig 11 the 4 tons at the center e separate there; 2 of them going up e i to i, and 
thence to the abut a as before. ’ ( 

In either Fig if there is a uniform load on each chord there is no 

difference in the second process; for after having by the “first process” divided each 
load among the points of support of its own chord, the portion at each point must 
be taken up as it occurs, and carried on with the others to the abutment as before. 


The third and last process consists in completing our sketch, or diagram, 

m such a manner as to enable us to measure by scale the strains produced along 
every member of the truss, by these vert strains thus accumulated at the diff points 
of support, a, h, c, l, k , etc. 1 


f* 1 . 1 .® iM , Fi S 10 ( thRt is > whenever the web members are alternately 
veitical and oblique) from each point of support of one chord only, beginning at the 

the vert P Rt^ !’ '“ lr ’^ r ’ etc -> tu represent by any convenient scale, 
the vert strain figured at said point; except at the center one i, where the vert line 

must represent only half the vert strain, inasmuch as that is all that goes to each 
abut Braw also the hor lines vu,vu, etc. Then will each oblique line t u, j u etc 

om e as y fi. d e 8 ti“o f C r 8trai “ ( T’ 6 - 7 ’ n - 2 . 15 - 7 ) "long its own oblique web mem" 
f V ■ T , ,nes R,ve tl,e 1,or strains exerted on the chords by each oblioue 

at both its ends We have figured all these strains (7,5, 3,1) at the head and foot of 
each oblique Each of these hor strains extends from the ends of the oblique, to 
the center of the chord; therefore the end stretches of the chords bear 7 tons hor 
stiain, the next ones i + 5 = 12 tons; the next ones 7 +5+3 = 15 tons - and the 
7 + 6 + » + ! = 16 tons; all of which are fi?ured + along the cboSL * * 
c ha\e said that the vert, hor, and oblique sides of the triangles give the strains 

b'danced bv the ntf re $° rre 5u tu , sa - v tlmt each of them gives a force, which being 
lion b> U tl two * thereb -> cullses a strain equal to itself, instead of mo- 


|® ,op strains at the centers of the two chords w r ill i»e <J 

In botb F'S 8 10 and 11, whether one or both chords be uniformly loaded • 
oi it the truss be inverted ; with only the exception in the foot note.* * 


Z IhVwebZm^fit 8 H F ! S F:i 0f Whic , h i « is one - In 6uch cases the strains 
those on the tie i,,! p 10atest ^. the ® en the span, and least at its ends ; while 
f ‘ f rafters Figs 14 and 16, and on the tic rods and hor chord 

in Figs 14 15ZfiVhv .» -T" and 1 ? 8t ftt its cen - This also is indicated 

of thf cj; of F p 10 and n lblck,1M “ ° f the ant * the exact reverse 


IS 

lil 

« 


meJVw 16 ' l Qp T *’ Fig 10 ’ the two eijds of the Obliques id, im which 
S: a?//«/ ang - e< ) 88 to t butt tight against each other, then the center hor 
selves- so that ? ot borne b .V the chords, but by the obliques tliem- 

t' a’*i S ° , 1 1 °. re then be that much less strain at the center point of that 

tbev d »w n a ° l ’ S !!' e c , ent , er stretch d m of the other chord. But if instead of thi? 

i 5 t against the chord, at some little distance from each other then the chord 
also lecenes the strain; so that the hor strains at the centers of’tlie "wo chords 
become equal, as we assume to be the case in all our Figs of uniform trusses uni 

of .5 ,Jf r.on .T«1. V n!" S8,UC lemark *» 









TRUSSES. 


561 


The strain along the center vert tie e i of Fig 10, will be equal to the 4 tons at e; 
md when the entire wt is assumed to be on the long chord , the vert lines at the other 
| points of support will give the vert pulling strains on the other verts, as 6, 10, 14. 

But. with loads upon both chords this last will not be the case; but 
| he strain on each vert tie will then be equal to the vert strain at its foot.* 

1 If the loaded truss is itiverted, the verts become struts or posts, and 
' :he obliques ties ; also the strain on each vert is then the one figured at its top ; but 
, ;he amount of strain on each part of the entire truss will remain as before. 

In Fig 10 all the uniform wt is on the long chord, and the resulting strains are 
| ill figured on the diagram. We add the strains that would occur in case there were 
i m additional uniform load of 6 tons on tbe’upper chord from l to t. This would give 
l ton at each point of support along that chord, except the two end ones l and t, at 
; *ach of which it would be but .5 of a ton. All these must be figured on the short 
diord of the diagram, as were 4, 4, &c, on the long one. The student may then 
l work out the case for himself. We repeat that uniform trusses and loads require no 
counter-bracing. 


i 


i 


i 


For Fiir 10. but for a load on eacli cliord. 


e i — 4. tons. 
d j = 6.5 “ 
clc — 11.5 “ 
b l = 16.5 “ 


id— 2.8 tons. 
jo= 8.39 “ 
kb — 13.98 “ 
l a = 19.01 “ 


a b or l k = 8.5 
b c or kj — 14.75 
c d or j i = 18.50 
d e or at i — 19.75 


tons. 

U 

u 

44 


M For the “ third process” in Fig 1 II (or when all the web members are 
xblique, whether equally so or not) after having found and figured the loads and vert 
strains at each point of support precisely as directed for Fig 10, then from every 
such point in both chords draw a vert line as ev, i v , d v, &c ; and on it lay off sepa- 
1 rately by scale both the vert strain that comes to that point through a web member 
from towards the center of the truss; and the one that goes from it through another 
web member towards the abut; except that at the very starting point e, Fig 11, there 
is but one vert strain (the one of 2 tons going from it); and at the very end a also 
there is but one, namely that of 14 tons coming to it along the oblique la; for the 
Z tons at a are not to be included, because they do not reach a by means of a web 
member. Therefore both at e and at a only a single vert strain is to be laid off.f 


* It is so in both cases, for in any of these trusses under stationary loads 
the strain along a web tie whether vert or oblique may be considered to commence 
at its lower end, that being the end at which the panel loads first act on their route 
to the abut,and up which they as it were work their way. But under moving loads 
the same member may have to act both as a tie and a strut; hence the remark 
:ivill not apply to such. Referring to what is said above, when the entire wt is 
Ion only one choi'd, the vert strains at the two ends of any tie in Fig 10 are equal. 
I Hence a line drawn to represent the upper one, may be assumed to repre¬ 
sent also the lower one. But when both chords are loaded, the vert strains 
figured at the two ends of any tie are unequal, and we must then have regard 
to the true principle. If the verts should be struts or posts (as 
if Fig 10 should be inverted) then any strain along them must be received from 
their tops, or the revei’se of the case with ties. It will aid the student very 
much in what follows to familiarize himself with the idea that strains pass 
only down the struts, and up the ties. 

f When all the wt of truss and load is assumed to be on the long chord as in our 
Fig 11, then the vert strain that comes to any point in the short chord by one web 
member is plainly the same in amount as that which goes from it by the other web 
member; and hence only one vert measurement need be laid off for it, as is seen at 
i v ,j v, Ic v, l v , in the Fig. But at the long chord (or at both chords when there is 
a load on both) the vert strain that comes to any point is less than the one that goes 
from it towards the abut, and is evidently the one last figured at that point, as tho 
2, 6, 10, Ac, tons at d , c, 1, &c, in Fig 11 ; while the one that goes from that point 
towards the abut is as evidently equal to the sum of the two strains figured at that 
point, as the 6,10, 14, Ac, tons at d, c, b , Ac. When both chords are loaded there will 
be two vert strains to be figured at each point of support in both chords, except at 
the very starting point, and at each abut, where will be but one in any case. 










562 


TRUSSES. 


Draw also the lior lines, thus forming a series of triangles (as ivu, ivu,jvu,jv < 
&c, of the upper chord; and a v u, bvu, b zu, &c, of the lower chord), each wit 
one vert, one hor, and one oblique side. Then the combined lioi* strain exerte 
upon either chord, by the obliques at any point of support and by the part load (i 
any) supported there; will he measured by the sum of the two hor lines directly opp 
site to said point, except at the center e, and at the end a. at each of which it will 1 
measured by the single hor line v u , or v u, opposite each.* These hor strains (7, 
3, 1 tons on the upper chord; and 3.5, 6, 4, 2, .5 on the lower chord) are figured clo; 
to the points of support at which they occur; and the total hor strains on the se 
eral stretches of the chords are figured midway of said stretches.f 

The strain along any oblique asy c, will be measured by the obliqi 

side ; u, or c u, of either one of the two triangles on either side of it; and this wi 
be the case whether one or both chords be loaded; or if the truss and load be ii 
verted. All the strains in Fig 11 (loaded on the long chord only) being figured c 
diagram, we give below the strains that would occur in case there were an a 
ditional uniform load of 7 tons on the short chord from Hot; which would give 
ton at each point of support along that chord, except the two end ones l and t, ■. 
each of which it would be but .5 of a ton. All these must first be figured on tli 
short chord of the diagram, as were 4, 4, &c, on the long one. The student mu 
then work out the case for himself. 


i 


II 


e i = 2.06 tons, 
i d = 3.09 « 
dj = 7.22 “ 
j c — 8.25 “ 


For Fig 11, but for a load on each chord. 


c k = 12.36 tons. 

a b = 4.38 tons. 

k b = 13.40 “ 

b c = 11.88 

U 

b l = 17.53 “ 

c d = 16.88 

» 

l a = 18.04 “ 

d e = 19.38 

44 


at c = 19.88 

(4 


I k = 8.63 ton 
kj = 14.88 “ 
j i = 18.63 “ 
i to center = 19.88 “ 


. 4*^ The strains in Figs 10 and 11 may readily be ealcr 

M .1 ( a . lter 1,ilvln g by the “first and second processes” found the vert strains i 
all the points of support) whether one or both chords be loaded, or if the truss an 
load be inverted. Thus, divide the hor stretch of an oblique by its vert stretch; tl 
qu°t ,e r«t Will l» e the natural tangent (.5 for Fig 10, and .25 for Fig 11) of tli 
angle (26 34 in Fig 10 and 14° 2' in Fig 11) which the oblique forms with a vei 
line. Divide the actual length of an oblique by its vert stretch; the quotient wi 
be the nat secant (1.12 for Fig 10, and 1.03 for Fig 11) of the same angle. The 
the strain along any oblique in Fig 10 or 11, is found by multiplying tl: 
vert strain that travels towards the abutment along said oblique, by the nat secan ' 
. . "® strain on either chord, caused by either end of anv obliqui 1 

is in b ig equal to the vert strain that travels along said oblique towards the aim 
ment mult by nat tangent. And in Fig 11, it is equal to the vert strain that conn 
to, added to that which goes from any end of an oblique, mult by nat tang- excel 
at the center and end, where the single vert strain must be mult by nat tang. 

All this is simply because that if wo assume the vert side of anv one of the tt 
angles to be radius or 1, then the hor side becomes by that same scale the nat tan 
o the angle which it subtends; while the oblique side becomes the nat secant o 
the same angle. 


iiT l lf 1>rl w e , iplC ? f fh °, finding the strains in Figs 10 an 

J is this. He know that if three forces are in equilibrium with each other at an 
point, the hues which represent them will form a triangle. Now at every point o 


Each of the upper hor lines uu,uu, &c, in Fig 11 is to be considered as con 
posed of two separate ones v u, v u, &e ; the right hand one of which measures tli 
hor strain caused by the vert strain that comes to i,j, &c; while the left hand on 
leasures the hor strain caused by the vert one that goes from i,j , &c, towards tli 
abutment. Such lines as u u, u u, &c, occur only when all the wt is on ono chord 
toi " hen both chords are loaded, the vert strain that comes to, and that which goi 
troni any point of support differ, therefore requiring two unequal vert, measurement 
and two unequal hor lines at each point of support of both chords, except at th 
center, and at the ends; at each of which will be but one. 


tThe common rn.e ™ ,"»<r»rm_w t X .pa n _ „ or ^ ^ 


, ... , , 8 times the height 

of either hor chord, is not strictly correct except when both chords extend the fu 
length of the span, and are both loaded throughout their entire leno-tli- or in tli 
impossible case of the entire wt being on the long chord. Still in ordinary cases 
is a sufficiently close approximation. On this subject see Art 19. ^ 














TRUSSES. 


563 



'upport in Fig 10 we have one set of three such forces; and in Fig 11, two sets. In 
'ig 10 it was not necessary to show those at the long chord. Now each set, or each 
triangle represents a vert force, a hor one, and an oblique one, keeping each other 
m equilibrium at the point of support. It is true that there are other forces acting 
ot the same point, but as they hold each other in equilibrium, they do not interfere 
dth the first ones. Thus, both the 7 and the 12 tons hor forces along the chord at 
are balanced or held in equilibrium by the equal ones from t and s, on the other 
4 alf of the truss; without disturbing the forces represented by the sides of the tri- 
ngles. Hence by measuring those sides we obtain the forces and strains themselves. 

The same principle either by diagram or by calculation 
pplies to Figs 12 to 13%, when 
niform and uniformly loaded. In those 
jigs all the weight is here assumed to be 
a the long chord; (but after what w r e 
ave said, no difficulty can arise when 
lacing loads on the other chord also.) 

,11 said wt is first to be properly dis- 
iJ ibuted among the points of support on 
dd long chord, and there figured, as 
down by the upper 4s and 2s along that 
lord in the Figs. This being done, we 
ave figured all the other vert strains, 
ius providing the data for drawing the 
;rt sides of the triangles; and these in 
irn give us the hor and oblique sides 
hich measure the corresponding strains, 
id all of w hich are drawn on the Figs. 

All these trusses being uniform and 
liformly loaded from the center each 
ay require no counter bracing. Bear in 
ind that the vert strains that accumu- 
fe at an abut must equal half the wt of 
le truss and its load. 

In Fig 12, with no oblique at the 
nter, the 4 tons at a having no oblique 
. contact on either side of them, go to b ; 
id on their way to b strain ab 4 tons, 
rom b all 4 go along the w’eb members 
the nearest abut e as figured. 

In Figs 12% and 13 the web mem- 

;rs of each are to be regarded in some 
>gree as belonging to two separate 
russes, namely abode and m n op e in 
ig 12%; and abode and m n o d e in 
ig 13; and the vert strains at their ends 
e to be found on that assumption, as 
gured. Iu Fig 13, o d is a vertical tie. 

In Fig 13% there being at none of 
le 4 ton loads on the long chord an ob- 
que in contact with them on either side, 
iey (like that at e Fig 10, or at a Fig 12) 
ass each by itself vertically to the upper 
lord, where figure them. Of those at b, 
go to the abut e by way of the oblique 
e ; but the other two 4s all go to e, each 
y its own oblique. Each of the three 
or lines gives the strain exerted upon 
ie upper chord at the respective panel 
oint; but the hor strain along the lower 
herd is uniform from end to end, because all the 
ods only. It is equal to the sum of the three hor 

40 




forces 

lines. 


that produce it act at ita 




























TRUSSES. 


£>t>4 




In Fig 13%, at the 4 ton loads at c and e, there is no oblique in contact wit 
them on either side; therefore they pass at once vertically to t and y, where figur 
them. Then begin with 2 of the 4 tons at a. 

All loads that have no oblique in contact on either 8i<le, whether sustaine 
by vert ties or by vert posts, are to be thus transferred at onee to the op posit 11 
chord in order to meet an oblique along which to travel to the nearest abut; a 
said vert ties or posts will be strained to the amounts of said loads. 



Fig. 13./ 


Art. 12%. Moving 1 loads and connterbracing. The foregoin; 
investigations refer to cases where the load is always in the same position, and o 
the same amount; and where, consequently, the strain on any given member i 
constant. But in many cases, as in railroad bridges, the amount, and mannero: 
distribution, of the load, change greatly from time to time, so that the st rain on 
given member may vary greatly in amount, and may even change from tensio 
to compression or vice versa. In the present article we investigate these change 
in the strains, caused by changes in the position of the load. 

To simplify the subject, we regard the weight of the locomotive as being vn\ 
forml \/ distributed over its length, as shown in Fig 13/; and this suffices to illus 
trate the principle. But specifications frequently call for calculations base 
upon the actual load on each pair of wheels of an actual or assumed engine am 
tender, as shown on p 546. 

For simplicity, also, we here seek onty the maximvm strain on each membei 
in order todetermine where counterbraces are required. In practice it is necessary t 
note not only the maximum but also the minimum strain on each member; fo 
when the strain on a member is subject, to great and frequent variations, esp< 
ciallv where it changes from tension to compression or vice versa, the ultima' 
strength of the member is reduced to an important extent. See p 435. 

'Y® Y 1 - 11 calcillate t,ie maximum strains in a bridge Fig 13/of 120 ft span m: 
divided into 6 panels, each 20 ft long and 30 ft high. For convenience of calct 
hit ion we will suppose the fig to represent the two trusses of such a bridge con. 
omed into one. We first calculate the max strain in each member of this sut 
posed double truss, and then divide each strain by 2 for the actual strain in th 
corresponding member of each actual truss. 

The ght of this double truss is 48 tons; or 8 tons per panel; or .4 ton pe 
inn. The floor, and its several timbers, such as cross girders, hor bracini 
«Sc, which, although not portions of the truss proper , are essential to the bridge 
and to be considered as so much permanent load, equally distributed along th 
truss, are assumed to weigh an additional 24 tons; or4 tons per panel; or .2 to 
per It run of the double t russ. Therefore, the truss and its accompaniments t< 
gether, or in other words, the bridge superstructure, weighs 72 tons; or 12 tor 
per panel; or .6 of a ton per ft run. Since each panel is 20 ft long, and 30 ; 


high, each oblique is 1/20* + 302 - 36 ft long: and the secant of the angle whic 
it fotms with a vert, (or of the brace angle,) is = length of oblique -t its vei 
spread — 36 -j- 30 — 1.2; and the nat tang of the same angle is = hor sprea 
vert spread = 20 30 = .6667. b 4 






















TRUSSES. 


565 


The moving load is assumed to consist of an engine, yz, weighing 36 tons; all ot 
lich is supposed to be so concentrated on its drivers, as to stand upon the length 
one panel. The engine is supposed to be fob 

>ved by a tender and cars, weighing 21 tons per panel; or 1.05 ton per ft run. 

IThe greater danger from engines, arises from the fact that in many 
them, especially in heavy freight engines, all the wt is concentrated upon their 
iving wheels; which are so close together that a 30 or 40 ton engine on 8 drivers, 
11 often have its entire wt sustained upon only 12 to 15 feet length of the truss; 
us bringing a great strain upon each individual web-member, as it crosses the 
idge. This point is one of the greatest importance in arranging the preliminary 
ta on which to form the basis for calculating the strains on a bridge truss; and we 
ould allow liberally for the greatest load that can possibly be brought upon one 
nel at a time. When the panels are quite short, each y>ne will have of course 
sustain only a smaller portion of the wt of engine; it may, indeed, have to be di- 
led among 2, 3, or 4 of them. 

The fact that the engine produces upon each panel in succession, a greater strain 
an an equal length of loaded cars can do, causes a modification ot the calcula¬ 
tes; as will be seen as we proceed. 

Such data must be prepared before beginning the calculations. Our assumptions 
the case before us have been made entirely with reference to ease of calculating 
r example with but few figures. Our truss, together with a load of cars extending 
>m end to end, will weigh 198 tons; or 1.65 tons per ft run ; or 33 tons per panel. 
Having prepared a diagram, begin by finding the strains caused in only the verts 
d obliques, by the bridge itself, half the wt of the truss alone being considered 
be on the top chord, while all of the weight (24 tons) of floor rests of course 
•ectly on the lower chord. 

By our “ first process,” p 559, there will then be 6 tons each at h, e , and c ; 3 tons 
h at i and a ; 8 tons each at n, n, p, q, and r ; and 4 tons each at in and x. By our 
j s ;ond and third processes, the strains on the web members will be as tollows: 

[if 


On ep . 

“ h o and c q, 
“in and a r 
“ e o and e q 
“ hn and c r 
“ i m and a x 


each. 


.. 8 

.15 

.29 

. 8.4 

.25.2 

.38.4 


tons. 


Sprite these strains at th e feet of the respective members. 

!if*Ve now have the strains on the web members, resulting from the truss itself,and 
m its uniformly distributed permanent load of floor, &c; and are prepared to 
jin with the additional strains resulting from the moving load. We find thoseon 
4’ counters, on one-half of the diagram ; and those on the main obliques and ver- 
foi *is, on the other half. First suppose the engine to be in the positiou y z , on the 
If diagram p x; with the train reaching to the farthest end m of the truss. In 
s position of the moving load, the vertical a r, and the end oblique a x, will be 
,re strained by it than in any other; and their strains will be produced by that 
u, tion of the load that may be regarded as being first concentrated at r; and as 
ising thence up r a to a ; and from a, down a x to the abutment x. To find this 
rtion, the truss may be considered as a simple beam or lever, marked off into 
ip eoual divisions at r, q, p, o, n. In that case we know (from the principle of the 



ait-sixth by the farthest abutment m. So also with the panel length y w of cars; 
y. weight being supposed concentrated at q , four-sixths of it must be borne bj x; 
it 1 two-sixths of it by w. Of the cars w v, three-sixths are borne by x ; and tluee- 
.01 ths by m. Of those along v u, two-sixths are borne by x ; and four-sixths by m ; 
to these along u s, one-sixth by x\ and five-sixths by m. The half panel ot cars, 
tint rests directly upon x, and produces no strains in the truss. 

)f jy this process, then, we find that five-sixths of the engine,or 30 tons; four-sixths 
Hthe cars y w, or 14 tons; three-sixths of the care to r, or 10.5 tons; two-sixths 
L v u, or 7 tons ; and one-sixth of u s, or 3.5 tons; making 65 tons in al 1, are borne 
^ the abutment x } when the moving load is in the position shown in our fig. I his 











560 


TRUSSES. 


load of 65 tons passes from r toa; and from a to x* The calculation is made 
shown under our fig below the heading “ With engine at r.” 

We next suppose the engine to hack into the position y w ; with the train reachii 
to to; and with no load between y and x. In this position the strains on qc, and c 
are greater than in any other; and by a process precisely as before, and as sliov 
under our fig, below the heading “ Engine at 7 ,” we find that 45 tons of the movii 
load go to the abut x ; passing from 7 to c; and from c to r, on their way to it. Tlx 
placing the engine at w r, the cars reaching to to, and no load between w and x, t> 
strains on p e and e 7 , are the maximum that they can be subjected to by aD engi 
and train ; and are found like the others, as shown uuder “ Engine at p .” We ha 
now reached the center of the truss, and have obtained data for the greatest strai 
that can occur on the verts and main obliques on one-half of it. So far as regar 
the corresponding members on the other half, we stop here ; for it is manifest th 
an engine and train, crossing the bridge in the opposite direction, must produce t! 
same maximum strains upon them as we have already found for the others. That 
» to will be strained the same as ax; ni the same as r a ; eo the same as e q, and so o 
by a returning train. Write these strains near the tops of the verticals. 

But as yet, we have not sufficient data for determining where counters will 1 
required. To obtain it, we draw the two lines hp, and i o, parallel to the oldiqu 
on the opposite side of the truss; and taking it for granted for the present, that the 
two lines are counters, we back the engine to v u, and then to ms; performing ea< 
time, calculations precisely similar to the former ones: and as shown under the hea 
ings Kngine at &c. We have stated in a former Art, that a counter is nm 
strained when the moving load extends from it to the nearest, abut. Therefore 
counter tip would be most strained with the engine on v u, and the train reaclm 
to TO; and 1 o, by the engine on u s, with a train to to. 


1? 


* With the engine at r, as in the Fig, the counters t o and hp are not require 
Consider them removed, and it will be seen that the several sixths of the panel loa< 
reach-their respective abutments by the following routes: 

Of m s, one-sixth to x, via nhoeqcrax; five sixths to m, via n i to. Of v 
two-sixths to x, via oeqcrax; four-sixths to to, via ohnim. Of w v thre 
sixths to x, via e q c r a x; three-sixths to m, via p e 0 h n i to. Of yw four-sixtl 
to x, via qcrax; two-sixths to to, via 7 e o h n i to. Of the engine z y five-sixtl 
to x, via rax; one-sixth to to, via rcqeohnim. 

The one sixth of engine going to to, in passing from e. to to, and the five-sixths g 
ing to x, plainly bring upon each member the same kind of strain as that broinrl 
upon it by the weight of truss and floor; i. e., tension upon the ties and ctrmprejsu. 
upon the struts; and thus increase their strains; but in passing from rtoe the on 
sixth going to to pulls the struts c r and e 7 , and presses the tie c 7 , thus diminishit 
the strains upon those members. 

And, similarly, in any position of the engine and train, each panel length of mo 
ing load, except w v, divides at its point of support; one portion passing up the ve 
tical tie (and increasing the tension upon it) on its way to one abutment • and t I 
other portion passing up the inclined strut (and diminishing the thrust upon it) ■ 
its way to the other abutment. ^ 1 ll) 

Caution. When a web member is subjected to its maximum strain no sue 
relieving strain can thus come to it from the moving load. tLs c 7 has its nnx 

W n n the en S ,n « is at ?> with train reaching to to ; and the 
°7 d at ?, t0 / e i e y e c q ,y 8en,, 'ng a compressive strain through it. 
anv mmnllryry«fv”V tnUn ( ) v ! ,ich ma V or may not be the maximum strain) upt 

mid compression uion iut P th ! 10 " ° f the tpftin » ia the difference bet ween the tonsil 
am compression upon it at that moment; and is of course a pull if the tension 
greater than the compression, and vice versa. 1 tension 

Thus, with engine ut r, as in the figure, the final strain on the tie a r is ^ 

, «J„“Ti og o. Co *™' 

floor, as per p 060 ^ ing to a ._ ttS al)OV6 0 w 94 tons, 

gine^tiiut^ris - 6 maximum 8trai “ on a r - B,lt c the final strain (with e 

Tension, Compression. 

I ive-sixths of en-v % of engine going to 1 


Tension, 

15 tons from 
bridge and floor 


+ 


( 


65 tons 
going to x 

T 
15 




gine which do not 
pass through c q 

T C T 

35 — 6 = 44 tons 


which presses c q, ai 
thus relieves its tensio 


sb 15 -j- 35 — 6 = 44 tons. 

But this is not the maximum strain one?; for this takes place as alreidv 
With engine at 7 , and amounts to 45 tons. P ,M Alreadj remdrke 













TRUSSES. 


567 



With Engine at p, 

there go to x. 

Tons. 

18 = 3-6 engine. 

7 = 2-6 v u. 

3.5 = 1-6 u s. 

28.50 go to x. 

1.2 nat sec. 

34.2 on e q 

from, load only. 


MAIN BRACES. 

With Engine at q, 

there go to x. 

Tons. 

24 = 4-6 engine. 

10.5 = 3-6 w v. 

7 = 2-6 v u. 

3.5 = 1-6 u s. 


45. go to x. 

1.2 nat sec. 


54. on c r 

from load only. 


With Engine at r, 

there go to x. 


Tons. 


30 

= 5-6 engit 

14 

= 4-6 y vv. 

10.5 

= 3-6 w v. 

7 

= 2-6 v u. 

3.5 

= 1-6 u 8. 

65. 

go to x. 

1.2 nat sec. 

78. 

on a x 


from load only. 


Vertical e p. 

28.5 go to x. 

tal on e p from load only. 


VERTICALS. 

Vertical c q. 

45 go to x. 

Total on c q from load only. 

COUNTERS. 


Vertical, a r. 

65 go to x. 

Total on a r from load only. 


Counter i o. 


Counter h p. 


Engine at n. 


Engine at o. 


6 = 1-6 engine go to x. 

1.2 nat sec. 


12 = 2-6 engine. 

3.5 = 1-6 u 8. 


7.2 on i o 
from load only. 


i 


15.5 go to x. 

1.2 nat sec. 

18.6 on li p 
from load only. 


' Supposing the calculations to have been made as above, we have thereby found 
i aeveral loads, say 65; 45; 28.5; 15.5; and 6; which go from train to x, from the 


























568 


TRUSSES. 


several points r, q, p, o, n, when the engine is at each of those points in successicf 
with a train reaching in each case to rn. Each of these loads, in travelling, as it we 
from its particular point, to the abut x, first ascends to the top of its own vert; o 
then descends along the adjacent oblique; producing in said oblique a strain as mu 
greater than the load itself, as the length of the oblique is greater than its v 
spread. Now, each of our obliques, as before stated, is 36 ft long, or 1.2 times as lo 
as its vert spread 30 ft, (or the height of truss;) which 1.2 is therefore the nat sec 
the angle which any of our obliques forms with a vert line. Therefore, to find t 
strain which each of our moving loads produces on the oblique next to it, on its w j 
to the abut x, mult each said load by 1.2, as in the above calculations. We thus c 
tain for these strains, 78 tons on ax; 54 on c r; and 34.2on eg; all which write nt 
the tops of the obliques. 

For the verticals and main obliques; Total max strain = truss stn 
+ moving-load strain. Thus: 


Verticals 


ep 8 + 28.5 = 36.5 tons 

ho, cq, each 15 -j- 45 = 60 “ 

in, ar, “ 29 + 65 = 94 “ 


llain obliques 


eo, eg, each 8.4 + 34.2 
h n, c r, “ 25.2 + 54 

irn, ax, “ 38.4 + 78 


= 42.6 tons 
= 79.2 “ 
116.4 “ 


For tile counters, we go to the other side,p m, of the truss. Beginni 
with the panel eh op, we examine its twodiags, hp, eo. and see that with theengi 
at v u, and the cars reaching tow, there is produced in the counter hp its inaxirm 
strain of 18.6 tons; which tends lo cause in the truss the kind of derangement shoi 
in Figs 9%. 9%, and 9 6. Now, to resist this derangement, there is nothing but t. 
8.4 tons produced by the truss, floor, Ac, upon the opposite diag e o of the same pan 
bince, therefore, the deranging effect of the load is greater than the preventive eflfi 
ot the weight of the truss, there must be a counter at h p, able to bear a pres eqi 
at least to the difference between the two, or to 18.6 — 8.4 = 10.2 tons; or else t 
strut c o must be made so that it can also act as a tie, capable of sustaining safeh 
pull of 10— tons. This last method relieves h o and ep from 10.2 tons each,hut tl 
relief comes w r hen they have not their maximum load. 

e now go to the next panel, h i n o. But here we find that the deranging eflft 
of the load on the counter i n, with the engine at u s , is but 7.2 tons; while the di 
veutive effect of the wt of the truss, exerted through the opposite diag h n, is 21 
tons. Hence, the moving load can produce no derangement of the truss; and c< 
sequently the counter i o may be omitted. On this same principle each panel on o 
side of the truss must be examined, when there are many of them ; and the inserti 
or omission of counterbraces be determined upon. When we thus arrive at a pat 
at which no counter is reqd, none will be needed between it and the nearest end 
the budge. Similar counters will, of course, be needed on tlie other side of the tru 
In piactice it is better to retain the first apparently unnecessary counterbraci 
counting from the center of the truss. Thus, although calculation shows i o to 
unnecessary it is well to retain it. The lighter a bridge is, in proportion to 
moving load, the greater will be the number ol panels requiring counters. 

The strains on the chords are greatest when the truss is load 
from end to end; and for Fig 13/, as well as for Fig 13g, mav readily he calc 
lated by Art 12; or iound by a diagram with a max load. Or sufficiently close 1 
most purposes, the hor strain along either chord at the center of the truss wlie 
the strum is greatest, will be equal to 

Total weight of truss and load X span. 


8 times the depth or height of truss 
198 X 120 23760 


which here is = 


99 tons. 


8 X 30 240 

Finally, each of all the strains in our double truss must be div by 2 ; for prop< 
tioniiig them among the tw’o actual trusses, which we have all along supposed (; 
convenience) to be combined into one. 

Art 12%. The Warren or triangular truss. Fig. 13 g. 

Here the dotted web-members which supplant the ties in Fie 13 f are r 

inclined to HlA Rum a pvfon f Q a tla/v 1_ rr . ^ ’ 


th , e 8trut ? or braces. Hence the hor stret 


c i 11--~rr : .—- att um ui uruceH. nence the hor stivt 

of each oblique will be but half the length of a panel, or 10 ft; or only half as great as 

l ( >g 13/. Consequently, the length of each oblique will be V 10 a + bO’ = 31 6 


and the nat sec of the brace angle will be ~= 1.05; and the nat tang of the sai 


10 


angle will be — = .333. 
30 


All the other data being the same as in the foregoing < 













569 



TRUSSES, 


niple, prepare a diagram, and from it find the strains on the obliques from the 
t of the bridge alone; one-half of the wt of the 

d c b a 


■mss only being supposed to be on the top chord, making, by our first process, p 559, 
8 tons each at e, d, c, and b ; 2.4 tons each at t and a ; 8 tons each at n, o, p , q, and r; 
»ad 4 tons each at to and x. By our second and third processes, the straius on the 
eb members will be as follows: 


On p d and p c, each. 4.2 tons. 

“ do and c q, “ . 9.24 “ 

“ o e and q b, “ .17.64 “ 


On 


e n and b r, each.22.68 tons. 

n i and r a , “ .31.08 “ 

i m and a x, “ .33.6 “ 


f rite these strains at th e feet of the respective members, as before. 

Having now the correct strains arising from the weight of the truss and floor: 
ext fiud, precisely as in the preceding example, how much of the moving load will 
0 to x when the engine is at r, as in the fig, with the train reaching to m ; and 
terward with the engine at q, p, o , and n, in succession. These loads will of course 
3 the same as in the former example, namely: 

Engine at n. Engine at o. Engine at p. Engine at q. Engine at r. 

6 go to x. 15.5 go to x. 28.5 go to x. 45 go to x. 65 go to x. 

Multiplying each of these by the nat sec 1.05, we get the compressing strains which 
ley produce on the end oblique strut a x\ and on the other obliques that are par- 
'Id to it; namely: 

On e o. On d p. On c q, 

6.3 16.275 29.925 

liich write near the tops of said obliques, 
une amount of strains, in the shape of tensions or pulls, on the respective dotted 
es which carry them to the struts between x and p ; and on the struts which carry 
'lorn to the dotted ties between to and p. Write them all near the heads of said 
bliques also.* 

Now, if on the half truss x p, we add together the strains written at the head and 
■ot of each oblique separately, the sums will be the total or maximum strain (com- 
ressive on the struts, and tensile on the ties) which said obliques will be subjected 
> on the passage of the train. They will of course be the same on the other half of 
be truss when the train crosses in the opposite direction. 

Finally, as to coiinterbracinsr, we go to the other half, m p, of the 
•uss. Beginning with the oblique d p, we see that with the engine at o, and 


On a x. 

68.25 


On b r. 

47.25 

But they also produce precisely the 


* The remarks in foot-note p 566 apply also, in principle, to Fig 13g. Thus, with 
le engine atr,as in the fig, the five-sixths of engine, going to a;, increase the strains 
a r a and ax. The one-sixth, going to to, diminishes the strains in rb,bq,qc, and 
p (but these members have not now their maximum strains), and increases those 
l p d, d o, o e, e n, n i, and i to. 

The final strains, also, are found in the same way as in Fig 13f. Thus, with engine 
t q, train reaching to to, we have final strain on q b (maximum) 

Tension. Tension. Compression. Tension. 

= 17.64 from truss and floor + 45 going to x X 1.05 = 47.25 — 0 = 64.89; 


nd on q c (not maximum) 

Compression. 

= 9.24 from truss and floor 

C 

= 9.24 + 


+ 


Compression. 

[(45 to x — % engine) X 1-65] 

C T 

22.05 — 12.60 = 


Tension. 

— ($ engine X 1-05) 
C 

18.69 tons. 
















570 


TRUSSES. 


the train reaching to m, the deranging compressive strain, 16.3 tons, of the mov 
ing load, is greater than the preservative tensile strain, 4.2 tons, of the trus 
and floor, acting on it at the same time. Therefore, d p, although a tie, the same a 
cp, is liable at times to be compressed rather than pulled. Therefore, it must be s> 
arranged as to act also as a strut; at least so far as to bear a pressure equal to th 
difference between tlie 16.3 tons of pressure from the load and the 4.2 tons of tensioi 
from the weight of the truss and floor; or to 12.1 tons. The same end would be ac 
complisbed by inserting a counter tie reaching from n to c. 

On the next oblique o d , which is a strut, the same as c q, the moving load on v % 
produces a pull of 16.3; while the truss and floor produce on it a pres of ouly 9. 
at the same time. Therefore, although it is a strut, it is liable at times to be pullc 
rather than compressed; and consequently it must be made able to bear a pull alst 
equal at least to 16.3 — 9.2 = 7.1 tons. The introduction of a counter strut reachin 
from e to p would answer the same purpose. On the tie e n, the deranging compref 
sivs load strain 6.3 is less than the preservative tensile strain 17.6 of the truss an 
fluor acting upon it at the same time. Therefore, it may remain as a tie only ; or ii 
other words, it requires no counterbracing. When this is the case, no other oldiqu 
between it and the nearest abutment rn needs counterbracing. It is almost needles 
to remark, that the half xp of the truss requires the same as the half mp, when th 
engine crosses in the opposite direction. 

The strains oil the chords are found as directed on p 568. 


Fig. 14 



Span 64 ft. 
Rise 16 ft. 


Art. 13. In Fig 14, as in Figs 10 and 11, the truss is of 64 ft span and 16 ft rise 
with an extraneous loadol 32 tons; of which our first process gives as before 
2 tons each at a and at the other end of the span, and 4 tons at each other panel poin 
b, c, d, e, / etc. The wt of the truss itself is neglected, as before 
Second and third processes.* Each panel load divides at its point oi 
support as explained more fully in Rem, p 573, one portion, r S "' etc, going cflrectl 
dowm the rafter to the nearest abut; the other, b s'" etc, by way of the web member 
*^e peak e of the truss, and thence via the rafters to the abuts. 

1 bus, beginning at the panel load b, nearest the abut, layoff by scale the vei 
br = that panel load 4 tons. Draw rh"> parallel to the rafter, and h"'s'" horizontal 
or parallel to the tie beam. Measure s'" r, which represents the portion (2 tons) of l 
going directly to a via the rafter, and write its amount (2) at a. Also m asura W'' 
the portion (2 tons) of b going toe via b kcj die, and write its amount Sc ovei 
the onginal panel-load ?4) of c; thus making the final load at c = 6 tons. Mak< 
c r this load, draw r h parallel to the rafter, and h" s" hor. Then s"r ( — 2 tons 
goes from c to a via the rafter; and c s" < = 4 tons) goes to e via c i di l wiTra X™ 

makeTr-glons^ fina] load at d th ’ ]S becomes 8 tons Therefor! 

make dr — 8 tons draw r h parallel to the rafter, and h' s’ hor. Then s' r (2 tons 

goes from d to a via the rafter; and d s' (6 tons) to / via d i e. Write tSese amount 


* ^VTienvFishing to know-merely the amounts of the several final panel loads in i 
truss like Fig 14, uniformly loaded and divided into panels of equal length : without car 
ing to trace the process of their accumulation from the original mi el loads • om 

second process may be shortened, thus: ’ 


original panel load at abut a 


M the entire wt of truss and load 


entire number of points of support a b c de /etc minus, 

atT'-C- M g «- i 37%i PI, i el k * d H ,t< ‘ “■e.fmal panel load, aro a. follow, 
at o 2 a at c — 3 a at d= 4 a ; and so on with any number of points of surnim- 

along a rafter, except at the apex e, where the final load is twice the final load a 
the nearest panel point (d in our Fig) or = lg the wt nf the “nai ioaa a 

The final load at « = half the weight of entire truss and load. 1 ,USS 110(1 ltS ° ad 






















TRUSSES. 


571 


at a and e respectively. Now it is plain that the same process, on the other half of 
the truss, brings another 6 tons to e. Write this at e also. Thus we get for the final 

15 load at e, 4 + 6 + 6 = 16 tons, or % the wt of the entire truss and its load. 

The 16 tons thus concentrated at e divide there, half going down each rafter to an 

16 abut. Draw er vert, and = these 16 tons. Draw rh parallel to ae, and As hor. 
111 Then e s and s r are the 2 halves (8 tons each) of ei, passing from e to the abuts via 
c- the rafters. Therefore write 8 at a. 

It is by mere accident that the two vertical lines b l and er, representing the loads 
* at b and e, happen just to extend to the tie a i in our Fig. 

We thus find, for Fig 14, 


Strains along* the verticals. 

W Along 6 l~ nothing, except weigbt of tie-bar 
from y to y. 

“ c k,~b « — 2 tons. 

“ dj, =c t = 4. 


Strains along the obliques. 

H I 

Along blt~b h~ 4.47 tons. 

“ cj—c A — 5.66. 

“ di=d A = 7.21. 


ei — 2d 8 — 2X6 — 12; for while each other vertical tie bears only the vert strain brought upon it by 
the oblique strut next nearer the abut; the center tie e i, of course bears those from 2 obliques: 
one on each side of it. 

Strains along the horizontal tie-bar a i. 

y 2 wt of truss and load X V\ span 


At i = s A = 16 tons; also 


height of truss. 


Fromj to i = s A + 5 A — 16 + 4 = 20. 

/ “A to j — sh-\-sh-\-s 4 = 16 + 4 + 4 = 24. 

r r r r r r r r r r r r 

“ a to k — jA + s A + i A+s A = 16+ 4 + 4 + 4 = 28. 

Strains along the rafter e a. 

From e to d = A r = 17.9 tons. 


“ dtoc-=hr + hr— 17.9 + 4.47 = 22.4. 

“ cto6 = Ar + Ar + Ar = 17.9 + 4.47 + 4.47 = 26.8. 

“ 6 to a = A r + A r + A r = A r = 17.9 + 4.47 + 4.47 + 4.47 = 31.3. 

It will be observed that the hor components A s, except the center one, have equal 
lengths; also those marked A r, parallel to the rafter; while the oblique ones have 
*iot. 

For a span of 100 feet, rise 20 ft, or ^ of the span; trusses 10 feet apart 
from center to center; loaded on top only; the following dimensions will be amply 
sufficient for a covering of slate. Rafters and tie-beam, each 10" X 12" deep. The 
rafters may, if preferred, be reduced to 9 X 12 at top. The verts of round bar-iron 
of good quality, % inch, %inch, 1 inch, and 1% inches diameter. The obliques or 
braces, 5 X 10, 6 X 19, 8 X 10; thus making the truss-thickness uniformly 10". See 
Table 2, p. 579. For shorter spans, see NOTE, p 573. 

Art. 14. In Fig 15, the process is the same as in Fig 14, except that the vert line? 
representing the strains at the points of support a, b, c, d, e, are to be drawn upward 
from l, Jc, j, i; and the strains l s'", k s", j s', are to be carried forward to the next 

8 




















572 


TRUSSES. 


panel-load. Fig 16 is simply Fig 14 inverted, and those members which resisted 
pressure in Fig 14, resist pull in Fig 15, and vice versa. In other words, the struts 
become ties, and the ties, struts. All the strains are equal to those in Fig 14, except 
those ou the verts, each of which is 4 tons greater than the corresjxmding one in Fig 
14, because the original panel loads of 4 tons each, instead of being applied directly 
to the ends of the obliques, as in Fig 14, have first to pass through the vert struts 
b I, c k, dj, e i; the total loads on which will be, respectively, 4, 6 , 8 , and 16 tons. 


/9G\ 
^ 64.6 \ 


Ji¬ 



ll) 


vert, strain at e 25.6 tons, instead of the 16 tons of Fig 14.* 

• ’ n !; lke e 7 '''- v sca * e > = 25.6 tons ; draw r h parallel to the rafter e a, and meet¬ 

ing the other rafter; also draw h s, parallel to the raised tie-bar i a. Then the strains 
along the members of the truss will be as follows, taken from a Fig on a larger scale. 

Strains along- the verticals. 


•Along the one at 6 = nothing, except weight of 
tie-bar for the width of 
one panel (8 ft.) 

Along c 0 = 6 s'" = 2 tons. 

“ d z = cis" ~ i •< 

“ ef ='2 da' -+-9.6 = 21 . 6 . 


Strains along- tlie obliques. 

Along 60 = bli'" ~6.43. 

“ rz = ch" =6.96. 

“ di = dh' =8.04. 


From z 
“ o 
“ a 


Strains along tlie raised tie-bar ai. 

At i = As = 26 tons, 
to t = hs -+- h ' s' = 26 -f 6.5 = 32.5. 
to* = hs 4- h ' s ' -J- h" s" = 26 4 - 6.5 4 
too = hs h ' s' h ” s " 4- h "' s '" =: 


4- 6.5 = 39. 

: 26 -f- 6.5 4 - 6.5 4- 6.5 = 46.S. 

Strains along the rafter ea. 

From d to e — hr = 28.5 tons, 
e to cl = 

6 to c 

o to 6 = nr -t- H r -4- h 'r- 4- h"'r=z 2 K.fi -U J is _l_ 7 13 7.13 = 49 9 


= h r = 28.5 tons. 

= hr A' r = 28.5 T.13 = 35.63. 

~ h r -f h' r h" r = 28.5 -f 7.13 4 - 7.13 = 42 76. 
= ,lr + A'r + A” r-f A"'r = 28.5 4-7.13 4-7.13 


* 11 J s probable that the tie rod is sometimes raised in this manner by nersous ienorant r,f n-o r 
that they thereby greatly increase the strains on the rafters, &c P ignorant of the fact 

All the strains in Figs 14, 15. and 16 may also be found bv me- 

cisely the same process as that for bowstring and crescent trusses in Art 19 J 1 

The tension in the tie-rod which brings the 9.6 tons additional load to e, causes at the same time 
an equal upwaid pull at a. Hence the final pres of the truss unon cuch ahut romnirw « ♦ 

(= half the weight of truss and load) as in Fig 14 P abut rema,ns 8 10118 


Art. 15. In Fig 16, the process is the same as in Fig 14, except that the lines h s, 
«c, must be drawn and measured parallel to the inclined tie a i : instead of being hor. 
As in Fig 14 b s'" is carried forward to c; c s" tod; d s’ to e. In this way, we find 
as befoie, the vert strain of 6 4~ 4 4 - 6 = 16 tons at e. But we must now add to 
these 16 tons, another strain generated by the obliquity of the tie-rod a i. This strain 
is found by mult the one at e, (16 tons, or halt the wt of the entire truss and its 
load,) by the vert dist nt, (6 ft,) which the center i of the tie-rod is raised above 
the horizontal^ a u, and div the prod by the dist i e, (10 ft.) 

ihat is, — = 9.6 tons; which also write down as in the Fig; making the total 














TRUSSES 


573 


Rem. The reason for measuring only parts of the vert lines which, in Figs 14, 15, 
16, represent the whole panel-loads, is that the rafter ae, Figs 14 and 16; or the tie 
a t of Fig 15, being incline.d, also bear a part of each panel-load ; and since that part 
does not go forward to the next point of support, but goes backward, along said in¬ 
clined member, to the abut at 
cl Jv 1) a, it must be omitted in the 

second process. Thus, in Fig 
17, if ba be an inclined rafter 
resting on an abut a; bg a 
strut; and ft r a vert line repre¬ 
senting the load sustained at 
ft by ft a and bg; if we com¬ 
plete the parallelogram bmrn 
of forces, then will ftm give 
by scale the strain along the 
^*4 strut; and bn that along the 
■— rafter. The strain along the 
& strut is made up or composed 
of the portion ft s of vert force; 
and the hor force sm. The 
vert portion bs alone goes to the next point of support; while s m strains the tie ag 
hor. So also the strain bn is made up of the other portion (bo or sr) of the vert 
force ftr; and of the hor force on; which, when the strain bn reaches a, become 
again resolved into two; one of which, bo, presses vert upon the abut; or, in other 
words, transfers to the abut the portion bo of the load resting on ft; while the por¬ 
tion on, which is equal to sm, strains the tie ag hor. 

But in Fig 18, where ft r also represents a load resting on ft, and supported by a 
strut ft g, and by a hor chord ft a, if we complete the parallelogram ft m r n, we have 
the strain ft to along the strut, composed of all the vert force ft r, aud the hor force 
r m. The whole of ft r is transferred to the next point of support; while r m aud ft n 
produce only hor strains along 6 a and g y. 



Art. 16. The roof truss, 
trusses, an e and e k l. Fig 21. 


Figs 19, 20, and 21, consists of two complete Fink 
It is supposed to be of the same span (64 l't) and lit 


NOTE. 

The following' may at times save trouble in designing roof 
trusses. After the dimensions of all the members of a roof truss of any span 
have been calculated, then those of any smaller span similarly arranged, and having the same rise 
! in proportion to its span, and the same extraneous load per sq ft; but with the trusses at the same 
dist apart as in the large span ; may be found safely ; and often near enough for practice, thus : 

Find the area of cross section of each member of the large truss, in sq ins. Theu make the area 
of cross section of each member of the small truss less than that of the corresponding member of the 
large truss in proportion as its span is smaller. The small truss thus obtained will be stronger than 
the large one.under the same extraneous load per sq ft. For instance, suppose a truss of 175 ft span 
has been designed to carry safely a total load (t e including its own \vt) of 40 lbs per sq ft of ground 
covered. Such a truss, by table, p 580, will weigh 8.05 lbs per sq ft of ground covered, leaving 
40 — 8.05 = 31.95 lbs per sq ft as its safe extraneous load of purlins, slate, snow, &c. Now a truss 
of one-fifth this span, or 35 ft, proportioned by the above rule, would sustain safely one-fifth the 
same total load; or (with trusses at same dist apart iu both cases) would sustain the same total load 
(40 lbs) per sq ft. But the weight of the small truss itself, per sq ft of ground covered, is only one- 
fifth of that of the larger one, or 1.61 tbs, leaving 40 — 1.61 = 38 39 lbs per sq ft as its safe extraneous 
load, while that of the larger one, as shown above, is only 31.95 lbs. Reductions will, however, 
rarely be made to as small as one-fifth ; and where the short span is not less than half the long one, 
the method will answer very well in practice. For examples of reducing, see p 581. 

Witli the same total load per sq ft (including the wt of the 
truss itself) and with trusses at the same dist apart in all cases, the strains on 
the several members of trusses proportioned by the above rule, will be in the same proportion as the 
spans; as will also the areas of cross section and weights per ft run, of each member, the wt of the 
truss alone, per ft of span and per s j ft of ground covered, and the total load on a truss, including 
its own wt; but the total wts of the trusses alone will be as the squares of the spans. 
















TRUSSES. 


(16 ft) as Figs 10, 11, 14, 15, 16; and to have the same number <9) of points of suf 
port for the weight (32 tons,) supposed to be uniformly distributed along its to] 
Consequently from our first process there will, as before, be 2 tons of pane 
load at each of the end supports; and 4 tons at each of the others. Write the* 


V 

9 

a 

o 


Scale oj tons J~or strail 

_ 


ns 


ltisc 16 ft. N 


FINK TRUSS 


tc-j> 

> 


H • 

O / 

$ 





































TRUSSES. 


575 


down as in Fig 20.* The part truss ex a maybe regarded as being composed of 
three separate trusses ex a, eg c, cm a; as will appear more plainly from ena, etc, 
and co a, in Fig 21. These may be called first and second secondary trusses. In Fig 
19, the half eyp exhibits a truss on the same principle, but having a greater num¬ 
ber of points of support for the uniform wt. That half truss consists of first, second, 
and third secondaries, as shown by e y p, eg i, and esw. However far this subdi¬ 
vision may be carried, if the struts occur at equal dists, each of the smallest divisions, 
as co «, Fig 21, is to be regarded as a complete truss in itself, loaded at its center 
only. One-lialf, therefore, of its wt i#ust rest upon each of its supports a and c. 

Thus, with our second process at one of the shortest struts, as b m, Fig 20, 
2 of its 4 tons go to c at the next longer strut, cx; and 2 of them to the end a of 
the rafter, as written on the Fig. Then, at the other of the shortest struts, d g, 2 
tons go to c; and 2 to the end e of the rafter. 

We will suppose, for the present, that the end e, and the corresponding end of the 
other half truss e k l. Fig 21, rest upon an abutment at e, as a rests upon its abut. 

We thus have 4 tons at b ; 4 at d ; and 8 at c. 

In like manner we now regard the first secondary truss axe (see ane, Fig 21) as 
loaded at its center, c, only, with 8 tons as just explained. Of these 8 tons, 4 are of 
course supported by the abut at a, and 4 by the similar abut supposed, tor the pres¬ 
ent, to be at e; both of these 4 tons are therefore set down as at a and e. When 
there are more points of support, as along the rafter ep. Fig 19, the process is pre 
cisely the same: we first adjust the strains of the four third secondaries, esw, wri, 
i v k, kup; placing them at e,w,i, k, and p: then we transfer those thus accumulated 
at w and k, to e, i, and p ; and finally transfer them from i to e and p, at the ends of 
the rafter. Now, returning to Fig 20, we see that in addition to the original panel¬ 
load of 4 tons at e, we have accumulated 6 tons of vert strain from the other panel- 
i loads; and it is plain that the same process, performed along the other half truss 
i e k I, Fig 21, would bring 2 + 4 = 6 tons more to e, as written in Fig 20. Thus it 
appears that we have 16 tons in all ate;f resting upon our supposed abut there. 
, But as this abut has no existence, the 16 tons really come upon the rafters them¬ 
selves at e, and of course half of this wt (or 8 tons) is transferred by a rafter to each 
abut. Write down the 8 tons at a as in Fig 20. We thus have for the total vert 
pressure at a, 2 + 2 + 4 + 8 = 16 tons = half the total wt of truss and load, and 
this is a further proof of the correctness of the operation. 

Having thus finished our second process of finding the additional strains at the 
several points of support produced by the original panel-loads on their way to 
the abutments, we have only by our third process, to complete the drawing, 
so that we may measure by scale the strains along all the members of the truss. To 
i do this, from the tops of the struts draw vert lines bv, cv,dv to represent the total 
vert strains accumulated at those respective points ; namely, 4 tons at b, 4 at d, 8 at 
c. Draw vo,vo,vo parallel to the rafter a e. Then ho, co, do will give the strains 
along the struts ; 3.6 tons on b m or d g ; and 7.2 tons on c x ; and vo,vo,vo will 
give those produced directly by the final panel-loads from their points of support, 
upon the lower halves of the rafters of their respective secondary trusses. Thus the 
i 4 tons at d exert a pres there of 1.77 tons (as shown by the upper no) upon the lower 
| half rfcof e c. The 4 tons at b exert an equal pres upon b a, and the 8 tons at c 
strain ca 3.54 tons. These strains v o of course become proportionally greater, and 
[ bo, co, do proportionally less, as the rise of the truss increases in proportion to the 
span. They will be referred to further on. 

Lay off m t, x i, g i respectively equal to b o, c o, do; draw ij, ij, ij ; and i y, i y, 
i y , parallel to the ties; thus completing the parallelograms of forces ijrny, ijxy, 
ij'gy. Draw the diags yj, yj, yj ; and the vert lines mu,xu,gu. Lay off the 
vert dist ef equal to the total vert strain (16 tons) at e; make ez — to half of ef ; 
draw z h lior; and hf. 

Now m y and rnj give the strains (4 tons each) along the ties m a, m c, caused by 
the 4 tons at b; which strains extend from m to a and c.J In like manner g y and 


# When merely wishing to ascertain the strains along the members of such a trusR, without caring 
to trace their progress, we may omit part of the following : and, after having made a correct diagram 
of the truss, we may at once write down the vert strains at the points of support, thus: At e (the 
apex or peak of the truss ) write on e-half of the entire wt of a truss and its load, (for which, per sq ft, 
see Table 4, p obi) at the foot n of the rafter, one half: at the center strut c one fourth; at h and d, 
one eighth at each: and when there are four intermediate subdivisions of the same kind, as along 
the rafter e p, Fig 19, one sixteenth of said entire weight at each of such additional points, &o. I hus 
the total load at any longer strut will be twioe as great as that at any next shorter one. 1 hen begin 
at “ Having thus finished our second prooess," , ^ . ... . . 

? This is precisely half the wt (32 tons) of the entire truss and its load: and as this will always be 
the case in trusses on this principle, it proves our work to be correct thus far. . 

1 When the main tie a t is hor, as in Fig 20, these strains along the ties will be equal to those at 
the points of support, only where the height of the truss is equal to M of its span ; as in ihecs.se 
before us. when the height is le»s than the strains op the ties will be greater than those at the 
points of support; and vice versa. 









576 


TRUSSES. 


g j give the strains (4 tons each) extending from g to c and e. In Fig 21, the shor 1 
ties, o a, o c, i c, i e, show this more distinctly.* 

Next a; ?/ and xj, Fig 20, give the strains (8 tons each) produced along x a and x , 
liy the 8 tons at c. This also is shown more plainly in Fig 21, by the ties n a an | 
H kbe bor Iine fi z &' ves tlie 8tra ' u (16 tons) produced along the entire hor ti j 
« l, Fig 21, by the lb tons at e. Fig 20 may be considered one-half of Fig 21. 

We have for the total strains on the ties as follows: 


Along c m and c g, strain — mj or g v = 4 tons. 


Along x e, from x to g, strain = x j 


n l or ?.Jiti. 

- o — -7 ■ —« — ;f , ...«i.J X J tOn S ■ 

Along x e, from g to e, strain — xj + gj = 8 4 = 12 tons. 

Along t a. from t to x. strain = ft z = 16 tons. 

Along t a, from x to m, strain = ft* + xy = 16 + 8 = 24 tons. 

Along t a, from m to a, strain = ft2 + xy + my = ]6 + 8 + 4 = 28 tons. 

The line/A or e h, Figs 20 and 21, gives the longitudinal pres (17.9 tons) brougli 
upon each rafter by the 16 tons vert load w'hich our second process brought to < 
Jins pres strains the entire rafter uniformly from end to end ; but the several poi 
tions of the rafter sustain, in additior, pressures arising more directly from th 
pauel loads. lor instance, the 4 ton load at d, produces at d, as already explainer 
a pres v n = 1.77 tons along d c. and one, do = 3.6 tons, along the strut do- an 
tins last exerts pulls, y g and gj, = 4 tons.each, along g c and g e. Now if on q j w 
draw the parallelogram g nj u, making j n and u g vert, and uj and g n parallel t 
the rafter, then,; noru; will give the vert load of 2 tons which travels to e fron 
the 4 tons at d; while,; u or gn will give the strain, = 2.7 tons, exerted alone th 
second secondary rafter e c by the tie g e. Similarly the parallelogram y u w a eive 
the vert load y w or u g, = 2 tons, which travels from d to c, and the strain' y u o 
w g, — 4.47 tons, exerted upon c e at c by the tie c g. The 4 tons at d therefore nrc 
duce a stram y m = 4.47 tons along cd, and one, uj = 2.7 tons along de; because it 
— 2.7 tons exerted at e, and vo = 1.77 tons exerted atd(=4.47 tons in all )both pres 
the lower part d c,and strain it against the equal pres of y u at c ; whereas this upwan 
jires, — 4.41 tons of y u, is diminished at d by the downward pres, 1.77 tons of v i 
leaving an upward pres of 2.7 tons to strain d e against the equal downward pres u 
at I'o i 1 ? * \ *? anie ' Vli y y u and u 3 °f t1' e lower parallelogram show r the strains (4 4 
and 2 7 tons) brought upon a b and b c respectively by the 4 tons load at b; and w r 
■uj, of the middle parallelogram give the corresponding strains (8.94 and 5 4 tons 
brought upon a c and c e respectively, by the 8 tons at c. 

The total strain upon any portion of the rafter is found by adding together th. 
uniform strain e h or fh, common to all parts, and the strains peculiar to such poi 

WhS 0 ? V * ” es y “ T\ uj - Taking the P a,t c d for instance ; as th 
lower half of c e it sustains y u of the upper parallelogram, = 4.47 tons; as part ol 

the upper half of a e it sustains uj of the middle parallelogram, = 5.4 tons • and a 
part of the eutire rafter it sustains e h = 17.9 tons; or, in all, 4 47 + 54 + 17 Q- 
li.li tons. 9 i . — 

Thus we have, for tlie total strains along a rafter. 


th 


It 


From e to d, strain = uj of upper parallelogram + uj of middle parallelogram + „ ft = 2 7 4- 5 
+ 17.9 = 26 tons. 1 * 

From d to c, strain = upper y n + middle w/ + e h = 4.47 + 5.4 + 17.9= 27 77 tons 
c to 6, strain _ ower uj + middle y u + e A = 2.7 + 8.94 + 17.9 = 29 54 tous 
h to a, strain — lower y u + middle y u + e A = 4.47 + 8.94 + 17.9 = 31.31 tons 


„ ,?T nter rr t e 1 may be omitted in short spans where the tie bar is hor a 
a l Fig 21; since it then sustains nothing but the wt of the half (y y) of the centra 
spread x x of the hor tie a l. K y ' centra 

Art ). 6 A * If fh « main or primary tie is raised at its center 

as p n Fig 19, proceed as at Fig 16, and, after having found the vert strain 

at all the points of support, as before, add to that (16 tons) at e, an amount 

Said x the vert ht t n, Fig 19, to which the 
16 tons tiep n is raised above the borp t. 


the remaining ht, n e, of the truss. 

This additional amount is the strain on the cen vert rod e n which is indisnensa 
ble in such cases Then, as in Art 15, lay off the vert ”/ Fig 20 eouaf to th 
total vert strain at «, thus found ; and after dr awing / h parallel to the raller, 5rav 


thLuW&XVt^ ei\ 

plainly that the central portion x x of the primary tie a ? needs onlc a*’ mer - e ^ to sh °w mo 
it to sustain the thrust produced by the Ifi'tTsJain at e whereas e " ab 

must be stout enough to bear, in addition, the pull along the first secondfrvlies na kl 
its ends m a, m l it must resist not only the two preceding forces but a ’ ^ wlllle 1 

secondary ties o a, o l. Likewise it is plain that the portion o e of the first J the SeC0IJ 
be stouter than the portion n g; because q e has to bear also the null alnmr tha ^ ar 7 t,e n mu 
«. In Fig 20, ,h.,o portions of the ‘ 












TRUSSES. 


677 


parallel to the inclined tie, instead of hor. fh gives the pres throughout the 
itter, due to (he total vert load at e; and h z gives the pull throughout the raised 
* e-r °d, due to the same load. Both ./ 7 h and h z are of course greater than the cor- 
“ }8 Ponding strains e h, h z, Fig 20. Like them, they are to be used in finding the 
terfal strains in the several parts of the rafter and tie-rod. 

If we omit the third secondary trusses e s w, Ac, Fig 19, the strains on the struts, 
’ V' 1 V t & jn > will be the same as those on the corresponding struts, d g, cx, bm, Fig 
D; but the strains on the rafter, corresponding to y u, uj , Fig 20, and those on all 
lie ties, corresponding to m y, mj, &c. Pig 20, although found by the same process 
s in Fig 20, will be greater than in that case; in addition to which the uniform 
trains,//}, hz, exerted throughout the entire rafter and tie-rod respectively by the 
nal vert load at e, will ateo be greater, as explained above, 
tt Art 16 B. If the main tie is raised only part way, asp y. Fig 19, 
t ,nd then continued hor, as y o; draw it as if it extended to n, as ‘in Art 16 A, and 
r . se the same hts t n, n e, for finding the additional vert pres at e. Find fh and h z 
ie 8 in Art 16 A. On p n lay off by scale the pull h z on p y found as above; and on 
,j his as a diag draw a parallelogram with sides parallel respectively to y e and y o. 
d'he latter (hor) give the total strain on y o, and the former give an additional stiain 
„ n y e, to be added to those found for each part of that member as directed in Art 16 A. 
0 In this case, as in Fig 20, the cen vert eo sustains only the wt of half the hor 
^tretch of the tie bar, and may be omitted in short spans. 

if Rem. 1. It is not necessary actually to draw all three of the parallelograms, as in Pig 20. The large 
, 8 r center one alone will suffice ; for we need only div the several strains measured along the strut cn, 
ig 21; and along the ties n a, n e, by 2, to get those along the struts 4 o and 4 t; and along the ties 
ir o, co, ci, ei. And these, in turn, div by 2, will give those along the smaller subdivisions shown 
Hetween e andp, Fig 19, if there are such ; and so on with any number of still smaller ones. 

j 

18 

d 

i.i : 

j 

| 

I 

I, 

) 

e 


e 

r 




578 


TRUSSES. 


Art. 16C. Remarks on king; and queen ; and on Fink truss* j 

for roof*- The following comparison is founded upon total spans, or lengij 
ot truss, of 154 ft. Rise 30.8 ft; or £ of the total span. Trusses 7 ft apart from c<, 
ter to center. Each rafter 83 ft long. Total load, including the truss itself. ! 
lbs per so ft of roof; or 20.8 tons to each truss. There are seventeen points of si; 
^ 20 8 
port in each truss; consequently a full panel-load (Art 11) is — = 1.3 tons. True 

as shown, half of each, in Figs 47A and 48, p 606. The strain in tons (calculai 
as if all the weight of truss and load were on the rafters) is marked on ea 
member. The assumed coefficient of safety for ties is 3. Iron is supposed to be u 
that will not break with a less pull than 20 tons per sq inch; the assumed safe alk 
• 20 

able pull being therefore here taken at g- =% tons per sq inch. The safe press’ 

along the rafters is taken at 3J4 tons persq inch. The struts are asstimed to be wrouj 
cylindrical tubes, with an outer diam equal to of their length; and of si 
thickness as will give them a metal area of l sq inch for each 2 tons of strain. 1 
rafters are in the present case supposed to be 9-inch rolled I’hoenix beams; l]/ 2 sq 
transverse area: weighing 25 lbs per foot run. The ends of all ties are suppo 
to be enlarged, or upset; so that the cutting of the screw-threads shall not dimin 
their effective area. The purlins are supposed to be at or near the “points of si 
port,” so as to produce no cross-strains on the rafters. 

Tabic 1. Weight of the Fink truss, of which Fig 47A sho' 

one-half. 

Length 154 ft. Rise I length. Trusses 7 ft apart. Load 40 lbs per sq ft of roof; including truss 
___(Original.) 


Name of part. 

Number 

of 

parts. 

Area of 
each part, 
sq ins. 

Lbs. per 
foot run 
of 

each part. 

Total 
weight of 
all the 
parts. 
Lbs. 

Lbs 

Rafter. 

2 










4150 


41 oO 

(n . 

2 



430) 

450 




2 


0.0 



Main tie.^ n . 

2 




1446 

lr 

ru . 

2 

4 

3.66 

12.22 

272 
294 J 




4 



78' 



1 „ . 

2 

2 



Ho 



Secondary ties....^ t . 



150 

► 

676 

| *. 

2 

8 



1 lo 





136 
76 J 



. 

C j . 

2 

vf. Zi) 



471 

Struts. 1 w . 


1.20 

4.0 

136 

1 

(i . 

8 

> 

Center vertical y . oav 

i 



40 

400 

400 


40 

400 

400 

Joint and splicing-pieces, nuts, &c, &c... say 
Shoes at ends of rafters, say. 








Wt of purlins not included. Total wt of truss 








=7588 

7588 


























































TFt< ; i the same total load per sq ft , including the trusses , (with trusses 7 ft apart; 
ise J span) the area of each part, its wt per ft run , and the strain upon it, are 
:s the. spans; but the total wts of the trusses alone are as the squares of the spans, 
lence, it is easy to deduce from the table the areas reqd for smaller spaus. The 
afters for small spans are frequently made of round iron rods from to ins 
of ordinary flat bars. Tubes with the same area of metal, would be better. 


TRUSSES. 


579 


lam; or < 


or trusses also of different spans, and rise of \ the span, 7 ft apart, in which the 
afters and struts are of wood, with ties of iron, the strains may be deduced quite 
losely from those in Figs47Aaud48. They will, however, be somewhat greater, 

IGCHUSti wooden struts, not heintr hnllnw 1 innr qssnmutl irnn nnoo mnct l, n 


ecause wooden struts, not being hollow like our assumed iron ones, must be heavier 
van the latter to prevent bending. The weight of the load, however, is generally so 
inch greater than that of the truss, that this consideration of the strut is not very 
iateria.1; so that a roof partly of wood may be assumed in practice to weigh, together 
'ith its load , but little more than an iron one; and the strains on the several parts 
ill be nearly the same in both cases. 


able 2. Weight of the king and queen truss, of which 
Fig 1 48 shows one-half. 

.ength 154 ft. Rise ^ length. Trusses 7 ft apart. Load 40 tbs per sq ft of roor; including truss. 

(Original.) 


Name of part. 


ifter. 


liu tie. 


irticals 


nter vertical. 


ruts. 


r !! 

P 

K 

D 

C 

B 

A 

f i 

3 

k 

<<■ 

I m 

b 


9 

?• 

* 

t 

u 

V 

l w 


int and splicing-pieces, nuts, &c, &c .. say 
oes at ends of rafters.say 

Total weight of truss = 
t of purlins not included. 


Number 

of 

parts. 

Area of 
each part, 
sq. ias. 

t.hs. per 
foot ruu 
of 

each part. 

Total 
weight of 
all the 
parts. 
Lbs. 

2 

7.5 

25 

4150 

2 

2.2 

7.33 

146 70' 


2 

2.44 

8.14 

162.80 


2 

2.68 

8.94 

178.80 


2 

2.92 

9.74 

194.80 


2 

3.16 

10.54 

210.80 


2 

3.41 

11.34 

226.80 


2 

3.65 

12.14 

242.80 


2 

3.65 

12.14 

242.80, 


2 

.0 

.0 

.0 I 


2 

.1 

.33 

5.34 


2 

.2 

.67 

16.00 


2 

.3 

1.00 

32.00 

c 

2 

.4 

1.33 

53.33 


2 

.5 

1.67 

80.00 


2 

.6 

2.00 

112.00J 


1 

1.4 

4.67 

150.00 

2 

.88 

2.92 

64.20 b 


2 

1.05 

3.50 

91.00 


2 

1.28 

4.25 

136.00 


2 

1.55 

5.17 

196.37 


2 

1.82 

6.08 

267.63 


2 

2.12 

7.08 

368.30 


2 

2.40 

8.00 

464.00 J 





400.00 




400 00 




8592.46 







4150 00 


1606.30 


298.66 


150.00 


1587.50 


400.00 

400.00 


8592.46 


8592 46 

Hence, the wt of the king and queen truss in this instance is equal to ‘ - 

i 588 

1.132 times (say 1^ times) that of the equally strong Fink; or the Fink is about 
part lighter than the K andQ. Theoretically the diffwould be less, because the rafters 
the K and Q truss being so much less strained at top than at foot, may be diminished 
ward their upper ends, instead of being proportioned throughout with reference to 
e max strain at their feet. If the theoretical diminution toward the tops of the 
fters, were made in both cases, the wts of the two forms of truss would be nearly 
ual. hut in practice, on the score of inconvenience, this is rarely done in roofs 
moderate span ; say less than about 100 ft. No such diminution, or but very slight, 
>uld be admissible even theoretically, when the purlins are not placed at the points 
support only. Willi same total load per sq ft, indiuling 
•usses llienmelves at same dist apart, flic total wts of trusses 
re as the squares of their spans; but their wfsperfl of span, 
• well as the cross areas, wts per ft run, and strains along; 

41 






































































580 


TRUSSES. 


imli\ 7 idnal member.*, are directly as the spans. 

When the dist apart of the trusses is 7 ft from center to center; the 
rise ^ of the span; assumed load, including the wt of the trusses themselves, 40 
lbs per sq ft of roof covering; and the various parts proportioned for the several 
strains per sq inch assumed in Tables 1 and 2; the weight of a properly con¬ 
structed Fink truss will be approximately as follows: 


Total wt in lb* of 
a Fink roof-truss 


square of span in ft 


and the wt in lbs per ft of span 


span in feet 


3.1 


A total K and Q truss, will be about ^ part more; or 

span in feet 


square of span in ft. 


2.7 


Or per foot of span, = 


2.7. 


23716 23716 

These rules give —— = 7650 lbs, for the foregoing Fink ; and ——- = 8784 lbs, 
o.l 

for the K and Q truss. From these rules we have drawn up the following 

Table 3. Approximate weights of roof-trusses of the Fink 

system. (Original.) 


Rise 4 span. Trusses 7 ft apart. Load 40 His per sq ft of roof, including truss. 


Total 

Span. 

Total wt of 
a Truss. 

Wt per ft. 
of Span. 

Wt per sq ft 
of ground 
covered. 


Total 

Span. 

Total wt of 
a Truss. 

W t per ft. 
of Span. 

Wt per sq ft 
of ground 
covered. 

Feet. 

Lbs. 

Lbs. 

Lbs. 


Feet. 

Lbs. 

Lbs. 

Lbs. 

20 

129 

6.46 

.92 


100 

3228 

32.3 

4.60 

25 

202 

8 08 

1.15 


105 

3557 

33.9 

4.83 

30 

290 

9.67 

1.38 


no 

3904 

35.5 

5.06 

35 

396 

11.3 

1.61 


115 

4267 

37.1 

5.29 

40 

516 

12.9 

1.84 


120 

4640 

38.7 

5 52 

45 

654 

14.5 

2.07 


125 

5041 

40.4 

5.75 

50 

807 

16.1 

2.30 


130 

5452 

42.0 

5.98 

55 

976 

17.8 

2.33 


135 

5880 

43.6 

6.21 

60 

1160 

19.4 

2.76 


140 

6336 

45.2 

6.44 

65 

1363 

21.0 

2.99 


145 

6782 

46.8 

6.67 

70 

1584 

22.6 

3.22 


150 

7260 

48.4 

6.90 

75 

1815 

24.2 

3.45 


155 

7750 

50.0 

7.13 

80 

2064 

25.8 

3.68 


160 

8256 

51.6 

7.36 

85 

2331 

27.5 

3.91 


165 

8782 

53.3 

7.59 

90 

2616 

29.1 

4.14 


170 

9324 

54.9 

7.82 

95 

2912 

30.7 

4.37 


175 

9879 

56.5 

8.05 


ar 


For king and queen trusses add part to the tabular wts; when the rafters 
as usual of the same size throughout. 

The wts in the 4th column will remain nearly the same, whatever m 
be tlie dist apart. For if this be increased say to 14 ft, each truss will sustain twi< 
as many sq ft of roof; and must itself be at least twice as strong and heavj 7 , i 
to do so. We say “ at least," because if the dist apart is increased, the wt of t 
lins will generally increase more rapidly than said dist. Thus, if the dist be double 
the purlins will not only be doubled in length, which alone would double their w 
hut they must also be deeper. In practice, however, long purlins are usually pr 
vented from becoming very heavy, by trussing them, as at 7, Figs 21^, 

The cost, at shop, of trusses alone for iron roofs and bridges, gener 
ally varies between 2 and 24£ times the cost of ordinary “refined” bar iron. Th 
putting up alone from % to J4 the cost of the iron. With roof trusses 7 ft apart, iroi 
purlins will weigh about 2 lbs per sq ft of ground covered by the roof. Therefore t< 
any wt in the 4th col add 2 tbs. Add for covering with tin, slate, or corrugated : ron 
See pp 404, 418, 429. 


Rem. 1. As to the proper total weight, or load, per sq 

of roof , {including snow and wind,) that should be assumed to be sustained by the trusses, eugiticei 
The French appear to consider 30 St>s as sufficient; while the English use 4 
subject to violent vibrations like bridges, they do not require so high a coefficiei 

nld lint hnu'PVPr in mir nnininn bo t n Iron ♦ 7 .no o 9 • «bi n ... ~ : J 


differ considerably 
Since roofs are not 

of safety; this should not, however, in our opinion, be taken at less than* 3 : and this we consid 
sufficient. The load is evidently influenced by the character of the roof-coveriug. Within ordinal 
limits, for spans not exceeding about 75 ft, and with trusses 7 ft apart, the total load per sq ft, inclu 
ing the truss itself, purlins, .fee, complete, may be safely taken as follows ; 





































TRUSSES. 


581 


Table 

Span 75 ft or less. 

Roof covered with corrugated iron, nnboarded.t 
„ If plastered below the rafters, 

“ corrugated iron, on boards, 

(< If plastered below the rafters, 

„ “ slate, unboarded, or on laths, 

,, ‘ “ .on boards, 114 ins thick. 

,, ‘ “ if plastered below the rafters, 

“ “ shingles on laths 


4 . 


8 lbs. 
18 “ 


11 

21 

13 

16 

26 

10 


If plastered below rafters or below tie beam. 20 


Wind 


and Snow.* 

Total. 

20 tbs. 

28 Ibs. 

20 “ 

38 “ 

20 “ 

31 “ 

20 “ 

41 “ 

20 “ 

33 “ 

20 “ 

35 “ 

20 " 

46 “ 

20 “ 

30 “ 

20 “ 

40 “ 


Exasnple of use of foregoing tables. A Fink roof 60 feet span; rise 1; 
-russes 14 ft apart; and to be covered with slate, on boards inch thick. Here we 
ie< at once from fable 3 that at 7 ft apart, its wt would be about 1160 ft>s; therefore 

•nr 14 i f ! P K Jt 232 1 ° fts ' But our table is for 40 P er s 0 ft of roof: While’ 

d- bf ? * hoards, 35 Ibs, or y part less, is sufficient. Therefore, we may reduce the 
height of the truss y s part, making it only 2030 Ibs. J 

) 3 * Roof as before, 60 span ; trusses only 7 ft apart. Turn to Table 1, where the 

ireas are given for a total length or span of 154 ft. But 60 ft is the — = sny the 

J? f 154 ft; •lJ 1 ? rc f ? re l tbe ar( ' as ', and the wts P er font run of each member of 
r he ®°. pt fP an ’ Wl11 . be A of those of the 154 ft one. Thus, the area of a rafter will 
>e 7.0 X -4 — 3. sq ins; which corresponds with a rolled T iron of 3 X 3JX ins and 
H 1 “ ch average thickness. Its wt per foot run will be 25 X .4 = 10 Ibs .The’area 
F the Pa*? n of the main tie will be 1.95 X -4 = .78 sq inch, which we see at once 
rom a table of circular areas, is equal to a round rod very nearly 1 inch diam Its 
vt per ft run == 6.5 X -4 = 2.6 Ibs ; and so wilh all the other members. But the total 
vts will be asjhe squares of the span. The square of 154 is 23716; and that of 60 is 

!600. And -716 = - 152 ; therefore, the total wts will be .152 of those in Table 1. 


the two rafters will weigh 4150 X .152 = 631 Ibs. The main tie, 1446 X .152 = 
.20 lbs, &c. Lastly, if for 35 Ibs per sq ft, reduce each area and wt y H part. 

Since the rafters are generally made of T or I iron, a pattern precisely adapted to the calculated 
trains, will not always be procurable; and in that case we may either take the nearest one in excess • 
r change the dist apart of the truss to suit the pattern on hand. Owing to the variety of modes of 
rrnngiug the details of the junctions, &c, an exact coincidence between the calculated a'nd the actual 
pts, is not to be expected; but we suspect that in properly proportioned roofs, the discrepancy will 
Barely be found to vary more than about 5 per ct from the results of our rules. 

It might be supposed that with iron of a tensile quality considerably higher than 

f ’ur assumption of 20 tons per sq inch; as say of 25 to 30 tons, the truss might be 
nade much lighter. But this is not the case; because the superiority would affect 
he ties only ; inasmuch as the compressive strength of iron does not increase with 
ts tensile strength ; but to a certain extent rather the reverse. Now, by Table 1, it 

ppears that the ties in a Fink roof-truss, constitute less than T 3 ^ of its entire wt. 
’herefore, iron of even 30 tons, would reduce the weight of the truss less than y 
■art of r 3 g part; or y 1 ^; and 25 ton iron, about ^ part. 


Short spans need not have as many subdivisions, or “points of support, ’ as a large one; and this 
ill lessen the number of parts of the truss; but inasmuch as the remaining parts will require to be 
voportionally stronger, this consideration will not materially affect the wts’ While on this subject 
■ e will remark that too few points of support are probably used at times; owing to either an uuder- 
aluation, or an ignorance of the effect of the transverse strains produced by the load on the parts 
f the rafter between said points. These parts must be regarded as so many separate beams sup- 
orted at both ends ; or rather, as firmly fixed at both ends, when the pieces composing a rafter are, 
s usual, strongly connected together; in which case the beam is about twice as strong as when 
lerely supported. If the separate parts be trussed, like the purlin at 7, page 583, to neutralize this 
•ansverse action, it must be remembered that additional compression will be thereby produced 
mgthwise along the rafter. The best practice is, as far as practicable, to Increase the number of 
oints of support, so that the purlins may rest upon them aloue, or near them ; and thus relieve the 
afters entirely, or in part, from transverse strain. 


Rem. 2. As to the effect produced on the weight of a truss, by 
hanging its rise, no short correct rule can be laid down. Although as*a 
oof becomes flatter, its area becomes less, so that each truss has less total wt of roof 
overing, snow, and wind, to sustain, still the strains on most of its members become 
reater; requiring greater wt of truss. To find this increase with accuracy, it is 


* See Snow and Wind, p 216, 221. 

t The corrugated iron itself will weigh from 1 }4 to 2 lbs per sq ft on the roof. If not plastered under- 
eath, the condensed moisture of the air, especially from crowded rooms, will fall from the iron into 
le rooms below. Mere boarding will not prevent this, even if tongued and grooved, unless the circu- 
ition of air against the under side of the iron is effectually cut off. 












582 


TRUSSES. 


necessary to make a diagram, and perform all the calculations. The strains on a 
1'iuk rafter become more nearly uniform throughout its length, as the pitch of the 
roof becomes less; while, with a rise of ]/ a span, the strain at its foot is about 1-j 4 ^ 
times that at its head. On the contrary, the strains on its struts remain nearly tkt 
same in amount for all ordinary rises. 

In the king and queen truss the strains at the heads and feet of the rafters retain the same pro¬ 
portions to each other, at all rises; the straius on the verticals become less as the roof becomes Hatter; 
while those on the obliques vary according to their several obliquities. Under these irregularities, 
which a£Tect the K aud Q, much more than the Kink, we can do nothing more than say that when it 
is merely wished at the moment to form a rough idea of the effect of changing the rise, we may 
assume the weight of a Fink truss to increase about in the same proportion as we diminish the rise; 
or to diminish as we increase the rise. Thus, if we increase the rise or the roof in Table l. oue-fourtb 

part, so as to make it equal to .25 or % of the span, instead of .2 or-j of the span, we may diminish its 

wt ]4 part; making it about 6000 lbs, instead of 8000. Or ir we reduce the rise from X to yU, making 
it only half as great, we shall double its weight, making it 1S000 lbs; as rude approximations. 


Figs 21 show a few of the many forms of the details 
of iron roofs. Every maker has his own modifications of them. Most of the 
figs explain themselves. They will serve as hints. 

R and P stand for rafter and purlin. In small roofs, with the trusses only 3 or 4 
ft apart, the purlins may, as at 6, he simple inch or % round rods, about 9 ius 
apart; and the slates may rest immediately on them, being tied to them by i 
wire. They may be bent down at their ends, and riveted to the rafters. As the 
between the trusses increases, these purlins may be made of flat iron, from 1 to 3 
deep, and *4 inch thick; or of light T iron, &c; and may be trussed, as at 7, so as If 
admit of being placed several feet apart. IVhen, however, they have to bear grea 


weight, the mode at c, Fig 7, of confining their ends to the rafters, will be too weal 


Sometimes they may be arranged as at y. Or the purlins, of either iron or wo 
may rest on top of the rafters, as at 1 and 5; or their ends may rest in a kind ol 
stirrup, as at t. Fig 2; and at P, Fig 4: in castings placed at the “points of sup 
port” of the truss; or they may be confined to the sides of the rafters by twoangl 
irons, as at P, Fig 9. Purlin* should, when practicable, be su 
ported only at or near the “points of support” of the truss; an 
as a general rule, it will be expedient to arrange the number of these points witl 
reference to this particular. The rafters are then relieved from transverse strains 
and may be proportioned with regard only to the compressive strain in thedirectioi 
of their length. Too little attention is sometimes given to this point, and the trails 
verse strain is overlooked, to the serious injury of the roof. It is well, however, t< 
bear in mind that thin deep rafters are liable to yield by buckling sideways; am 
that this tendency is diminished by purlins well secured to them between the “ point 
of support.” Sometimes castings similar to 2, are used at the heads: and 3, at th 
feet, of the struts and vert ties; which last have their ends cut into right and lef 
hand screws, for insertion into corresponding female screws cut in the castin 
At 3, 11 is the main tie passing loosely through the lower opening through the cas 
ing. Below it, is seen the head of a small set-screw, for tightening together th 
casting aud the tie; to prevent the former from slipping out of place. There mi 
be different patterns of these castings, to suit the obliquities of the several obliqu 




or, in small roofs, the parts a a may be made with hinges, for the same purpose. 




<r 


if 














TRUSSES. 


583 


^ t8S(*lroii shoes for supporting tlie ends of the trusses 

upon the walls. With the exception, perhaps, of these shoes, it is better that the 
details generally should be of wrought iron. 

At 8 is a mode of confining- thin metal roof-covering t, to 

I*****1*>*S P, by means of short (about an inch) U-shaped pieces (c c 11 is one 
ol them) of the same metal; to which i is riveted by an % inch rivet through each 
flange 11. lhis may be adopted with corrugated iron covering, which, by its strength, 
allows the purlins to be placed several feet apart. See Corrugated Iron. Flat 
sheets require boards beneath them. 


At 1© is a mode of confining a wooden purlin P on top of an iron 
one p, by means of a crooked spike s s x ; which, after being driven from below, is 
clinched or bent on top. Wooden purlins are sometimes thus required, for nailing 
slates or plain sheet metal. At 11, c, is a stick of timber inserted between an iron 



Fifjs 211 


urlin P and the corrugated roof-covering a a. To such sticks plastering-laths may 
e nailed, when the roof is to be plastered beneath, to avoid condensed moisture, 
here is room for much ingenuity in all these details Fig 12 is a rafter made of two 
tiannel-bars riveted together; with a web member c c between them. Two angle- 
ars are often thus riveted together for a rafter. 


Fig 13 shows a turnbuckle or arm swivel tb, for shortening a tie-bar 
lade in two lengths. If tb is made of a pipe or solid bar it is called a double 
iiut or pipe swivel, and, at least for a part of its length, it is made square or 
[exagonal, so that it can be turned with a wrench. In either case a female screw is 
lipped in each end of the nut, right and left hand respectively; and corresponding 
:rews are cut on the ends of the rods. WHien the swivel is revolved, the two ends 
f the rods are drawn nearer together. In the o?-w swivel, one rod-end may be left 
lain and round, as in fig, and furnished with a head c. 


Fig 14 is a mode of tightening four lengths of tie-bars crossing 
ich other, by means of a ring. The ends inside of the ring are cut into screws, and 
rovided with tightening nuts, as in the fig. The rings are usually % to 1% inch 
lick ; 3 to 5 deep ; and 7 to 10 diam. 

























































584 


TRUSSES. 


Horizontal Fink Truss, 

uniformly loaded. 

Strains on posts = final 

panel loads : at c — xi~ half the total wt 
of truss and load ; at 6 — in i, and at d ~ g i, 
each = half x i. Strains on ties: 
on m c and g c, each = m s ~ g t>; on x m 
and x g, each — x o ~ xn \ on m a and g e, 
each — xo-\-my — xn-\-gw. Strain 



OH chord a e (uniform throughout) — half o n -J- half y s ~ half o n -f- half v to. 


In the Fink truss, the effects of a moving- load may be 

calculated as for a full uniform maximum load from end to end. Thus assuming 
at first, as in the preceding cases, that everything is borne by one truss only; then, 
when the load is upon the top chord of the truss, each vert post may in practice be 
regarded as upholding one-half of that portion of the entire wt of bridge and dis¬ 
tributed load which is between the two extreme ends of the two obliques which 
uphold said post. Thus, in Fig 46, the half-way post dc, bears half of all between 
a and b. The post m g, half of all between a and d; the post h o, half of all between 
vi and d. This is equivalent to saying that the half-way post bears half the entire 
wt of the bridge and load: each quarter-way post, one-quarter; each eighth-way 
post, one-eighth, &c, &c, of this same entire wt of bridge and load ; and these consti¬ 
tute theoretically the strains on the several posts. But after having got thus far, it 
is necessary to examine whether some of the smaller ones may not have to be in¬ 
creased, for the following plain reason : Suppose we have assumed our max load to be 
a string of engines, weighing 1 ton per foot run; or, including the wt of the 

bridge itself, say 1.4 ton per ft; and suppose our posts to be as close together as 5 
ft; then the least loaded posts would each bear 5 X 1.4 = 7 tons. But we know that 

from 16 to 20 tons may 

OAnoAn fro nrii>1 



Fig. 21 h. 


be concentrated within a 
length of 5 ft, on four 
drivers of an engine; and 
half of it will have to be 
supported by each post in 
succession as a train passes 
When we thus find by trial 
which posts will be more 
strained by an engine than 
by our assumed max per ft 
run of the whole truss, wo 
In the Fink, and Bollrna 


must increase the load first found, correspondingly, 
trusses, the verts are always struts or posts. Having fixed upon the load for each 
post, as p o, Fig 21 h , then for the strain which said load will produce upon each of 
the obliques, or ties, p c, p h, upholding said post, take any distp d on the-post, to 
represent the load by scale; and draw dw,d n, parallel to the ties; then p w, p n, 
measured by the same scale, will respectively give the strains on each ; whether they 
be equally inclined as usual, or not. The two hor lines n a,w a, by the same scale, 
give the two hor forces which the load at the top o of the post, acting through the 
ties, produces upon the chord at c and A; which two equal and opposing forces pro¬ 
duce along the intermediate stretch c h of chord, a strain equal to one of them. In 
other words, either n a, or w a, gives the hor strain produced along c h , by the load 
at o only. 

Strain oil the chord. This, from a uniformly distributed load, is the same 
throughout the entire length of a Fink chord. To find it, observe which obliques 

ord 


(as m e , g e, o e , Fig 21»,) of one-half of the truss, terminate at one end , e, of the cbo 


Then, having previously found the loads on the posts, c o, u g,t in, which pertain t( 





















TRUSSES. 


585 


those obliques, ascertain by the pro¬ 
cess in Fig 21 fi, the hor force n a, 
in both figs,) which each of those 
oads produces on one oblique. Add 
together these forces n a, (there 
.will be but three of them in Fig 21 i, 
is marked by the dark lines ;) their 
mm will be the strain along the en¬ 
tire chord. The obliques m u, u r , 
c, g c , do not terminate at e; and 
ire, therefore, omitted 
;he chord strain. The 
ength or not. 



in finding 

process is the same whether the verts are all of the same 


Or the hor chord-strain produced by each of the loads on the posts c o, u g, t m, 
nay be calculated thus, and added together. 

Horizontal entire load X ’T di f fr T, 
strain = 071 ^ osi 0 end e c ^ or< ^ 
twice the length oj the ]>usL 












586 


TRUSSES. 




’o/> 


& 

e* 

o|)^' 

if 

-.y 


3 

o 

53 


ed 


Art. 17. FIs* 22 represents a suspension truss on the Boll- 

man plan ;* the whole weight supposed to be along the top a p. 

In this, the strain from eacl 
panel-load, as for instance tha 
at d, passes down to the foot of 
its supporting post dj ; and fron 
there is transferred to the tw< 
ends a and p of the chord, by 
means of two ties, as j a, j p 
upon which the post stands. Ir 
this manner the vert strain fron 
each panel-load is separately 
sustained; and transferred di 
rcetly as a hor strain to the ends 
of the chords, by its own pos> 
and pair of ties; without pro 
ducing, as in the foregoing cases 
an additional vert strain ai 
the points of support of the 
other panel-loads. So omil 
our 2nd process; and 
having divided the uniform wi 
of the truss and its load, amonj 
the several points of support a 
b, c,d, e, &c, as before, we pro 
ceed at once to draw the parallel 
ograms of forces v v l g, v u k g 
&c, for measuring the strains 
To do this we have only to sel 
tip the equal vert dists l v, k v 
j v , Ac, each to represent by 
scale the 4 ton panel-loads or 
top of the respective posts; then 
complete each parallelogram by 
drawing v u . v g parallel to the 
two ties which support each 
post. Then the lines l u, lg\ 
ku,kg,& c. give by scale the 
strains along the respective ties 
The end a of the hor chord i< 
pressed hor by the seven hoi 
forces uo, u o y , u o„, u o & c .. 
equal to 1.75 -f 3 -f 3.75 + 4 + 
3.75 + 3 1.75 = 21 tons ; and 

the other end p is in like man¬ 
ner pressed by the seven corres¬ 
ponding forces not shown ; and 
these two sets of equal op¬ 
posing forces produce a strain 
equal to one of them ; or to 
21 tons, uniform throughout tht 
entire chord. The tie la car¬ 
ries to a so much of the weight 
of the 4 tons at b as is rep¬ 
resented by l o, or 3.5 tons; 
k a carries to a a weight equal 
to k o Jy or 3 tons; j a carries j o„, 
— 2 5 tons ; i a carries i o n , — 2 
tons; w a, w o = 1.5 ; x a, x o = 
1; and y a, y» = .5 ton. All 
these amount to 14 tons; which 
with the 2 tons of half panel¬ 
load at a, give 16 tons ; or half 
the entire weight (32 tons) of 

* Invented by Mr. Wendel Bollman, C. E. 



00 


50 


N 


OJ 


50 

d 

% 

* 

CO 

- 

o 


o 

' ei 
o 

co 
































TRUSSES. 


537 


The strain l u — 3.91 tons. 

ku = 4.25 “ 

" ju = 4.52 “ 

“ iu = 4.47 “ 

Each post or vert is of course 
le whole wt is supposed to be 


strained to the amount 
on top of the truss. 


The strain l g = 1.82 tons. 

“ £0 = 3.17 “ 

“ j g = 4.05 “ 

“ i g = 4.47 “ 

of a full panel-load, when 


In the Bollinaii, for a moving; load, having first pre- 

, ared the working diagram, determine the max weight that can come upon a post, 
i his will be the same for each post. If the moving load is on top of the truss, this 
i>ad on each post will consist of the greatest wt of engine that can stand upon one 
ranel-length of truss; together with (approximately enough for practice)one panel- 
-wgth of door; and the half of a panel-length of truss. If the load is at the bottom 
jt the truss, the posts bear no part of either the moving load, or of the floor; but 
! ich of them will be strained to the amount of the wt of half a panel of truss. 

The loads on the posts may then be written upon the diagram. 

The obliques or ties, however, when the load is at the bottom, bear fas in the 
; ink) the same amount of strain from the moving load and floor, as when it is on top. 
herefore, when it is at the bottom, each pair of ties sustains not only the load rest- 
’ ig on the post which they uphold; but the wt of one panel-length of floor, and the 
1 ax panel-weight of engine. In other w r ords, whether the load he on top , or at bottom , 
1 le two ties at the foot of each post, sustain a wt equal to a full panel-length of truss 
'id floor; together with the max panel-wt of engine. Having added these wts to- 
jther, lay off their sum by scale at each post, as shown at / v. k v,j v, % v. Fig 22 ; com- 
lete Igvu, kgvu, & c; and measure the strains lu,lg,ku,k g, &c, along the ties. 

The strains on any pair of ties, may also be calculated thus; having 
1 ie load they sustain. 


Strain on 
short tie 


T sy h or dist from post to 
farthest end of chord 

total length of truss 


length of 
short tie 

length of 
post. 


7 , v It or dist from post to length of 

Strain on ° CU ' nearest end of chord long tie 
long tie total length of truss ^ length of 

. post. 

The hor strain on the chord will be uniform throughout, as in the Fink 
uss; and will depend upon the max uniform load that can cover the whole bridge; 
id not, as in the case of the ties, upon the greatest load which each pair of ties may 
eve to sustain in succession; unless we assume our max uniform load to be a string 
engines which may bring a max panel-wt of engine upon every pair of ties at 
ce. In that case we have only to measure upon one-half of our working diagram, 
e several hor lines corresponding to u o, &c, in Fig 22; and their sum will be the 
qd hor strain on the chord. But if we take our max uniform load on the whole 
uss, to be a string of cars , w r e must diminish the chord-strain thus found, in this 
anner: Add together a full panel-weight of truss, floor, and cars; then, as the full 
nel-wt of truss, floor, and engine , (which we before assumed as the straining load 
each pair of ties,) is to the panel-wt of truss, floor, and cars, just found, so is the 
>r chord-strain before found, to the one reqd. 

We will repeat, that chords must be strong enough to bear not only the hor pull 
push to which they are exposed ; but also to sustain safely, as beams, the trans- 
r*» strains from the floor, and from the moving load, when these rest upon them 










588 


TRUSSES. 


BOWSTRING, AND CRESCENT TRUSSES, UNIFORM] 

LOADED. 

Art. 18. Before attempting to find the strains on either a uniformly loa 
bowstring or a crescent truss, Figs 23, 23 6, 23 c, by means of a diagram, the student should fatnil 
ize himself with the following remarks : 

Rem. 1. The basis of the entire process is that at every point of s 

port, beginning at an abut as the first one, we have acting one or more known forces, balance 
held in equilibrium by either one or two unknown ones ; and the object at each point is first 
means of the parallelogram of forces, to find the resultant of the known ones; and second bv 
same pnnciple, to resolve this resultant iuto two components in the directions or the unknown o 

This is all that is required in either the bowstring- or t 
crescent truss.* 

Rem. 2. While more than two unknown forces exist at i 

point of support, their amounts cannot be found. If one force is know 
and two unknown, the three bulaucing each other, draw a line by scalt 
represent the amount and direction of the known one ; and, considering it as one side of a trian 
from its two ends draw lines parallel to the two unknown ones, to meet each other, thus comolei 
the triangle. Then these last two will by the scale give the two unknown ones ; because when tl 
forces meeting at one point, balance each other, three lines representing them both iu amouut 

in direction, will form a triangle, t 

If I here are two or more known forces, first find the single 

sultaut of them all and. taking it as one side of the triangle, find the other two si 

(that is, the two unknown forces) as before. After a little practice the student will find it uune 
saiy to draw more than half the sides of the parallelogram of forces. 

Rem. 3 The bow Is to be considered straight from apex 
apex. 11 actually curved it will be much weakeued. 

Rem. 4. At each point of support, or apex, consider every member that meets there to be a f< 
either pushing towards said point, if along a strut; or pulling fM 
ifo 8, on ^ H * le * ^* us 18 shown by the arrows in Figs 23 23 a • thus 
F,g 23 ato, the force, along the strata o, S o, and Z,%so’- 

pushes towards q ; while the lorce along the tie r o palls from 0 as also fro’m r 1 
loads on the bow are forces pushing vert downwards. 

Rem 5. If the known forces are not all alike at anv nnint ft 

is, neither all pulls nor all pushes,) then while constructing the parallelogram, of r ’ 

of the force, must be changed, and be regarded as actinga?the < 

sstf !SSis before: 80 as * 

three known forces acting at c, namely ec pushing towards c- a lna/Wn ^ a8wl seen hereat 
pushing vert downwards towards c ; ludp cpuiufgf rot S in thticZ ft* ^ ‘ 

the sides of the parallelograms, either consider the pull p c'to be changed to a push in the direc 

* The same process applies equally to all our figs from 

^ a * s .°? w j let ‘ er uniformly loaded on one chord or on both • also to the 
verted bowstring, and to Fig 23d. In this last i. In th« „ , 1 > <«iso to me 

which prevents the truss from spreadiug, and thus causes it to exert onutlvTp^sZuiletthell 
1 his is a distinctive character ot all so-called truss gJruers including all 
figs hack to Fig 5. But when »«» is converted into an arch Joi snstalni 
compression, it ceases to be a tie. and the truss then exerts horas wefl m , 
force against the abuts; and becomes what is called a braced arch The i 
cess requires a slight modification before it can be used for such, as shown in Art 20 ! 

though strictly speaking the process does not apply to the Fink truss m v. 

there encounter three unknown forces at anv point where three vefiJi?.. 20, 21 * beca , u ® 
meet at a rafter as at c, still as we can readily de enr ine theTtrafn onZuZf ZT' C \ C5 ’ 
the entire vert strain at such point, (which strain can be found in a F^k truss ST “ 
calculation) we thus reduce the unknown forces to two and th»r»fni. A tru , 8 b > a mere 

for such trusses. The student would do well to tbe proce88 ' 

t Any one of the three forces is then the auti-resultant or 

X\ a h ncer ’ ° r h opposer, of the other two; and if three arrow-hen 

S \ showing their directions be added to the sides of the triangle ® “thUFh thfv 

4 ^ a ? vvete ’ chase each other around the triangle • that is ’th</» 

“ of an J one of them will touch the butt end of the one next to it • l6 ’ t,le 1 

w.ll « B,. (his i» ..„( »„ when ,h£ e 

forces and their resultant, or equivalent in effect • for the arr,fwhl P H?t! 
sultaut will then meet that of one of the other forces. ’ ai how-head ol the 











TRUSSES. 


580 




































TRUSSES. 


^ccordinl^the 1 known t ,T?, *° be similarl >' changed to pulls. The first of course is the easiei 

Kem. 6 . To decide whether an unknown force is a pull 01 

ftr!? U h av Ll ‘ 8W h *' 11 ‘ e r th , e “® niber alt)11 K which said force acts is a tie or 

tion Then ,u resu f t0f the k ? own forces .’ add “> a » arrow-head to show its dire. 

. . . , £ this resultant completed the triangle by means of the two unknown force- 

the trlZ.e Th t0them a,S0 ' P lac * n 8 them so that the thfee arrows shall chast each . her aroun 

last oue. For the present we will C alM*t the et m ri *' a,n ^ b ® a st ^ ut in the first case, and a tie in th 
web member. Nowitcan Zllys be Leu « J b u 0th . er I orce wi “ be a '«-‘K 

to displace the apex at which thev »e. r hat the resultant and the chord-force together ten 

truss f and if weTmpiy^nsider^atIt % X^t^f’VT lnwards ’ frf ' m towards th 

tendency, we shall have no trouble in deciding whether it must'fo^thlt'n.rn“ teraC li th,S displacin 
apex, or in other words be a tie or a strut 8 t0er U 11)1181 for that P ur P ose P*»H or push at th 

k, °'” '»"**• ““ ‘“™ '»«*■ themselves me, 

***** **** an< 

mil m.uy errors. A little practice however wiil roctitAhi.*"* fim attem P M probably com 

ssss. i; rn d h r„“^r£i‘ s tf “ s --*>« 

sss^f 

ot the web members are less than .1 of a ton • so that the’ P 592 ’ tha ‘ the 8tra >“* on sorn. 
us to the extent of 100 per cent, or more in these small s.r^I V “7 eVen , w \ th great care wislen. 

mark* to ,uTZK!siri»s .r ‘“sTil SP* 1 *,'-?* .*»"*««..* «* 

ft. The bow is divided into 8 equal parts ■ and tin* lnwe^ni^I Its span 18b0 > ,ts nse 1< 

the upper ones. The trusses aie assumed'to be 7*ft apart from^enter^® r,zo “ talll y_half-way betweei 
truss aud its load is supposed to be equally distributed afon^ me h ce , nter - T he total wt of th. 
tons, which corresponds to 40 lbs per square ft of roof covering o“ d to amount to 10 .. 

load ; and half as much, or .66 of a ton resting directly tor whhont ret. 8 *™ 1 i‘ S ,on f for a ful1 P anf 

sisf **• <&■«•. them z-srjass srrs wsszz 

line represents the vert upward reaction of t h« n Ji « ° °“ e 4 J but » or . to 5.2 — .60 = 4.55 tons. Tbii 

and load that causes the strains which we are about tefeeek t ^This reacting Wt °7 ? e balf trusi 
one. balances the two unknown forces « P alone the !m .„h J , react ‘ ng fo . rce > wh,ch > s » knowt, 
have only (Rem 2; to consider a v as one side of a trianel’e anH ? a, °? 1g the string - To «“ d these w. 
iug lines a n aud vn parallel to or ?n the same d reel ons «« m“ ,to w tW0 ends t0 draw two meet 
give by the same scale 9.94 tons strain along thehorstrine “ 8, d tbe M “ ak “°. wn for , ces - Then will „ , 

rjf rss* ,i;r *<“™ * »”• £ 

•I* b5‘«cg*“A*^£*" wn"^ k "°r «... 

r:SmrSheV zi sr tP- s 

forces, namely ec along the bow^; and ep along the fihlim,? ! ^^ Glance 1 w t O Unknowi 
I 1 ig W) to represent a e ; and x e to represent tlfe 1 I »ni' < | Ue A H ®? ce we onl J to draw /« fse 

^ rf e^by 

tiir;;“?m, irs,r, kl rb. l! i“ 


rormlv distributed are eq^aMhat^L^aTie^Soa^'/re^soeaual 8 ^ 610 ^ 8 & l° ne Which a load is UD| 

rectly on an abut is just half a panel load. q ’ ° 1 ' that the P orl iou which rests d\ 

inasmuch as that por tfotf of s'aid t* w h fc^ rms' dir ect L*on lV< h h' f "h ° f the trUSS and its load ? ho 
of the web members, and therefore has nothin^ abut, does not reach the abut by wa 

ure set up at the abut for a 8 certa?ning sa?d s trains WUh thGlr 8train8 ’ U is omitted ^om the pries 

supported only"at'iu^ds, U wHI VoTdo'to assmnlth-u thMm^ig'^c‘ h . e raf / er ’ and thp latter i: 
a rafter presses, at its foot, in the direction H a and nl in .w D ^ 0DCe .'‘ trated Said ''P d *. Sucl 
fore if a ar in that Fig be drawn to represent thf ini’wlnl relctiol ofThf ?" f W Ie,,gth 0 a - There 
angle a H a- (not aox) ; and H .<■ (not o r) gives the pull o, the tie he.m Rn.’iT 0 m, ! st draw th e tri 
the strums are more readily found by the method given in connectten^ith Fig" 










I 


TRUSSES, 


591 


p, thus completing the triangle des. Then ds gives by scale the strain 10.5C along cc; and ei 
lives .lt.of a ton along ep.* 

Now going again to the apexp, we fiud that we have two known forces p a, p e, both pulls, balanc- 
ng two unknown forces p q, p c. Therefore as at Fig Y. take p f and p s to represent the two known 
orces ; complete the parallelogram fp s d ; draw its diagonal pd ; and taking it as one side of a tri- 
1 A.ngle, draw d o parallel to p q, and p o parallel tope, t hen is do by scale 10.01 tons strain along 

andjio is .10 of a ton along pc. doing to c, we have three known 
forces, c«, cp, and the 1.3 ton panel load, all of them pushes, so that none of 
hem need be reversed; aud balancing co, cq, both unknown. In this case, as shown at Fig V, we 
oust draw two parallelograms, beginning with any two of the known forces. Say 

re begin with cx representing the 1.3 tou load, and ce, representing the 10.56 tons. On these two 
Iraw the parallelogram c etx. aud its diagonal c t. Then draw ca to represent the third known force 
I pof .10 of a ton ; and on it and the diagonal ct draw the second parallelogram c t s a, and its di- 
gonal cs. This last diagonal is the resultant or single force which would balauce the two unknown 
orces co, cq; therefore take it as one side of a triaugle, and from its two ends draw two meeting 
ines parallel to said unknown forces, and measure them by scale to obtain the amounts of those 
arces. On account of the smallness of our scale we have not shown these two lines. In practice, in 
rder to prevent confusion, it is well to rub out the sides of the parallelogram after having found the 
rst diagonal. In this manner proceed until the final strains on o s, sr, and r t complete the whole. 

If the entire uniform wt of truss and load is assumed to 
>e on the string or lower chord, as in Fig ‘23 a, then all the web mem- 
ers become ties; but the process remains unchanged. Therefore first distribute the entire wt 
mong the lower apices and abuts by our “ first process.” Theu, as in Fig 23, draw the vert line a v 
= half the entire wt, minus what rests directly on one abut) aud from it fiud the strains on ae and 

р. Then going to e we have one known force ae balancing the two uukuown ones ec and ep. 
.fter finding these go top. Here we have three known forces, p e, p a, and the panel load; the first 
iro of which are pulls, while the third (resting on top of the string) is a push vert dowuwards. 
herefore we will reverse this last, and consider it as being a vert pull pi; aud draw the first par- 
Uelogram on p i and p a. After finding the resultant of the three pulls, and by it the two forcesp q, 

с, we go to c. Here of the two known forces, ec, being a push; and pea pull, we must reverse 
ne of them, say p c; representing it by an arrow t c; and complete the parallelogram on tc aud ce. 

We have now had an instance of all the cases that occur, and have shown how to ruauage them. 

Proofs of accuracy of the work. The resultant of the strains on 

lose members (o s and r«), Fig 23, of the half truss that meet at the center s, of the bow, and of half 
16 le panel load, s, at the center, should come out to be a hor line, and equal to the hor straiu along 
le center stretch, r t, of the string. With all the care, however, that can be taken, it will be very 
ifficult to make the coincidence exact. In some trusses the half truss will have three members 
leeting at the center of the bow ; one of them (a center vert one) belonging partly to each half of 
le truss. Then only half the strain on this one, as well as half the center panel load, is to be used 
i the proof. All this applies to any form of truss however loaded. 

b Again, if all the uniform wt of tlie truss and its load is as- 

anted to be on the hor string, and if the string is divided into equal, 
“ »r nearly equal, parts by the web members, the hor strain at the center 

iould be equal, or about equal, to 

Wt of half truss and load X .25 of the span 

Depth of truss. 

ur ut with very unequal divisions of the string, such as will rarely occur, this formula is not even 
>ughly approx. 

With all the wt uniformly on the bow, unequal divisions of the 

ring have no effect on the center hor strain ; and if the rise does not exceed about one tenth to one 
ghth of the span, and the bow is about evenly divided, the above formula will be nearly as approx 
i when the load is on the string. For greater rises, however, multiply the span by the following 
R ultipliers (.instead of by the .25 of the above formula), when the load is on the bow; and the half bow 
jout equally divided into at least two parts. 


.1 or less 

.15 

] 

.2 

Rise, in Parts of tlie Span. 

.25 | .3 | .333 | .35 | .4 

(Original.) 
.45 | .5 

.246 

.243 

.239 

Multipliers. 

.233 | .226 { .222 j .219 | .211 

.202 

.192 


Reus The multipliers for intermediate rises may be taken in simple proportion. When multiplied 
i r the span they give the hor dist from the abut to the center of gravity of one half the loaded 
>w, (as the .25 or the formula gives that of one half the loaded string,) assuming the load to be 
.nncentrated at the points of support on the abut and at the apices. It is a his 

enter of gravity of half load so concentrated that must be 
ised for finding* the strains in the truss, and not that of half load as 

■tuallv evenly distributed; for these two centers of gravity under these two aspects may differ 
•eatlv from each other, not only in the evenly loaded bow, but in the string, if divided into very an¬ 
imal parts by the web members. Even the number of divisions of the loaded bow makes some dif- 
rence in this respect, but not so great but that the multipliers in the above table will be correct to 
ithin about three per ct at most in any case in which the entire bow has at least four nearly equal 
visious. 


It is usual and far more convenient, to draw all such lines at the apices of the diagram itself: hut 
account of the smallness of the scale of Figs 23, and 2.3 a, we have drawn them at R, 1. and T, to 
vent confusion. Also our lines are not drawn to scale because some of the forces are too small 
>e appreciable with so small a scale. 

















592 


TRUSSES, 


If both the bow and the string; are iisiiformly loaded, it ia 

plain that the multiplier for any given rise must be somewhere between the .25 of the formula, and 
•the decimal for that rise in our table; and this furnishes an easy method of finding, approx enough 
for practice, the hor dist from the abut to the cen of grav of a half truss thus loaded. Thus after 
allotting to the bow and string their respective proportions of the entire wt of the truss and load, 
find the hor dist for each of the two separately ; and then combine them. 

Table of approximate strains in tons on Jtowstring trusses 

of 80 ft span. Trusses 7 ft apart from center to center. Load (including wt of trusses) 40 lbs per 
square ft of roof covering; all assumed to be uniformly distributed on the bow. Bow divided into 8 
equal parts; like Fig 23; and straight from apex to apex. Lower apices half way hor between the 
upper ones. Each column of the table commences near an abut, or end of truss. The first or end 
web member in the columns is a tie, the next one a strut, and so on alternately towards the center, 
as in Fig 23. Below the table is given the wt of each entire truss and its load. 


Rise 20 ft, or % Span. 


Rise ft, or % Span 

Bow. 

Tie. 

Web. 


Bow. 

Tie. 

Web. 

Tons. 

Tons. 

Tons. 


Tons. 

Tons. 

Tons. 

7.01 

4.85 

0.38 


8.80 

7.50 

0.22 

6.02 

5.18 

0.35 


8.30 

7.72 

0.18 

5.56 

5.34 

0.25 


8.03 

7.78 

0.15 

5.34 

5.38 

0.25 


7.90 

7.88 

0.15 



0.10 




0.10 



0.10 




0.10 

Total wt = 11.6 tons. 


Total wt ~ 10.75 tons. 


Rise 10 ft, or J4 Span. 

Rise 8 ft, or j-Q Span 

Bow. 

Tie. 

Web. 

Bow. 

Tie. 

Web, 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

10.9 

9.94 

0.14 

13.3 

12.5 

0.07 

10.6 

10.01 

0.10 

13.0 

12.6 

0.05 

10.4 

10.16 

0.07 

12.8 

12.7 

0.07 

10.3 

10.22 

0.07 

12.7 

12.7 

0.07 



0.05 



0.03 



0.04 



0.03 

Total wt = 10.375 tons. 

Total wt zr 10.25 tons. 


Art. 20. Tlie braced arch. Fig 23 W. Find the center of gravity of 


m win 



either half, as one, of the truss and its load. Then 

Weight of said half ^ Hor dist of said cen of grav 
truss and load, in tons * from the nearest abut 

Vert dist c s from abut c to cen o of truss 


Ilor pres at the 
cen o of the truss. 


Having thus found the hor strain 1c at the center, and knowing the wt t) c of the half truss and 
load minus the part panel-load that rests direotly on the abut, draw them as showu, and find theii 
resultant r c. Then use this resultant precisely as the vert line a v. Figs 23 and 23a, is used 
namely, by drawing from its ends two Hues for finding the first two unknown forces; then proceei 
precisely as in those figs. 


Proof of accuracy of f lic work. If all has been done correctly, th 

last resultant near the center of the truss will be in a straight line with the last member; the strai 
along which it represents. Also, the resultant of those members of the half truss which meet at th 
center of the truss, and of half the center panel load, should come out a hor line equal to the ealcu 
la ted hor strain. But as in the bowstring. Ac., it is almost impossible to secure a perfect agreement 
It would be well to test the strains near the center, especially the very small ones, by the principl 
indicated bv the following remark. 


Rem. It is not necessary to begin at an abut in order t 
work out the strains; for after having disposed the load properly amon 

the apices, and calculated the hor strain at the center of a truss, we may employ said strain, and oi 
half of the panel load at the center, as two known forces acting against'the half truss at the cente 
find their resultant; and resolve it aloug the two members of said half truss that meet there - ar 
thus work on down to an abut. The line representing the hor strain must evidently be drawn’as 
pushing against the half truss whose strains are sought. This remark an 
plies also to bowstring trusses, and others, as Figs 10 and 11 &c 

in which we have two or more known forces acting against our half truss at the center; and a 
more than two unknown forces of our half truss meeting there also. In an actual bridge there wou 
be a hor member w l, supporting the flooring, and a short vert one extending upward from o a 
helping to support wl. But these do not form part of tne truss proper. 

















































TRUSSES. 


593 


S. ti n evers l Suppose the half o, n, c, of the braced arch. Fig 23 W, to be a 
ssed cantilever, within and c flimly built into, or attached to a wall; and loaded either alone the 

er endnranex ven° k calcula , te the strains > be 8'n w '’h only the load concentrated at the 

j f ? r .* pe j °’ as f, g,ven kn ° wn force, and resolve said force along oiandoji; aud so on to n 
.1 c, taking in on the way the loads at the other apices as before. The uniter 
lord will be in tension; the lower in compression. A revolving 
uss drawbridge, when open, assimilates to two cantilevers joined back to 

krt. 21. _ This fig represents an a 

tened swing-bridge support- 
on rollers on the pierp, and by de¬ 
ls a o, at, &c. All the wt of ne is up- 

d by a n ; that of e s by a i, &c ; and \ 

it of s t directly by the rollers. Draw / .t C .8 \Jl 

and i Ic vertical to represent the wts 

oe and cs; and draw bm,kn hori- |JD| 

>tal. Then will o m and i n give the 




- — •• •»! >IV M'UU C. tv T c tliu 

ains along ao and a i. Also b m will give a hor compressive strain reaching from 
p c: and k n one reaching from i to c. 
















594 


TRUSSES. 


Art. 22. Fig 28, shows the general arrangement of a niiih 
wooden Howe bridge-truss; Fig 29, some of its details ; and Fig 30. tho 
of an iron truss. High trusses are sometimes made as in Fig 1. The top and Lotto 
thords of the wooden one are 

i 


each made up of three or more 
parallel timbers c c c, placed a 
6 mall dist apart, to let the vert 
tie-rods r r pass between them. 
The main braces, o o, are in pairs 
or in threes. The pieces com¬ 
posing them, abut at top and bot¬ 
tom, against triangular angle 
blocks, $; which if of hard 
wood, are solid; and if of cast- 
iron, hollow; as shown at T, 
Figs 30; strengthened by inner 



vm/// Il 


■*>&////■////// 


HOWE 

Eic,23 


ribs. 


D 


These extend entirely across the three 
or more chord-pieces. Against their 
centers, abut also the counterbraces e. 

These are single pieces in small bridges • 
or in pairs, in large ones; and pass be¬ 
tween the pieces which compose a main 
brace. W here the wooden braces and 
counters cross each other, they are 
bolted together. For wooden chords, 
the angle-blocks are cast,* as at T. The 
dotted lines show the strengthening 

x serves to keep the block in place. 


t 



'H 

it 

c 

■vs| 

% 

W/, 

1 

c ] c j 

! 


s tM 

0 

□ 

0 



r ‘ 

t 

Erf. 2 9. v 


nbs; and x serves to Keep the block in place. The vert tie-rods r r of iron are 

end ’ The heads^nlTfeet of C " S U> 8i ? f bridge; ' vith a 8Crew and nut at 

InH !! and feet of the braces and counters, butt square against the angl 

blocks and are kept in place only by the tightening of the screws of the vert ti 

.L™m « *■ ! B -?• e " d W. vd: and the end, giZ *» /! . 

the upper chord, muy be omitted; also tcaudiy; but it is seldom done. 



In Figs 30, of an iron Howe truss, the top chord P, M, and W, is cast in one pic 
transversely, as at P. Its separate lengths are connected together by flanges a 
bolts, somewhat as shown at W; where a a are cast longitudinal flanges for strenal 
emng the transverse bolting-flanges g. Instead of separate angle-blocks at the upr; 
chord, solid ones may be cast in the same piece with the chord itself, as shown at 
lhe lower chord usually consists, as in other iron bridges, of four or more flat bar6 
rolled iron c, placed on edge; and some dist apart, as at R. On top of them rest t 
lower angle-blocks 5 , which have shallow channels below, for receiving the cho 
pieces; and thus securing them from lateral motion. A cast washer, a, below t 
chords, is provided with similar channels on top, for the same purpose. The brae 


.. lD , t0 prevent , the Pressure of the heads and feet of the obliques from crust 

the Jingle-blocks are cast with deep projecting flanges under their bases'*; and which i 
ng between the pieces which compose a chord, extend to the opposite face of the chord There 

^ UP ° n , br °^ W l S , her8 at ,he ends of the "xta. By this rne^ns the vert componei 
of the strains along the obliques are transferred directly to the verts without at all 
«hords. Angle blocks of course have openings for the passage of the vert rod*. 8 







































































TRUSSES. 


595 


n id counters, o, e, in moderate spans are usually cast in a star-shape, as at j* The 
ti« Uo , wln ? table gives dimensions sufficient for a strong Howe bridge; although in 
toi ^oden bridges it is customary to add arches when the span exceeds about 150 feet. 

Dimensions for each of two trusses of a Howe bridge for a 
[ingle-track railway. Timber not to be strained more than 800 ibs per so 
cn; nor iron more thau 5 tons per sq inch. Iron supposed to be of rather superior 
lalitv, requiring from 25 to 27 tons (60480 lbs) per sq inch to break it. The rods 
be upset at their screw-ends. To each of the two sides of each lower chord is sup- 
•sed to be added, and firmly connected, a piece at least half as thick as one of the 
t °i'd-pieces; and as long as three panels; at the center of the span. 


! 

'*3 

a; 

v 

' fc* 

Rise. 

Feet. 

No. of Panels. 

An upper 
Chord. 

A lower 
Chord. 

An End 
Brace. 

A Center 
Brace. 

A Counter. 

End Rod. 

Center 

Rod. 

No. of 
Pieoes. 

j Size. 

No. of 
Pieces. 

Size. 

No. of 
Pieces. 

Size. 

No. of 

Pieces. 

Size. 

• | 

No. of 1 

Pieces. 

Size. 

No. of 

Rods. 

a 

J _eJ 

3 

V-. . 

O on 

o o 
S5« 

a* 

a 

3 






Ins. 


Ins. 


Ins. 


Ins. 


j 

Ins. 


Ins. 



. 

5 

6 

8 

3 

5X 6 

3 

5X12 

2 

5X 8 

2 

5X 6 

1 

5X 6 

2 


2 

i 


0 

9 

9 

3 

6X 9 

3 

6X14 

2 

6X 9 

2 

5X 8 

1 

5X 8 

2 

\\ 

2 

i <4 


|5 

12 

10 

3 

ex 12 

3 

6XH 

2 

6X11 

2 

6X 8 

1 

6X 8 

2 

i \4 

2 

144 


0 

15 

11 

3 

6X14 

3 

6X16 

2 

8X12 

2 

6X10 

l 

6X10 

2 

25* 

2 

154 


3 

18 

12 

4 

6XU 

4 

6X16 

2 

9X14 

2 

6X12 

1 

6X12 

2 

s 

2 

1 % 


0 

21 

13 

4 

8X14 

4 

8X18 

3 

8X14 

3 

6X10 

2 

6X10 

3 

2'K 

3 

144 


d 

24 

14 

4 

10X16 

4 

10X20 

3 

8X13 

3 

8X10 

2 

8X10 

3 

3 

3 

154 

t 

0 

27 

13 

4 

12X16 

4 

12X20 

3 

9X16 

3 

8X14 

2 

8X14 

3 

35* 

3 

1« 




The same dimensions will serve for a double road for common travel. 

For bridges of iron, assuming the safe strain for iron to be 5 tons per sq inch, or 
times as great as the 800 Ibs assumed for wood; the areas of the cross-sections 
the individual members will as a general rude approximation, lie about oue- 
irteeutU part as great as those of wooden ones. E<gunlly strong wooden, 
id iron bridges of the same span, will not differ very materially in weight. 

Art. 23. Fig. 31, 
y X Wi M shows in like manner, 

' ' a wooden Pratt 

truss: and Fig 32,some 
details of a small iron 
one.* After the forego¬ 
ing, they do not need 
much explanation. Since 
the angle-blocks support 
tension rods instead of 
struts, they are placed 
above the top chord, and 
below tbe bottom one. 
The main obliques are 
in pairs: and the smaller 
single counters pass be¬ 
tween them, as in the 
Howe. In large bridges 
they are in threes, fours, 
Ac. The vertical posts, 
which, when of iron, are 
hollow,*are retained in 
their positions both hy 
the strains on the 
obliques, which termi¬ 
nate above and below 
their ends; and by being 
let into the chords. In 
large spans, the details 
generally vary more or 
3 from those in the Figs. In Fig 31, cccare the main braces ; andooo the counters. 



: Cast iron Is now rarely used in trusses, except for short blocks sustaining compression, such as 
;le-blocks etc. Rolled iron and rolled steel have taken its place. 















































































































596 


TRUSSES. 


When the roadway is helow, as in Fig 31, the ends, rb,y r, of the npper chord ; t 
end verticals p and u ; and the two tension obliques in each end panel, may be om 
ted ; and two diagonal struts from b and y must then be substituted, extending 
the abutments, for upholding the upper chord, Ac. In <tl«» Pratt the chor 
may be of the same dimensions as in the foregoing table for Howe’s. The posts m 
have about Ath less area than the main braces of the Howe. The main brace ro< < 
and others, (of the same number as the main brace pieces of the Howe,) may ha i 
the lollowing diams in ins; allowing the safe strain to be five tons per sq inch. 


For each Trttss of a Pratt Bridge. 


25 Ft. 


50 Ft. 


75 Ft. 


Spans. 

100 Ft. ; 125 Ft. 


150 Ft. 


175 Ft. 


200 Ft. 




• 

End Main 

-brace Rods. 



Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

lot. 

Ins. 

lD8 

2 of 1ft 

2 of 2ft 

2 of 2ft 

2 of 3 

2 of 3 

3 of 3 

3 of 3ft 

3 of 39 



Center Main- 

-brace Rods. 



Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Tns. 

Ins. 

Ins 

2 of 1ft 

2 of 1ft 

2 of 1ft 

2 of 1ft 

2 of 1ft 

3 of 1ft 

3 of 1ft 

3 of I? 



Counter Rods at Center. 



Ins. 

Ins. | 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Tns 

lof 1ft 

lof 1ft 

1 of 2ft 

1 of 2ft 

1 of 2ft 

2 of 1ft 

2 of 2 

2 of 2> 


In Pratt s truss the directions of the main braces and counters are respective 
the reverse of the Howe. Many of the remarks in the preceding Art, apply equal 
in this. Neither the Howe nor the Pratt possesses any special advantage over t 
other as regards ease of adjustment, Ac. In both trusses, arches are frequently add 
in wooden railroad bridges when the span exceeds about 150 ft. 

Art. 24. Town’s lattice truss. Fig33, as originally introduced, and ve 
extensively employed, was of extremely simple construction'; being comnosed * 
tirely of planks from 2 to 3 ins * v 

l X 


thick; and from 9 to 12 wide; de¬ 
pending on the span. Two sets 
of these were placed crossing 
each other at angles of about 
90°; and were connected to¬ 
gether at tlieir intersections by 
either 2 or 4 treenails of locust, 
or other hard wood, about 2 
ins diam. At the top and bot¬ 
tom, similar planks, a a, c c, 
were treenailed hor, to form the 
chords; or in large spans, (many 
exceeded 150 ft,) there were two 



upper and two lower chords, (sometimes of timber 6 ins thick,) as shown in the fi 
by n n o o 1 lie transverse section A shows the two npper chords on a larger seal 
each chord consisting of two planks; one on each side of the lattices. Two truss 
ol this kind, with a depth equal to % or A of any clear span not exceeding about 1 
ft; pianks of 3 X 12 white pine; the open squares ft on a side, in the clear, were co 
Bidered sufficient for a common-road bridge 20 ft wide. Many of these bridges warn 
sideways very badly; and when applied to railroad purposes, failed entirely, 
some cases the better mode of three lattices was employed; two of them running 
one direction ; and the third in the other direction, passing between them. A fund 
rbmlli that the parts were of equal size throughout the span; whereas t 

clioids should be stoutest at the center; and tlie lattices, near the ends. These d 
fects caused the lattice to fall into unmerited neglect, in the United States • where 
























































TRUSSES. 


597 


W*ma 

« rod 
w hai 

ch. 


MU 


Europe it is, when properly proportioned, highly esteemed; especially for iron 

bridges; some of which, on this principle, have been 
constructed of more than 300 feet span. The tendency 
to warp, owing to the thinness of the trusses, is obviated 
in the large bridges referred to, by placing a double 
truss, Fig 34, (instead of a single one,) at each side of 
the bridge. The trusses T, D, composing a double one, 
are placed a foot or more apart; and are connected to¬ 
gether at proper intervals, by short pieces riveted to 
each one, for stiffening them. At T and D are seen the 
three lattices or lattice-bars, of each truss; two of 
ich, on the outside, constitute the main braces; while the center one is the coun¬ 
brace. Such a double truss bears some resemblance to the Fairbairn box-girder; 

diflf being chiefly that in the former the sides are composed of lattice-bars ; and 
;he latter of solid plates. 



















598 


TRUSSES. 






• .. *. 


The Bowstring truss, Fig 35, is an excellent one as regards strength, 
economy of construction. It has, however, the disadvantage, in large spans, of a ( 
culty in connecting together overhead the two trusses of a span, so as to be as 

~ _ _ from lateral vibration as a hr 

with parallel upper and lc 
chords. In the latter, this 
nection can be made from en 
end of the span; but in the I 
string it can be done only 
some distance each way from 
center; from want of head 
near the ends. In short sj 
with low trusses, this defet 
not felt. For a graphical met 
of finding the strains in 1 
string and similar trusses, se 
583 to 592. 




































TRUSSES 


599 


i 

to 

loi 

3 cl 
fDlU 


* 


’he Lock Ken viadnct, England, of 130 ft clear span, and 18 ft high, 
l two trusses, 13 ft 8 ins apart clear, for single-track railroad, on the Bowstring 
jiciple, Fig 35, omitting only the verts; which, however, affects the strains on the 
i ques. For convenience of construction, it was built chiefly of rolled channel* 
i, (see 11,) of 8 ins by 4, by 4, by % inch. At Fig 35, d shows a transverse section 
^te arch or bow; which is uniform throughout. That of the uniform chord or 
jug is of the same fig and size as the arch. Each has an area of 33 sq ins in each 
is. The top strip, a, a, of rolled iron, is 24 ins by %; and is riveted to the upper 
gesofthe two channel-irons 11 ; which are 8 ins apart, so as just to allow the passage 
A'een them of the obliques d ; which also are of the same channel-iron. The panels 
I about 12 ft long near the center of a truss; and 8 ft at its ends. The wt of the 
’’ 1 trusses alone, without the roadway, is very nearly 50 tons. The w r t was increased 
ond the theoretical requirements, to save the trouble and expense of preparing 
fitting together many pieces of diff dimensions; yet, although the bridge is a 
ng one, the trusses alone, weigh together but .37 of a ton per foot ruu. Where 
obliques cross, one is in two pieces, riveted to the other by straps. 

the State of New York aro many much-used bridges for common 
ivel, by Mr. Whipple, of 100 feet span,and 12% ft rise; with two trusses 19 
Ijlpart from center to center; for two roadways; and having two outside footways, 
6 ft wide. Each truss has 9 panels, braced altogether by vert and oblique tie- 
3^1 (no struts) arranged as in Fig 35. The verts next the center of each truss, consist 
Ji of two rods of ins diam, welded together at top; and straddling 2 ft at bottom. 

other verts are single, and 2 ins diam. Theobliques are all single, and l%ins diam. 
ieti arches are of cast-iron. The transverse area of metal of each arch is 18 sq ins at the 
vn; and 21 sq ins at the spring. The shape of arch transversely resembles a channel- 
with its back upward; the total depth of flange 7 ins; the width of arch on top, 11 
at center of span; but increasing uniformly by means of wide open-work on top, 
ft at springs. Each consists of 9 straight segments, held together at their but- 
flanges, by the verts themselves; which pass through them, and have screws and 
u at their ends. The screw-ends are not upset. The thickness of metal in the 
ics nowhere varies much from % inch. Under the floor, and between the trusses, 
horizontal diagonal braces of rods •% inch diam ; two of them to each panel; each 
|hem with a tightening swivel. The chord of each arch consists of 4 rods of 2 ins 
n. In the same State, are also many similar common road bridges of 72 ft span. 
I, 9 ft; two trusses, 19 ft apart from center to center ; * and two outside footways 
ft each in addition. Each truss has 7 panels, with vert and oblique ties, as in Fig 35. 
h cast-iron arch is in 7 straight segments, of tlie same shape as the foregoing; 
hi a cross-area of metal of about 12 and 15 sq ins. Its width at center of truss 10 
at springs, 30 ins. The two verts next the center of each truss, consist each of 2 
|) of 1% diam ; the other verts are single, each 1% diam. The obliques are all 
■le. 1 inch diam. The chord or string of each arch, is 4 rods of 1% inch diam. 
i izontal diag bracing of 3 4 inch rods under the floor, as in the foregoing. 

©me cast-iron bridges of the Severn Valley Railroad, 
g land, of 200 ft clear span, consist of arches rising 20 ft, and supporting the 
road cm a level with the tops of the arches, instead of above them as in Fig 35%. 
Ire are no diags between the arches and the roadway, as in that fig; but cast- 
verts only, placed 4 ft apart. The railroad is double track ; and there are four 
les one under each line of rails. The transverse section of an arch is I; each 
ge ’is 15V' ins wide, by 2 ins deep ; the web is 2 ins thick. Total depth at center 
pan, 4 ft; and at the skewbacks, 4 ft 9 ins Transverse area of each rib at crown, 
sq ins. Each arch is cast in 9 segments of equal length. 

'he cast-iron bridge across the Schuylkill at Chestnnt St, 
lila, Strickland Kneass, Esq, Engineer, roadway on top, has two arches of 185 ft 
r span each, and 20 ft rise. Clear width, 42 ft. Each arch has 6 ribs, about 8 ft 
rt in the clear; and of the uniform depth of 4 feet, including a hor top rib 8 ina 
e • and a similar one at the bottom. Thickness everywhere 2% ins; thus giving to 
ti rib a transverse area of 147% sq ins. The standards are vert, with ornamcnta* 
i It is a city street bridge. The roadway consists of cast-iron plates, which sup- 
t a pavement of cubical blocks of granite, laid in gravel. The arches are cast in 
nents 12 ft 10 ins long; each with end flanges 12 ins wide, for bolting them to¬ 
iler with four \% inch diam screw-bolts at each end. For a change of tempora- 

- from 12° to 99° Fall, the crowns of the arches rise 2y\ ins. Under a uniform 
raneous load of 100 &>s per sq foot, the greatest pres on the arches is but 3G00 lbs 


irith onlv two trusses the width between them, In the dear, should not be less than 16 ft. to 
r two Ordinary vehiciesto pass each other readily ; but 18 or 20 ft is dill better; more would bo 
icessary when there are outside footways. The headway should not be less than 13 ft. 








600 


TRUSSES. 


per sq inch of their cross section; or not more than °f the ultimate crush L 
strength of average cast iron, in short blocks. 

The Moseley Bridge, Figs 35%, by Thos. W. II. Moseley, of Kentucky t 
essentially a wrought-iron Bowstring, with a hollow plate-iron arch of trianguiea 
cross-section, apex up; and formed of three plates riveted together: the two sii-r 
plates, ab, ac, having their top and bottom il 

edges bent to form flanges for this purpose. 

The chords o e; the verts; and the two counter¬ 
arches tt\ are also of iron. These counter¬ 
arches are intended as a substitute for the ob¬ 
liques of Fig 35. Each of them consists of two 
angle-irons, back to back, riveted together, 
and to the verticals, which pass between them. 

Each of them has a sectional area equal to 
half that of the main arch. The verticals are 
placed about 2 ft apart. They are flat (not 
square) bars, for convenience of riveting. They 
pass through holes in the bottom-plate of the 
main arch, (see dotted line of top Fig,) and are 
fastened at a by the same rivets which connect 
the upper flanges of the two side-plates, ab, ac. 



„ . - . -■ , *■-’ The chords, oe , are also flat ba 

and have a transverse area half as great as that of the main arches. At their ei 
they are attached to strong wrought-iron shoes upon which the feet of the m 
arches rest and abut. 1 he rise ol the main arches, measured to the bottom of 
arch, is -§■ or yy of the clear span. 

™ e ,£ l l?7 ing a T. the prin „ cipa ! dimensions of a single-track bridge of 93 ft cl 

It was 9 h.fu mTiSCO t0 T- 1 f arch >) canTing the “Iron Railroad” at Ironton, 01 
It was built in 1860, and is traversed by heavy engines with trains of pig iron c< 

?6V n S , C : b J t0, \ 0f ^ l °Y ft; to , middle of arch 11 Bottom-plate s s°of ar 
Tn<f fl by ‘t* 1UC V .^de-plates a b, a c, in clear of flanges, 14^ ins, by .29 in 

at a’o r. 1 3 1US ; Bottom flanges b and c, each 1.88 ins. Vert rods, 3 i 
two flat Tn / fr ° m cen t er t0 center - The chord of each arch consists 

? b a’ a , h °/ 4 Y\ ins of cross-section. The bridge was tested for tli 

foot k run y f Y L t0U and a . rollin S load of 1 ton at the same time, • 

’ an d deflected only % inch. With a load of 1 ton per foot, the pressurt 

inore at the “^ hep v, ’? uld be about 5% tons per sq inch of metal; and a tr 
at the feet, the sectional area of metal in each main arch is 18% sq ins. T1 
bridges are of easy construction, and consequently cheap. For long spans - 
'agona 1 bracing (see tig 35) would probably be essential for preserving the fo 
of the arches under heavy moving loads* 6 

An iron arch roof in Philadelphia, clear span 80 ft rise 16 ft c f 

inl! S -’O f fts U n f ‘r°f n ot arC 'n 0 ^ S1 '‘ gle 74nch P . hoenix beam of 6 sq ins sectional area; w’ei ' 
ing _0 lbs per foot. This rests on cast-iron shoes on the walls. The lior chord or 

rV 8 £ U l° r ° d \°J diarn - At the center of this tie is an arranges 
nilar to No. 15, of page 583, from which diverge upward, to the arch, a central v 
rod 1 inch diam ; two struts of 6-inch Phoenix beam, 20 ft apart at the arch • and t 

the arch ^eVTare^O^nf &rCh at l ' albwa / between the struts and the feet 
tne arch. 1 here are 10 such trusses, each of which by itself weighs about 3900 1 

icy are placed 16 ft apart; and by means of purlins resting upon them support 
entire weight of the roof, which is of inch hoards, covered bj'tUn .hSi’tST 

.The iron roof of a rolling-.mill near Boston nf oimnt 

t span, and 16 ft rise, has arches of the Moseley section a, ft, c. Fig 35%-’but with 

7 ins. nnn r fl eS - Th %*™**™ *** 12 ft apart ‘ Sides of thearches,'clear of the flam 
2 ins; lower ones. 1 inch; total of each side, 10 jns, by ,19 inch thi 
I ottom plate 8% inch by % inch thick. Total sectional area of an arch 5.925 so i 
There are besides, a chord; and 24 vert suspending rods ; but no obliques The rc 
s covered with corrugated iron, on purlins. When required, the heavy bon roil 
the mill aie lifted by tackle supported by a roof-truss. 

Figs 36, represent the Burr truss; which was formerly more u 


to*exceed , Two t tf)ns^o*f^couqfress^ve^strai'n 0 per V S('/inch on IZTYl ^ th l Writer W0Uld P™** 

lattices. instead of vert bars income of his bridge's' stil .b has als o introdi 

meat first described. At 1 ton per ft run, the pull ou tfe ^ arrai 






















TRUSSES. 


601 




an any other in the United States. It is at present regarded with disfavor by some, 
cause many early ones failed under railroad traffic, in consequence of bad propor- 
ms, and the absence (as in our Fig 36) of counterbracing. When properly con¬ 
ducted it makes an excellent bridge. The common objection to it, and not without 
alason, is that .a truss and an arch cannot be so combined as to act entirely in con- 
art; yet, as soon as any ordinary truss begins to fail, the almost invariable remedy 
to add an arch. When, however, the two are to be united, it is better to so pro- 
rtion the arch as to be capable by itself of safely sustaining the max load at rest; 

Jl 

Jh 



id to confine the duty of the truss to preventing the arch from changing its form 
ider a moving load. Counterbracing may be effected by strapping the heads and 
et of the braces to the chords; or by iron rods parallel to the braces; two to a 
•ace; with screws and nuts, as at v f. Or by similar rods across the other diags of 
le panels. Tile following' dimensions answer for a single-track R R 
ridge of about 150 ft span. Rise from out to out of chords one-eighth of the span; 
jout fourteen or sixteen panels. Width in clear of arches, 14 It. Six arch-pieces 
f of 10" X 13" each, to each truss. Upper chord c, 14" X 16". Lower chord a a , 
vo pieces each 10 " X 15"- Posts p, 14" transversely of the bridge, as in the right- 
ri and fig; by 10 "; except at the heads and feet, where enlarged to receive the ends 
!l ff the braces. Braces 10" deep, by 13" wide. Floor girders o, 10" X 16"; and 
'b 3 ft apart from center to center. Suspending rods (shown at s; and dotted in x) 

\ vC diam. Counterbrace rods in pairs parallel to braces, about diam. Bolts 
>r arches, lower chords, &c, 1>£" diam. Theoretically, the posts, braces, and arches 
a lould gradually diminish from the ends, toward the center of the truss; while the 
ij lords should increase; but in practice, the additional labor of getting out and fitting 
ieces of diff sizes, frequently makes it more economical to use uniform sizes.* The 
uue amount of arch would answer also, if trussed, as in Fig 35; and the arch by 
,i self with a full max load, would be strained less than 800 lbs per sq inch, at its feet, 
dor a span of ‘200 ft, with the same proportion of rise, the transverse areas of the 
t iveral arch and truss pieces should be increased 33 per cent; and for one of 100 ft, 
ley may be diminished the same. None of these dimensions are the result of close 
ilculation. The dimensions just given will answer for common travel , for a span of 
X) ft; with a depth from out to out of chords, of % the span ; panels 10 to 12 ft long, 
[any such spans have been built with timbers of about 3 ; less transverse section; 
ml without counter bracing. The heads of the posts are notched about 2" to 3" into 
ie bottom of the upper chords; and are moreover tenoned into it some ins further: 
ith two wooden pins through the tenon; see n, Figs 36. Their feet are notched 
oth into and upon the lower chords, so as to leave the two chord-pieces a a only 
bout 2" apart. Through these and the post pass two bolts of about 1" to 1(4 diam. 
Since the upper chord resists compression only, its pieces may come together with 
plain butt joint, d. To this may be added fishes e d, of stout plank, on the sides 
f the chord, bolted through by 4 or 8 bolts. 

The lower chords resist pull; and the pieces composing each lower chord must 
tie re fore be joined together These pieces shoul d be as 

1 It, will be borne in mind that our examples are not intended to illustrate perfectly proportioned 
uctures. None of them would endure strict criticism. There is more waste of timber in an arch 
it with a uniform trausverse section throughout, than in the straight upper chord of a Howe or 
itt similarly built; for both must be proportioned to the greatest strain. This is at the center of 
, tw0 i aM t ; but while that at the center of the arch is as great as iu these, that at its feet is muoto 
ater. See Example 2, Art 38, of Force in Rigid Bodies. 

































































602 


TRUSSES. 


the g m wKf b theo < !her? ll,d ** ° PpOSite t0 otlier ’ but one °™ osit 

The braces are merely cut to fit to the hearts and feet of the posts, after these las 
have been fixed m their places; and usually have no other connection to them tha 

,Dd - for sm *" •>'“*«! .> r Kn-w-bolt. forTrgeo„™ 
Iheenrts of the timbers composing the arches, butt full square against each other 

I? 80 h . ave a wooden rtowel. The joints should occur at the posts as show: 
T , be arcb f® are screw-bolted to the posts, as shown in the figs, by bolts of 1 1 
1^ n s diam. U here the arches pass the lower chord, both are notched and wel 
“If together. The feet of the arches abut against cast-iZ JILT ' 

" hen suspension rods (dotted in Fig 36) are used for assisting to support the roar’ 

r^n b nf y r e P,HCed aS , Sh0W, ‘ ats ?; » a strong block of load SSSy notch* 
ti° P m t ‘. e l,I> P® r arch-pieces. The roils are suspended by a washer and nut on to> 
ot the block; and after passing down between the arches and chord have a siinila 
arrangement, but inverted, below the last; as shown at g. ’ 




« 


- 

| j 

















TRUSSES. 


603 




If a Ion ^ beam, a b, Figs 45 and 47, requires to be strengt hened, 

iis may be done by adding a vert post dc; and 2 inclined tie-rods ca, cb. And if, 
!y t er this, the two halves, d a, d b, of the beam still are found to be too weak, addi- 
iijU onal intermediate posts, oo, may be introduced; with other ties, ij, to sustain them. 

In Figs 45 and 47, the roadway 
is at the chord a b; no parallel 
lower chord being necessary, it 
may be omitted. The inclined 
ties act as substitutes for it. But 
if the bridge is so near the water 
as not to allow the posts and ties 
to be placed beneath the roadway 
a b, we may raise the entire truss 
upon two posts or piers * s, Figs 
44, 46 ; and place the roadway 
n n, at the lower ends of the posts 
id ties; instead of letting it rest on top of the chord, as in Figs 45 and 47. In Figs 
and 46, the truss and its load do not then rest directly upon the abuts y y, but 

upon the tops of the posts 
s, s; and the only part that 
does rest directly on the 
abuts, is one-half of that 
small portion of the road¬ 
way comprised at each end, 
between e and n ; in other 
words, only one half the wt 
of the roadway of the end 
panels; the other half being 
sustained by the inclined 
ties which meet at e. 

When the tie-rods all pass 
from the feet of the posts to the 
ends of the chords, as in Figs 
44 and 45, we have the Boll- 
man truss. And when,as 
in Figs 46 and 47, only those 
which sustain the center post 
d c, both pass to the ends of 
the chord, while the others 
are disposed as in said figs, the 

ink: truss is the lesult. 


C J 0 

BOLLMAN 

Fig. 45. 































































604 


TRUSSES. 


I 


The following 1 dimensions for single-track Fink bridges, ] 

with chords and posts of wood; and iron suspension bars; are on the assumption 
that all the bars deflect )/% of the span ; that the road is on top; that the bars shall _ 
not be strained more than 10000 lbs, or 4^ tons per sq inch, under a weight of bridge 
and load, amounting in all to two tons per running foot. Assumed wt on each 
driving-wheel of engine, 5 tons. Screw ends upset. 

Dimensions for one truss only, of a single-track Fink bridge. 


The spans are in feet; the other dimensions are square inches of cross-section of 
each member. 


Areas in sq ins. 

Areas iu sq ins. 

a 

u 

o 

•6 

© 

M 

*6 

o 

« 

T3 

© 

AS 

4> 

m 

© 

At 

• 

a 

a 

•6 

© 

•6 

o 

AS 

T3 

O 

AS 

•6 

© 

AS 

*6 

© 

AS 

OB 

© 

At 

Os 

m 

£3 

O 

06 

Ct 

CO 

to 

Ck 

0Q 

JZ 

O 

CO 

e-t 

eo 

-«j 

03 

H 

35 

275 

s% 

2 H 


56 

00 

470 

21 

5?i 

1 % 


14t> 

40 

300 


2H 

. , • 

64 

100 

505 

23 « 

6M 

2 

... 

160 

45 

322 

10H 

2 % 

• . . 

72 

125 

680 

29H 

8 

2H 

i% 

225 

50 

340 


3% 

... 

81 

150 

650 

35 

OH 

3 

i% 

280 

60 

375 

14 

3% 

1% 

05 

175 

730 

41 

11 

3H 

IX 

335 

70 

410 

ir>H 

4% 

l% 

110 

200 

800 

* 6 % 

12H 

4 

1 % 

300 

80 

440 

im 

5 

1% 

126 









Wc have given the area of the 1st post only. For that of the 2.1, we may take % of the first; for 
the 3d, % of the second; and for the 4th, % of the third; without pretending to any great accuracv. 
The iron rods may be flat, square, or round, so that the proper area be maintained, it will usually 
be best to have them flat. 


A width of 18 feet is necessary for allowing two ordinary vehicles to pass 
each other readily. It should never be less than 16 ft; and nothing is gained by 
exceeding 20 ft. 

It will of course be understood that each member, especially in large spans, will consist of two or 
more pieces, side by side. Thus, in a spau of 200 ft, the 800 sq inches of each chord, will probablv 
consist of four beams or about 10 1 ' X 20 ' , or 12” X 16 ", placed side bv side ; but with sufficieut iuter- 
vais betweeu them to allow the several oblique bars to pass. Or it mav cousistof six beams of smaller 
size. So also, the 1st, or maiu rod, of 46Jf sq ins, will probably be made up of from 4 to 8 bars, of 11.7 
or a.85 sq ins each, placed side by side; occasionally some inches apart. And so with the others’ 
The feet of the opposite posts of the two trusses of a Fink spau, are connected together by ties of wood 
°r lr ? a ’ P revenI ' lateral motion; and for the same purpose, diagonals are carried from each uppei 
chord to the foot of the opposite post; as is usually doue in top-road bridges of any kind. These hah 
better be tie-struts. J 


1 

Oi 


h 


The weights of bridges of the same span, designed by T different persons, 
vary considerably, from several causes; such as the form of truss; quality of iron- 
coefficients adopted for safety, and for strength of materials; whether the roadway is 
on the top chord, or on the bottom one, &c, &c. 

For a mere approximate weight to be assumed as a preliminary in 
calculating the strength and proportioning the parts of a bridge, or for forming some 
rude idea of the quantity of iron required, we suggest the following purelv empirical 
formulas. They give the weight in pounds per foot run of span, of only the two 
trusses or main girders together and their lateral bracing, for a single-track railroad 
bridge of standard gauge (.4 ft sy 2 ins). The weight of cross-beams, flooring rails 
etc, is not included. ’ ’ 

For spans not exceeding 75 feet. 

(Plate girders.) 

Weight per foot run (see above) = 5 X span in feet + 50 X V span in feet. 

For spans of from 75 to 250 feet. 

(Pratt, Whipple, or Warren trusses.) 

Weight per foot run (see above) = 4.5 X span in feet + 22 X V' span in feetT 

For spans exceeding 250 feet.* 

(Trusses of various designs.) 


T 

: f 

i it 

lest 

p« 

pit! 

on 

Ijm 

T 

V 

V 

\ 

»l 
hi 
ik 
toil: 

■ T 
*8 

|.*i 


Weight per foot run (see above) = __ 8 P an2 in fe et 


60 


+ 500. 


* In spans longer than about 300 feet, the differences in truss designs are so m eat 
that the actual weights may differ widely from those obtained by simple practical 
formulae ol any kind. v F 


«j!V 

8t> 

iilli 







































TRUSSES. 


605 


Table of approximate weights of single-track iron railroad 
bridges of standard (4 ft 8 1-2 inches) gauge. 


Span. 


Feet 

20 

30 

40 

50 

60 

70 

80 

100 

150 

200 

250 

300 

400* 

500* 

600* 


Weight of the two trusse8, or main girders, and their lateral bracing. 

Lbs per foot run of span. 

~ 325 

425 
520 
600 
690 
770 
650 
670 
940 
1200 
1500 
2000 
3200 
4700 
6500 


Lbs total 
6500 
13000 
21000 
30000 
41000 
54000 
44000 
67000 
141000 
240000 
375000 
600000 
1280000 
2350000 
3900000 


Plate girders. 


Pratt, Whipple, or Warren 
trusses. 


Trusses of various designs. 


For double track bridges add 80 per cent to the weight of single-track ones 
as obtained by the formulae. 

For narrow gauge bridges take 75 per cent of the weights for standard gauge 
ones as given by the formula}. 

Iron floor systems, with two stringers under each track and adapted for 
heavy loads, such as A p 546, may be taken as weighing approximately as follows: 


Span, in feet. 

Weight of iron floor system, in pounds per foot run. 


Single track. 

Double track. 

20 to 100 

200 to 275 

550 to 700 

100 to 250 

250 to 350 

700 to 850 

250 to 300 

325 to 400 

750 to 900 

300 to 400 

375 to 450 


400 to 500 

425 to 500 


500 to 600 

475 to 575 

800 to 1000 


Two iron safety stringers will together weigh about 150 lbs per foot run. 
For wooden floor systems, we may, as a rude average, set down for spans 
not exce&ding*bm!t 200 feel, cross floor girders of about 7" X 15", and about 2^ to 3 
feet apart, from center to center; clear span 14 feet; together with substantial stnng- 
nieces of about 10" X 12", for supporting the rails; the rails themselves; and a plank 
Sway between the rails, all complete! at about .14 ton, or 314 lbs, per foot of span ; 
or with a full floor of 3" plank, 14 feet wide, about .2 ton, or 448 lbs For greater 
spans with the trusses farther apart, increase this to .25 ton, up to 300 feet; .3 ton, 
4.00 fppf • 35 ton to 500 feet; and .4 ton, to 600 feet. 

Two safety stringers will together weigh about 100 to 150 lbs per foot run. 
When a bridge is to be roofed and weather-boarded, an addition must, of course, 

be made to our weights. . „ , ,_ 

Wooden bridges weigh about the same as iron ones of stiengt . 

In rile Northern Pacific R R bridge over the Missouri River, at Bismarck, Dak 
built 1881-2 the two inverted bowstring trusses of the 115-foot approach spans, weigh 
(together) .38 ton per foot run; two Pratt trusses ot the 400-foot channel spans, 1.09 

t0 Two trusses of the Newark Dyke bridge, England, Warren girder 240)4 foot span, 
i 02 tons The single-track tubes of the Victoria bridge, at Montreal, 244 feet 
2 1 14 tou"!'tb; it» foot span. 2 ton.. The single-track Britannia tube Eng, 460 
feet span 3 43 tons. Two trusses alone, of the Penna R R, at 1 hiladelphia, 1 foot 
f. et x b Mr Linville, .52 ton. The fine 320 foot span across the Ohio 

by Mr Linville, 1.6 tons per foot. All these are of iron; 

single track. ______ 


* See foot-uote (*), page 604. 


































606 


TRUSSES, 


The greatest load that can come upon a bridge. 



If for a single-track rail¬ 
road, can scarce!} - exceed 
that of a striug of heavy 
locomotives coupled to- 
gether, without their ten¬ 
ders. Such engines will 
weigh Iron) one to two 
tons per foot of their ex¬ 
treme length but a long 
striug of such, without 
their tenders, is hardly 
probable. 


Our table on page 
605 makes sufficient al¬ 
lowance for the greater 
loads per ft run that may 
probably come upon small 
spans. The diffs between 
• urtth and 8th cols will 
give our assumed max 
loads. 

On very small spans 
the loads cannot be as¬ 
sumed to be equally dis¬ 
tributed. On bridges Tor 
turnpikes and common 
roads, no probable contin¬ 
gency could crowd people 
upon them to such an ex¬ 
tent as to weigh more than 
80 lbs per sq ft of floor. 
The French standard in¬ 
deed is but half of this, 
or 42 lbs per sq ft; and 
is sufficient for probabil¬ 
ity, but not for possibility. 
The latter way increase it 
to 80 lbs; and this may 
safely be taken as the 
maximum load on spans 
of 20 or more feet. To 
compensate, however, 
for momentum, we re¬ 
commend to adopt 100 lbs, 
or .015 of a ton, as the 
limit for crowds, t A 
bridge for a single-track 
carriage way; with room 
between the trusses for a 
footway also, should not 
be less than 12 ft wide in 
the clear; in which case 
its greatest load at 100 
fi>s per sq ft, would be % 
a ton per ft run ; or if 24 
ft wide, with but two trus¬ 
ses, the load would be full 
1 ton per ft run. 


But in a com¬ 
mon bridge also, the great¬ 
est load per ft run, on a 
very short span, will be 
greater than in a long 
one; as in the case of two 
wheels of a truck hauling 
a large block of stone, &c ; 
and this m-ust be taken 
into consideration in 
building such. 




to 


■C 


Z 


bridle® wifhTresuUof 84 fts^r ™ picUd mrtn u P on the P^tform of a wei^b 

others. Keault, 120 fts per sq ft. See foot-note p 028. beiDB lowerea aovFn tr0I “ above, among tin 























TRUSSES, 


607 


It must also be remembered that each transverse floor-girder must bear at least all 
the weight resting upon two wheels; no matter how close together the girders may 
be placed. If they are farther apart than the dist between two axles of a vehicle, 
they will have to bear more than the load on one pair of wheels. 

The allowance for safety in a truss bridge. 

As the result of a long-continued series of deflections applied to an experimental plate-iron girder 
of 20 ft span, Mr Fairbairn concludes that a bridge subject to 100 deflectious per day, each equal to 
that produced by % of its extraneous breaking load, would probably break down in about 8 years; 
while, with 100 daily deflections equal to that arising from but J4 of its breaking load, it would last 
fully 300 years. We are of the opinion that a bridge should not have a safety of less than 4 for its 
max extraneous load, and Its own weight, combined; nor do we see any use in exceeding 6. From 
4 to 5 may be used in temporary structures, or in those rarely exposed to maximum strains; and 6 in 
more important ones frequently so exposed. The last will (roughly speaking) generally give a safety 
of about 2 against reaching the elastic strength, which is the true guide in such matters. But 4, 6, 
&c, usually refer to the ultimate or breaking-down strength; so that a truss with such a safety 
of 2 would in fact be very unsafe. 

On the camber of truss bridges. In practice, the upper and 
lower chords of bridges are not made perfectly straight, but are curved slightly up¬ 
ward ; and this curve is called the camber of the truss or bridge. Its object is to 
prevent the truss from bending down below a lior line when heavily loaded. A cam¬ 
bered chord is of course longer than a straight line uniting its ends ; hut in practice 
the camber is so small that this diff is inappreciable, and may he entirely neglected. 
But when the chords are cambered, (see y s and c d , Fig 51,) they become concentric 
arcs of two large circles, of which the center is at t; and the upper one plainly be¬ 
comes longer than the lower, to an extent which, although much exaggerated in our 
fig, cannot be overlooked in practice. The verticals, instead of remaining truly vert, 
become portions of radii of the aforesaid large circles; and although their lengths 
remain the same, yet their tops become a little farther apart than their feet; and 
this renders it necessary to lengthen the obliques or diags a trifle. Therefore, we 
must find how great is this increase of length of the upper chord beyond the lower 
one; and divide it equally among all the panels, along said choi’d; otherwise the 
several parts of the truss will not fit accurately together. 

1st. To find tbe amount of camber of the lower chord. Di¬ 
vide the span in feet, (measured from center to center of the outer panel-points,) 
by 50. The quot will be a sufficient camber, in inches; as shown in tiie following 


Table of cambers for bridge trnsses. 


Span. 

Feet. 

Camber. 

Ins. 


Span. 

Feet. 

Camber. 

Ins. 


Span. 

Feet. 

Camber. 

Ins. 

25 

0.5 


100 

2.0 


250 

5.0 

50 

1.0 


150 

8.0 


300 

6.0 

75 

1.5 


200 

4.0 


350 

7.0 


Rem. 1. It is by no means necessary to adhere strictly to this rule; and the 
camber by experienced builders of iron bridges is often but one-half the 
above, or 1 inch per 100 ft of span. 

Rem. 2. A well built bridge of good design should not, under its greatest 
load, deflect more than about 1 inch for each 100 feet of its span. The deflec¬ 
tion is frequently much less than this. 

2d. To find the increase of length in tbe npper chord. 

< beyond the lower one, having the span; the depth of truss; and the camber; (all 
in feet, or all in ins.) 

With any camber not exceeding of the span ; (whioh, however, is about 7 times as great as is 

usually given to trusses;) mult together the depth of truss, the camber, and the number 8; div the 
prod by the span. The quot will be the inorease, in ft, or in Ins, as the case may be. Or as a formula. 

Increase in ft, — de P th X camher X 8 a u i n ^et; or 
or in ins, span, all in inches. 

This rule may bo considered practically perfect with any camber not exceeding 
of the span. Based upon this principle, we have prepared the following table, which 
may be used instead of making the above calculations. 
























608 


TRUSSES. 


Table for finding' increase of length of upper chord beyond 

lower one. 


Depth 

of 

Truss. 

Mult 

Camber 

by 

M span. 
1-9 “ 

1-10 “ 

1-11 “ 

1.00 

.888 

.800 

.727 


Depth 

of 

Truss. 


X span. 
1-5 " 

X “ 
1-7 “ 


Mult 

Camber 

by 


2.00 

1.60 

1.33 

1.15 


Depth 

or 

Truss. 


1-12 span. 
1-13 “ 
1-14 “ 
1-15 “ 


Mult 

Camber 

by 


.666 

.614 

.571 

.533 


Depth 

of 

Truss. 


1-16 span. 
1 17 “ 

1-18 “ 
1-20 “ 


Mult 

Camber 

by 


.500 

.470 

.444 

.400 


Ex. How 
the truss is f 



C 


Fu/ao" 


>w much longer is the upper chord than the lower one, when the depth of 

—-is j of the span; and the camber 5 ins ? Here in the table, and opposite \ 

span, we find the multiplier 1.15. Therefore, 5 ins X 1.15 = 5.75 ins, Ans. If the 

5.75 

truss has say 8 panels, then —— = .72 inch of this increase must be given to each 
panel, along the upper chord. 

The length of a diag, or obliqne, b c, Fig 50, may readily be found 

thus: Let asnc in this fig represent a panel when there 
is no camber; then o b n c will represent a panel when 
there is a camber; and oa and sb together are the 
portion of the increased length of upper chord given to 
each panel; but to an exaggerated scale. Now, to find 
b c, we have the right-angled triangle a b c, in which we 
know the side a c, (the depth of truss;) and the side a b, 

(equal to the panel width cn on the lower chord; added 
to s b, or half the portion of the increased length of 
upper chord given to one panel.) Hence, we have only 
to square each of those two sides; add the two squares 
together; and take the sq rt of the sum. This sq rt is 6 c. 

Example. Span 200 ft. Height (a c) of truss % of the 
span, or 25 ft, or 300 ins. Camber 5 ins; 10 panels each 
20 ft, or 240 ins, (c w,) measured on the lower chord or 
span. Now, the height being of the span, the increase 
of length of upper chord will be equal to the camber, 5 ins; and this divided among 

10 panels, will be — = .5 inch to each panel ; or ob will be .5 inch longer than cn; 

and o a and s 6 will each be .25 inch. Hence, in the right-angled triangle a be, we 
have a b = 240.2o ins; and a c — 300 ins. Hence, 

fcc = y/ a/i 1 -fac , = 67720.0625 + 90000 = y/ 147720.0625 = 384.344 ins; 

ft • 3 i 4 inS - an ? earner, b c would be in the position g c, which is 32 

ft, .18i ins long; or .15/ of an inch (about y & inch) shorter than b c. 

An error to this extent would prove seriously inconvenient if the oblique were a cast-iron 
w th carefully planed ends, intended to fit closely hetweeu planed bearincs at the chords-^ ^ * 
with a drilled hole at each end for fitting over'pins whose position wa ^xed Lrf unaHerabl" in 
many cases, as when the obliques are merely rods with screw-ends, it is only necessary\o b e sure 
that they are long enough; because their exact length can then be adjusted when nn< , 8U , ra 

means of the nuts on their ends. So also when the obliques or other pieces' are flit ba?s intended t i J 
bolted or riveted to the sides of the chords; for the final rivet-hole” msy be madewhen t e n * 

redifced by STil VJSJST* ^ ** °“ e ° f W ° 0<lea ° bIi ^ they S 

When the panels are all of one size, as is generally the case, it is usual for builders 
parts together! f '‘ 8126 ° n a b ° ard platform or floor . to g«>de in fitting the 

In raising* a truss, or in other words, when putting its parts toeether in 
their proper position on the abutments and piers, a scaffold or falseworks 
must first be erected for sustaining the parts until they are joined together so as to 
form the complete self-sustaining truss. Upon the false works the bottom chords 
are first laid as nearly level as may be : and the top chords are then raised upon tern 
porary supports which foot upon the one that carries the lower chord. The unner 
chords are at first placed a few inches higher than their final position or than the 
true height of the truss, in order that the obliques and verts may be remldy shnwd 
into place. After this is done, the top chords are gradually let down until all rests 
upon the lower chords. The screws are then gradually tightened to brinu all the 
surfaces of the joints into their proper contact; and by this operat.oi /tlm nn er 
chord being supposed to have the increased length giveii by the foregoing rule) the 





































TRUSSES, 


609 


nd 

- 


imber, as it were, forms itself; and lifts the lower chords clear off from their false* 
orks ; leaving the truss resting only upon the abuts or piers, as the case may be. 

As a support, for the falseworks themselves on soft bottoms piles 
iay be driven, to which the uprights of the falseworks may be notched and bolted 
r banded. In some cases, as of rock bottom in a strong current, it may become 
spedient to sink cribs filled with stone, as a support for the falseworks. 

The falseworks should be well protected by fender-piles or otherwise from passing 
oats, ice and other floating bodies, especially in positions liable to sudden floods; 
nd numerous accidents have shown the expediency of guarding (lie unfinished 
uss itself against high winds. This last remark applies as well to roofs as to 
ridges ; and is too frequently neglected. 


ion 


si' 

md 


To prevent an overturning tendency in a whole truss when it is 
ot high enough to admit of being horizontally braced overhead, we may introduce 
ooden knees, or short straight struts or ties, of either wood or iron; which may 
>ot upon the cross-girders of the floor; and head against either some of the web 
lembers, or the upper chord. These braces or ties may be placed either between 
le two trusses of a span ; or outside of them: or both. When outside, some of the 
oor-girders may be lengthened out a few feet beyond the lower chord, for receiving 
le feet of the braces or ties. 


1 


The clear distance apart of the trusses in railroad bridges for 4 ft 8^ 
nch gauge, is generally made not less than 14 or 15 ft, and 26 or 28 ft, respectively, 
^r single and double track, in through bridges; and 11 or 12 ft, and 16 ft, respec- 
ively, in deck bridges. For lateral stability, in long spans, the distance be* 
ween centers of trusses is generally made not less than one-twentieth of the span 
A headway of 18 to 20 ft should be allowed for clearing smoke-stacks, &c. 


'I f 





610 


TRUSSES. 






Floor-girders not exceeding 14 ft clear span, may be 8 ins, by 15 ins deei 
and placed not more than about 2% ft apart from center to center. Upon thei 
should be notched and spiked stout string-pieces, say 12" wide, by 9" deep, to carr 
the rails; and to distribute the pressure of the load. 

Ifor diag bracing for diminishing lateral motion, can be used only undt 

the floors of low bridges; but in high ones it is introduced also at the top of tt 
trusses. When of timber, these braces are about 4 to 6 inches thick; by 6 to 9 deej 
and form a lior cross between each two opposite panels of the two trusses. If tl 
bridge is roofed, and has girders r, Figs 36, 56%, upon and well secured to the upp< 
chords c c, the upper lateral bracing may consist simply of 4 iron rods n n, passir 
through the chords about midway of their depth ; and having heads and washers ( 
their outer sides. At the center of the cross the rods terminate in an adjusting-rinj 
see No 14, of page 583. In a bridge of 150 ft span, these rods need not exceed : 
inch diam at the center panel, and 1% at the end ones. If the bridge is high, ai 
not roofed, but open at top, then cross-struts r r. Figs 56%, must be inserted pu 
posely, when this rod-bracing is used. If it is also used at the lower chords, the floe 
girders perform the duty of these struts. Iron-bracing is not liable to catch ii 
from the locomotives. 

A favorite mode of lateral bracing, W, resembles a Ho\ 

truss laid flat on its side. In it the diags of the cross are struts of timber; and t! 
pieces r r are round rods. One of the struts is whole, with the. exception of a slip 
mortice on each vert side, at its center, for receiving tenons cut on the inner ends 
the two pieces which compose the other diag. At the sides of the chords, the eu 
of the diags rest upon a ledge, (shown by 
the dotted line i t,) about 1% ins wide, cast 
at the bottom of the cast-iron angle-block. 

The tie-rod rr, passing through the chords 
of both trusses, being tightened by means 
of the nut s, holds the diags firmly in place; 
and in case of their shrinking a little in 
time, can be again tightened up by the same 
means. 

Various modifications of these methods 
are in use; but we cannot afford them space 
here. The cast angle-block is as deep as a 




brace; its thickness need not exceed % inch, in a large bridge, 
a top view of it. It has holes for the passage of the rod r r. 


Fig. 56%. 
The dark triangb 


Art. 25. Lengthening-scarfs, splices, or joints. Thelowercho 

of bridges, being exposed to great pulling strains, require much care in connect 
together the ends of the several pieces of which they are composed. There is mi 
uncertainty regarding the strength of the joint-fastenings in common use for f 
purpose. Experiments on the subject are much needed. When only two pieces 
t and y, Fig 57, or 58, are joined by any of the ordinary methods, it is probably 
safe to depend on their possessing more than % of the tensile strength of a sii 
solid beam of equal cross-section. When the chord, as in Fig 59, is composed of 
parallel parts a a, n n, made up of long pieces, breaking joint with each other, a 
j j j, each of the two parts may be made somewhat stronger than either on* 
them would be by itself. This is owing to the opportunity afforded of connec 


ftn m r ., v » ig jrrmm , rT rf v ag gaa n . - - r 11 ■ ir ,r i n e* 

































TRUSSES. 


611 


em also by bolts l b, and packing-blocks, cr, of wood or iron, intermediate of the 
•mts jj , Ac. By tins means the strength of the entire chord may probably be urao- 
jaUy rendered equal to one-half of what it would be if solid. If the chord con¬ 
its ot 3 or 4 parallel parts, of long pieces, breaking joint, and connected in the 
me way, it will probably have about % of the strength of the solid. Care must 
course be taken that the serviceable area of the pieces shall not be reduced at any 
terinediate point, to less than it is at the joints. 


TOP 


TOP 



Fig. 57. * ,DE 
a Fig. 59. 


TOP 


SIDE Fig 

a 



SIDE 


Fig. CO. 


Fig. 60 a. 


TOP 



C 

Fig. 60 b. 

ig 58 is a simple and efficient form of scarf. Its length i i may be about 3 to 4 
js the greatest transverse dimension of the beam. At the center is a block t of 
1 wood, with a thickness equal to % that of the beam ; a width of 2 or 3 times its 
kness; and a length just sufficient to reach entirely through the beam. The 
ns are connected by 4 screw-bolts n n ; or by 8 of them, if the length requires it. 
es of stout rolled iron, aa,cc, with their ends bent down into the beams, are 
.sionally added. These require bolts o o, beyond the ends i i of the scarf. These 
s are not shown in the side view. 

g 57 is another excellent joint with splicing-blocks e e, instead of the block l of 
58. The indentations, v v, may each be about % as deep as the beam is thick, 
length of each splice-block, about 6 times s s. From 4 to 8 screw-bolts, as the 
may require. Length of each indent about that of the block itself, 
g 60 is a joint formed by two Hat iron links or rings, 11, let flush into the tim- 
, and retained in place by spikes. The iron may vary from to 1 inch in thick- 
; from 1 to 4 or 5 ins in width; and 2 to 6 ft in length, as occasion may require, 
g 60 5, is a joint formed by two blocks, c c, of bard wood, passing through the 
>ers; and connected by bolts, a a, n n. 

Fig 60 «, s s are cast-iron packing-blocks, sometimes used instead of plain wooden 
, at points l b, Fig 59, intermediate of the joints 47 . The openings in the centers 
le blocks are needed only when vertical truss rods have to pass through those 
ts. At e c, of the same fig, is shown another form, much used in chords composed 
co or more parallel strings. Both these are as deep as the chord; and their 
-sections, or end views shown in the fig, may be from 4 to 10 ins long; 2 to 4 ins 
; and from \/ z to 1 % * ns thick; according to size of bridge, &c. 
m. In selecting hard wood for splicing-blocks, treenails, or for any part of a 
;e, it is well to remember that the oaks when in contact with the pines, expedite 
ecay of the latter; therefore, it is generally better to employ the best southern 
w pine heart wood for such blocks, &c, or interpose sheet iron.* 


tie tendency of some kinds of timber to produce rapid decay when brought into close contoot 

43 

































































612 


TRUSSES. 


Eye-Bars and Pins. The lower chords of iron bridges usually 

consist of flat links or bars c and o, W and H, Figs fil on edge and connected by tight-fitting wrought 
iron pins b and P. After deciding on the size of the body W or H of the bars to bear safely the pul 
upon them, the proper proportioning of their heads or eyes and pins is an abstruse and difficult poiu 
upon which much has been written. It was formerly supposed that the diam of the pin should m 
governed by its resistance to shearing, but experience has shown that this was entirely insufficient 



We give a table of practical conclusions arrived at by that accomplished expert, Chs. Shaler Smitl 
from ill experiments by himself on a working scale. The table shows some irregularities, for as M 
Smith remarks “ the bars declined to break by formula.” The pin is more strained at the outer linl 
o o than at the inner ones c c c, so that the latter would not require so large a dtani, but that th 
must be uniform throughout in order to secure tight fitting for all of them. When web members i 
well as chords are held by the same pin the diam and head must be proportioned for that bar of the 
all which is most strained. When the heads are made by pressure in one piece with the body, tl 
metal v s and uzat the sides of the pin 6 must be wider than when the heads are first made in sep 
rate pieces by hammering aud then welded to the body. But the welded one W requires more iri 
lack of the pin as shown at 11. This width l t must be equal to the diam of the pin.* * 

The links are supposed to be of uniform thickness. 

Having drawn a circle b for the pin, lay oflT on each side of it as at v s, u x, ha 
the width of metal in the table for the head of W or H as the case may b 
Then for forming the head of H use only the rad b s as shown. For the head < 
W lay off also 11 = diam of pin. Find by trial the rad g n or g t and use it, exce 
for uniting the head to the body, where use a rad = 1.5 g n as shown. 


Width 
of bar. 

Thicks, 
of bar. 

Diam. 
of pin. 

Metal i 
across 

W. 

n head 
pin. 

H. 

Width 
of bar. 

Thicks, 
of bar. 

Diam. 
of pin. 

Metal i 
across 

W. 

n bead 
pin. 

H. 

1. 

.2 

.67 

1.33 

1.50 

1. 

.55 

1.28 

1.50 

1.6 

1. 

.25 

.77 

1.33 

1.50 

1. 

.60 

1.36 

1.55 

1.7: 

1. 

.30 

.86 

1.40 

1.50 

1. 

.65 

1.43 

1.60 

1.7' 

1. 

.35 

.95 

1.50 

1.50 

1. 

.70 

1.50 

1.67 

1.8 

1. 

.40 

1.04 

1.50 

1.50 

1. 

.80 

1.64 

1.67 

1.9 

1. 

.45 

1.12 

1.50 

1.53 

1. 

.90 

1.77 

1.70 

2.0 

1. 

.50 

1.20 

1.50 

1.56 

1. 

1.00 

1.90 

1.76 

2.2 


■ 

- 


Art. 26. Figs 62 exhibit joints adapted to most of the cases tl 1 
occur in practice with wooden beams, Ac. They need but little explanation. Fi, 
is a good mode of splicing a post; in doing which the line o o should never be 
dined or sloped, but be made vert; otherwise, in case of shrinkage, or of gr 
pressure, the parts on each side of it tend to slide along each other, and thus br 
a great strain upon the bolts. When greater strength is reqd, iron hoops may 
used, as at b, h, and.;, instead of bolts. Fig b, a post spliced by 4 fishing piec 
which may be fastened either by bolts, as in the upper part; or by hoops, as in 
lower. The hoops may be tightened by flanges and screws, as ats; or thin i 
wedges may be driven between them and the timbers, if necessary. Fig C shov 
good strong arrangement for uniting a straining-beam A, a rafter l, and a queen-j r 
u ; by letting A and 1 abut against each other, and confining them between a don j- 
queen-post t t ; n n are two blocks through which the bolts pass. A similar arrai jt 
ment is equally good for uniting the tie-beam w, Avith the foot v, of the queens; \a r 
the addition of a strap, as in the fig. Fig e is a method of framing one beam i 
another, at right angles to it. An iron stirrup, as at/, may be used for 
same purpose ; and is stronger. Figs g h, ij are built beams. When a he 
or girder of great depth is required, if we obtain it by merely laying one beam 




with other kinds, is a subject of great practical importance; but one which hitherto has received 
little attention. Black walnut and cypress are said to cause mutual rot within a year or two. 
served cases of this kind should be reported to the leading professional journals. 

* The strength of a given hinged-end pillar (see p 439) is increased to an important extent bj 
larging the diameter of the pin. 

























































TRUSSES. 


61 3 


Jon another, we secure only as much strength as the two beams would have if 
: a ' a }1 e - But 1 \ we prevent them from sliding on one another, by inserting Irani 
[ S e blocks or keys, as at gi or by indenting them into one another, as at!?; and 


Figs. 62 . 






































































































































































































614 


TRUSSES, 


then bolt or strap them firmly together to create friction ;we obtain nearly the strong! 
of a solid beam of the total depth ; which strength is as the square ot the dept 1 . 


U DvliU UCtVUl VUV -x — t —' 

^Tredeold "directs that the combined thicknesses of all the keys he not l< 

is thus more equalized over the entire area of the joint; or cast iron maj be used. . . v . 

Frequently a simple strap will not suffice, when it is necessary to draw the two timbers at 
tightly together In such cases, oueendof each strap may, as at x, terminate as a screw , and a! 
pushing 1 through a cross-bar Z, all may be tightened up by » ““t »t *^Or thepr^to rf Aeg 
blk Ktr show n at K, may be applied. Sometimes, as at A, the hole for the boll is nrst bored , u 
a hole is cut in one side or the timber, and reaching to the bolt hole, large eu ° u h ^ h l ° h ^, ?' V h ‘ “ 
nut to be inserted. This being done, the hole is refilled by a wooden plug, which holds the nut 
place. Then the screw-bolt is inserted, passing through the nut. By turning the screw the timL 

B1 When'the 6 ends * 0 ? beanfs, joists, &c, are inserted into walls In the usual square manner then 
danger that in case of being burnt in two, they may, in falling, overturn the wall, fins may 
avoided by cutting the ends into the shape shown at in. . , . . 

When a strap o, Fig R, has to bear a straiu so great as to endanger its crushing the tim er p 
which it rests, a casting like v may be used under it. The strap will pass around the backo 
casting. The small projections In the bottom being notched iuto the timber, will pretent the ca, 
from slidiug under the oblique strain of the strap. The same may be ca* 

below a limber as well as above it. When below, it may become necessary to boltgr <ppikei the cas 
to the under side of the timber. When the pull on a strap is at right ',.V 

is much strain, a piece of plate-iron, instead of a casting, may be inserted between the strap an 
timber, to prevent the latter from being crushed or crippled; see I and I. 

Art. 27. Expansion rollers, or planed iron Slides, or rot 
ers, or suspension'links, must be provided when an iron span excc 
about SO feet: in order to allow the trusses to contract and expand fn 
under changes of temperature, without undue strain upon some of its meml 
Figure 63 shows the general arrangement of roll¬ 
ers; which are cylinders of cast iron or steel, from 3 
to C ins diam ; and 1 to 4 ft long; planed smooth. From 
4 to 8 or more of these are connected together by a 
kind of framing, n«; and one such frame is placed 
under at least one end of the truss. The rollers rest 
upon a strong planed cast bed-plate no; bolted to the 
masonry below. Under the end of the truss is a sim¬ 
ilar plate s s, by which it rests on the rollers. Since 
a truss of even 200 ft span will scarcely change its 
length as much as 3 ins by extremes of temperature, 
the play of the rollers is but small. They are kept in 
line by flanges cast along the side of the bed-plate, 
downward from s s, so as completely to protect 
the rollers from dust, rain, &c. 


l. 



Fig. 63. 
Flanges should also pr 
L 


In Fig 64, r r gives a general idea of a rocker; and Fig 65, 
e *, of a suspension-link. U U in each fig is aside view of 
a cast-iron Fink upper chord, through each end of which 
passes arouud pin o, which sustains the entire weight of the 
truss and its load; and which is sustained by from 4 to 6 
rockers or links, as the case may be. In a railroad bridge of 
205 ft span, across the Monongahela, the links are 3J4 ft long; 
and the pins 5 ins diam; and in others of the same size, over 
Barren and other rivers, the rockers are a foot wide from r 
to »■; and about 5 ins wide transversely on the curved tread 
or rim. For the accommodation of these several links and 
rockers; as well as of the various bars b b , which constitute 
the oblique ties of a Fink truss; the ends of the octagonal 
cast,iron upper chords are widened out, as shown by Fig 66; 
which is atop view of a longitudinal section of such an end. 
The rockers, or links, and bars, b, occupy the spaces n n, &e, 
between the several partitions of the chord; and the pin oo 
passes through them all, except when it is expedient to 
attach some of the bars to the sides, or to the top of the 
chord, as at t. These figs are intended merely to illustrate 
the general principle, without regard to detail of construction. 

In some English bridges of considerable size, such as the 
Crumlin Viaduct, of 150 ft spans; and the Newark Dyke 
bridge, of 240 span; (both of them Warren girders,) and 
sustained like the foregoing, by the ends of the upper 
chords; no further precaution is taken with regard to expan¬ 
sion and contraction, than merely to rest the ends of said 
chords upon smoothly planed iron plates, upon which they 
may slide. So also several American bridges. 


















SUSPENSION BRIDGES, 


615 


r | 

4rt. 1. Table of data required for calculating the main 
If tains or cables of suspension bridges. Original. 


all 

IK 

.til 

scr 

.t' Qttj 
until 

ere 

m 

7< 

of 

ast 

4, 

ast 




SUSPENSION BRIDGES. 


i parts 
f the 
!hord. 


40 

35 

30 

25 

20 

19 
18 
17 
16 
15 
14 
13 
12 

1 

10 

9 

% 

7 

20 

% 

5 

►4 

3 


theChord. 


m 

4 

i 20 


.025 

.0286 

.0333 

.04 

.05 

.0526 

.0555 

.0588 

.0625 

.0667 

.0714 

.0769 

.0833 

.0919 

.1 

.1111 
.1 25 
.1429 
.15 
.1667 
.2 

•225 

.25 

.3 

.3333 

.4 

.45 

.5 


Length of 
n Main Chains 
between Sus¬ 
pension Piers 
i. in parts of the 
Chord. 

Tension on all 
the Main 
Chains at 
either Suspen¬ 
sion Pier, in 
parts of the 
entire Sus¬ 
pended Wt. 
of the Bridge, 
and its Load. 

Tension at the 
Center of all 
the Main 
Chains ; in 
parts of the 
entire Sus¬ 
pended Wt. 
of the Bridge, 
and its Load. 

Angle of 
Direc¬ 
tion of 
theChains 
at the 
Piers. 

Natural 
Sineof the 
Augle of 
Direction 
of the 
Chains, at 
the Piers. 




Deg. Min. 


1.002 

5.03 

5.00 

5 

43 

.0995 

1.002 

4.40 

4.37 

6 

31 

.1135 

1.003 

3.78 

3.75 

7 

36 

.1322 

1.004 

3.16 

3.12 

9 

6 

.1580 

1.006 

2.55 

2.51 

11 

19 

.1961 

1.007 

2.43 

2.38 

11 

53 

.2060 

1.008 

2.30 

2.25 

12 

32 

.2169 

1.009 

2.18 

2.12 

13 

14 

.2290 

1.010 

2.06 

2.00 

14 

2 

.2425 

1.012 

1.94 

1.87 

14 

55 

.2573 

1.013 

1.82 

1.74 

15 

57 

.2747 

1.016 

1.70 

1.62 

17 

6 

.2941 

1.018 

1.57 

1.49 

18 

33 

.3180 

1.022 

1.46 

1.37 

19 

59 

.3418 

1.026 

1.35 

1.25 

21 

48 

.3714 

1.033 

1.23 

1.12 

23 

58 

.4062 

1.041 

1.12 

1.00 

26 

33 

.4471 

1.053 

1.01 

.881 

29 

45 

.4961 

1.058 

.972 

.833 

30 

58 

.5145 

1.070 

.901 

.750 

33 

41 

.5547 

1.098 

.800 

.625 

38 

40 

.6247 

1.122 

.747 

.555 

42 

0 

.6690 

1.149 

.707 

.500 

45 

00 

.7071 

1.205 

.651 

.417 

50 

12 

.7682 

1.247 

.625 

.375 

53 

8 

.8000 

1.332 

.589 

.312 

58 

2 

.8183 

1.403 

.572 

.278 

60 

57 

.8742 

1.480 

.559 

.250 

63 

26 

.8944 


Natural 
Cosine of 
the Augk 
of Direc¬ 
tion of the 
Chains at 
the Piers. 


.9950 

.9935 

.9912 

.9874 

.9806 

.9786 

.9762 

.9734 

.9701 

.9663 

.9615 

.9558 

.9480 

.9398 

.9285 

.9138 

.8945 

.8726 

.8574 

.8320 

.7808 

.7433 

.7071 

.6401 

.6000 

.5294 

.4855 

.4472 


ese calculations are based on the assumption that the curve formed by the main chains is a 
pH b°' a 1 which is not strictly correct. In a finished bridge, the curve is betweeu a parabola and a 
larv ; and is not susceptible of a rigorous determination. It may Save SOIUC 11*011* 
t ill making; the drawing's of a suspension bridge, to remember that when the 
ction does not exceed about yy of the span, a segment of a circle may be used instead of the 
curve; inasmuch as the two then coincide very closely ; and the more so as the deflection be- 
^ ;s less than y^-. The dimensions taken from the drawing of a segment will answer all the pur- 
of estimating the quantities of materials. 

% r some particulars respecting wire for cables, see pages 412 and 413. 

he deflection usually adopted by engineers for great spans is 

"f 1 ? t0 To tlie span - As much a3 Yfi is generally confined to small spans. The bridge will 
rouger, or will require less area of cable, if the deflection is greater; but it then undulates more 
ly ; and as undulations tend to destroy the bridge by loosening the joints, and by increasing the 
entum, they must be specially guarded against as much as possible. The usual mode of doing 
s by trussing the hand-railing; which with this view may be made higher, and of stouter tim- 
than would otherwise be necessary. In large spans, indeed, it may be supplanted by regular 
e-trusses, sufficiently high to be braced together overhead, as in the Niagara Railroad bridge, 
b the trusses are 18 ft high; supporting a single-track railroad on top; and a common roadway 
ft clear width, below.* 

he writer believes himself to have been the first person to suggest the addition of very deep 
as braced together transversely, for large suspension bridges. Early in 1851, he designed such 
ige, with four spans of 1000 ft each : and two of 500; with wire cables ; and trusses 20 ft high, 
s intended for crossing the Delaware at Market Street, Philada. It was publicly exhibited for 
al months at the Franklin Institute, and at the Merchants’ Exchange; and was finally stolen 
the hall of the latter. Mr Roebling's Niagara bridge, of 800 ft span, with trusses 18 ft high, was 
irmnenced until the Latter part of 1852; or about 18 months after mine had been publicly ex- 
id. 

















































610 


SUSPENSION BRIDGES. 


Another very Important aid is found in deep longitudinal floor timbers, firmly united where their 
ends meet each other. These assist by distributing among several suspender rods, and by that 
means along a cotisideuible length of main eable, the weight of heavy passing loads ; and thus pre¬ 
vent the undue undulation that would take place if the load were concentrated upon only two opposite 
suspeuders. With this view, the wooden striugers under the rails on the Niagara bridge are made 
virtually 4 ft deep. The same principle is evidently good for ordinary trussed bridges. 

Another mode of relieving the main cables is by means of cable-stay*; which are bars of iron, or 
wire ropes, extending like c y, Fig 1. from the saddles at the points of suspension c, d, obliquely down 
to the floor, or to some part of the truss. In the Niagara bridge are 64 such stays, of wire ropes of 
] y s inch diam; the longest of which reach more than quarter way across the span from each tower. 
They transfer much of the strain of the wt of the bridge and its load directly to the saddles at the top 
of the tcwers: thereby relieving every part of the main cable, and diminishing undulation. They 
end at cand d, where they are attached, not to the cables, but to the saddles. They of course do not 
relieve the back stays. 

The greatest danger arises from the action of strong* winds 
striking; below the floor, and either lifting the whole platform, and letting 
it fall suddenly ; or imparting to it violent wavelike undulations. The bridge of 1010 ft span across 
the Ohio at Wheeling, l>y Charles Ellet, Jr, was destroyed in this manner. It is said to have uudu- 
lateil 20 ft vertically before giving way. It had no effective guards against undulation ; for although 
its hand-railing was trussed, it was too low and slight to be of much service iu so great a span. 
Many other bridges have beeu either destroyed or injured in the same way. When the height of the 
roadway above the water admits of it, the precaution may be adopted of tie-rods, or anchor rods, 
under the floor at different points along the span, and carried from theuce, inclining dowuward, to 
the abutments, to which they should be very strongly confined. In the Niagara Railroad bridge 56 
such ties, made of wire ropes 1}± inch diam, extend diagonally from the bottom of the bridge, to the 
rocks below. They, however, detract greatly from the dignity of a structure. 

Mr Brunei, iu so'me cases, for checking undulations from violent winds striking beneath the plat¬ 
form, used also inverted or up-curving cables under the floor. Their ends were strongly confined to 
the abuts several ft below the platform; and the cables were connected at intervals, with the plat¬ 
form, so as to hold it down. 

Art. 2. The angle adg, or a c i, Fig 1, which a tang dg or ci to the curve at 
either point of suspension c or d, forms with the hor line c d or chord, is called the Illic it* of 
direction ot the msiill distill*., or cables, at those points. Frequently the ends 
c h, and dr, of the chains, called the bnckKtaj’S, are carried away from the suspension piers 
in straight lines; in which case the angles l dr, e ch, formed between the hor line e l and the chain 
itself, become the angles of direction of the backstays. 



Sine Of ailgie of direction a (1 ff = _ Twice the deflection a 6 __ 

j/ (twice the deflection^ -+- (Half the chord)? 

Note 1. The direction of the tang d g or c i, can be laid down on a drawing, thus: Continue the 
line a h. making it twice as long as ah-, then lines drawn from d and c to its lower end, will be tangs 
to the parabolic curve at the points of suspension. 


Note 2. li the chord c tl he not lior, as sometimes is the case, the angle 
must be measured from a bor line drawn for the purpose at each point of suspension ; as the two 
angles will in that case be unequal, the piers being of unequal heights. 

lcnsdon OI1 all the instill Half the entire suspended weight of the clear 
chains or cables, together, _ _ span and its load 

at either one of the itiers, ~ 
c or cl, Fift-1. 


Sine of angle of direction a d g 


or 


V m Span) 2 -f a Defl)2 Ha,f '^ entire sus- 

-— . - L v pended weight of 

2 Deflection t he clear spau and 

its load. 


Tension on all the main 
chains or cables, together, 
at the middle, b f of the 
kI»hii, Fig- 1. 


Half the entire suspended 
w eight of the clear span X 
and its loud 


Cosine of angle of 
direction o d g 


Sine of angle of direction a d g 


Half the entire suspended weight of v Half the 
W| . __tLe clpur span and its load * span 

Twice the deflection 

The dlff between the tensions at the middle, and at the points of suspension, is so trifling with the 
proportion of chord and deflection commonly adopted in practice, viz, front about to JL, that it 
is usually neglected ; inasmuch as the saving in the weight of metal would he fully compensated for 
by the increased labor of manufacture in gradually reducing the dimensions of the chains from the 
points of suspension toward the middle: and in preparing fittings for parts of many different sizes. 
The reduction has, however, been made in some large bridges with wrought-iron main chains - but 
in none with wire cables. 




























SUSPENSION BRIDGES 


617 


Art. 2A. As it is sometimes convenient to form a rough idea at the moment, of 
the size of cables required for a bridge, we suggest the following rule for finding approximately the 
area in sq ins of solid iron in the wire required to sustain, with a safety of 3,* the weight of the bridge 
itself, together with an extraneous load of 1.205 tons per foot ruu of span; which corresponds to 100 
lbs per sq ft of platform of 27 ft clear available width. This suffices for a double carriage-way. and 
two footways. The deflection is assumed at yV of the span; and the wire to have an ultimate 
11 strength of 36 tons per solid square inch, as per table, page 412; and which can be procured without 
difficulty. For spans of 100 ft or more. 

Rule. Mult the span in feet, by the square root of the span. Divide the prod by 100. To the 
; quot add the sq rt of the span. Or, as a formula. 

Area of solid metal of aU, span X sq rt of span 

the cables ; in square ins ; — - + sqrt of span. 

for spates over 100 feet 100 

l; For spans less than 100 feet, proportion the area to that at 100 ft. 

If a defl of y ( y is adopted instead of -yVi the area of the cables may be reduced very nearly w part- 

The following 1 table is drawn up from this rale. The 3d col 

gives the united areas of all the actual wire cables, when made up, including voids. (Original.) 


Span 

in 

Feet. 

Solid Iron 
in all the 
Cables. 

Areas of 
all the 
Finished 
Cables. 

Span 

in 

Feet. 

Solid Iron 
in all the 
Cables. 

Areas of 
all the 
Finished 
Cables. 

Span 

in 

Feet. 

Solid Iron 
in all the 
Cables. 

Areas of 
all the 
Finished 
Cables. 

i 

Sq. Ins. 

Sq. Ins. 


Sq. Ins. 

Sq. Ins. 


Sq. Ins. 

Sq. Ins. 

1000 

348 

446 

400 

100 

128 

150 

30.6 

39.2 

900 

300 

385 

350 

84 

108 

125 

25.2 

32.3 

800 

254 

326 

300 

69 

89 

100 

20 

25.6 

700 

212 

272 

250 

55 

71 

75 

15 

19.2 

600 

171 

219 

200 

42 

54 

50 

10 

12.8 

500 

134 

172 

175 

36.4 

46.7 

25 

5 

6.4 


Having the areas of all the actual cables, we can readily find their diam. Thns^suppose with a 

span of 500 ft, we intend to use four cables. Then the area of each of them will be — = 43 sq ins-, 

and from the table or circles, we see that the corresponding diam is full 1% ins. 

The above areas are supposed to allow for the increased wt of a depth of truss, and other additions 
necessarv to secure the bridge from violent winds, and from undue vibrations from passing loads. 

When these considerations are neglected, and a less maximum load assumed, the following descrip¬ 
tions of the Wheeling and Freyburg bridges show what reductions are practicable. Weight, suffi¬ 
ciently provided for, is of great service in reducing undulation. 


* The writer must not be understood to advocate a safety of 3 against 100 lbs per.sq ft, in addition 
to the weight of the bridge, in all cases. He believes that limit to be about a sufficient one for a pro¬ 
perl v designed wire suspension bridge for ordinary travel; but for an important railroad bridge he 
would (according to position, exposure, Ac) adopt a safety of at least from 4 to 6 against the greatest 
possible load, added to the wt of the bridge. A train of cars opposes a great surface to the action of 
side winds; and trains must run during violent storms, as well as during calms , but a larg« °P ea 
bridge for common travel is not likely to be densely crowded with people during a severe storm. 

































618 


SUSP ENSION BRIDG ES. 


Art. 3. Tension on the back-stays, e h and dfr* Fig 1. and 
strains on the piers, or towers, or pillars. If the angle of direction adg and 
the angle 2 dr, between the back-stays and the horizontal, are equal to each other, the tension on the 
hack-stays will be equal to that on the main cables at the tops of the piers j and the pressure on the 
piers will be vertical; but if the two angles are unequal, then these tensions and pressures will de¬ 
pend, to a very important exteDt, upon the manner in which the chains or cables are fixed to, or laid 
upon, the tops of the piers. 



Art. 4. In Figs 

2, 3 and 4, the piers 
dnm are supposed to be im¬ 
movable ; and the cables 
k d u, passing over them, 
rest immediately upon hori¬ 
zontal rollers, which have no ' 
other motion than that of re- ' 
votving about their horizon- ‘ 
tat axes ; the frame to which 


they are attached being bolt¬ 
ed to the top of the pier. On 
these rollers the cables slide, 
when changes of loading 
or of temperature produce 
changes in their directions. 

In this case the ten¬ 
sion on the back¬ 
stays is equal to that on 
the main cable. See Funic¬ 
ular Machine. 

To find the direction and 
amount of theprCSSUre 
on tbe pier; from a, 

Fig 2. 3 or 4, lay offds and 
d r, each equal, by scale, tc 
the tension, in tons, on tht 
maid chain at d ; and from f 
and r lay it otf to r. In othei 
words, draw the parallelo 
grain dsvr, and its diagona 
d v. Then will d v give th« 
direction and amount of tb< 
pressure upon the pier. 

When, as in Fig 2, th< 
angles a d g and Ida an 
equal, the pressure d v wil 
be vertical, and equal to th< 
entire weight of the clea 
span and its load. 

When, as in Figs 3 and 4 
the aDglcs a dg and 2 d n an 
unequal, the pressure d 
will not he vertical, but wil 
incline from d toward th 
smaller angle. 

than the entire weight of tb 


When, as in Fig 3, a d g exceeds l d u, the pressure d v will be less 
clear span and its load. 

When, as in Fig 4, I du exceeds a dg, the pressure d o will be greater than the entire weight of th 
clear span and its load. 

If we suppose symmetrical piers, d n m, to be used in each case, tbe base m n of that in Fig 2, ma 
be much narrower than in tbe other two figs ; because, the direction of d v being vertical, the pressur 
has no tendency to overturn the pier. In Fig 2, the masonry of tbe pier should be laid in tbe usna 
horizontal courses, in order that its bed joints may be at right angles to the pressure upon them. 

But. iu Figs 3 aud 4, if the bases were made as narrow as in Fig 2, tbe lines of direction d v. of th 
pressure, would fall outside of them ; and the piers would consequently be in danger of overturning 
Also, the stones of the masonry, if laid in horizontal courses, would have a tendency to slide on eac 
other.^ To prevent this, the beds should be at right angles to d v. 

In fig 3. tbe obliquity of the pressure would tend to slide the base of the pier outward as show 
by the arrow ; but in Fig 4, inward. This tendency is produced by the horizontal component of tb 
force dy. The amount of this may be found thus, in either fig: From d downward draw a vert lin 
as m Fig 4; and from v a hor one. meeting it in z, then v z. measured by the same scale of tons a 
before, will give this horizontal force, and d z will give the vertical component of the pressure d i 
The effect upon tbe pier, of tbe one pressure d v is precisely the same as would be produced upon it b 
one vertical force equal to d z and a horizontal one equal to vz acting at the same time, as explaiue 
under Composition and Resolution of Forces. 

If, in either fig, we draw tbe vert lines s v and ro. see Fie 4. then d o. measd hv the foreenimr sent 



their difference will eqnal v z. It is this difference only that tends to slide, or to upset, the pier ; ti 
other portions of d o and p d neutralizing each other in that respect. 

The foregoing strains may all be calculated, thus: 


Horizontal pull in word by the main ehain rr Tension x Cosineof ad. 

v , * outward by the back-stay -Tension x Cosineor i di 

Vertical pi essu re l»y mam ehain = Tension x Sine of a a g 

“ buck-stay = Tension X Sine of 2 d « 




























SUSPENSION BRIDGES, 


619 


Fi<r.4 


,^; rt y' 5 * .l r l he / ab?e8 pass free, y over a loose pin, d, Fig4 A, supported !»y a link ri¬ 

nging from the fixed pin z, and capable of moving freely about both v 9 

its pins; the tension in the back stay will, as before, be equal to 
iat lu the main cable; and the direction and amount of the strain 
i the piers will be found in the same way as for Figs 2, 3 and 4; 
unely : lay otf d 8 and d r, each equal to the tension, and draw the 
irallelogram ds vr. Then will d v give the amount and direction of 
e strain ou the piers. This last will, of course, be transmitted through 
e pins and the link. The amouutof tension ou the link will be given 
the length of d v ; and the link (being free to move) will be in line 
ith this tension. The shearing strain on each pin is also given by d v. 

Art. 6. But if the euds of the cable and back-stay, 
gs 4 B, 4 C aud 4 I), at the top of the pier, be made fast to a truck 
»• agon which is supported by rollers on a smooth platform on top 0 r the pier, the axles of the 
llers being fixed in the truck ; then the strain on the back-stay will not be the same as that on the 
hie, unless the angles a d g and l d u are equal, as in Fig 4 B. 

If “ dy exceeds Id' u, as in Fig 4 C, the strain on the 
ck- stay will be less than that on the cable, and vice versa 
ig*D). 

But, in either case, if the top of the pier is horizontal, 
is usually the case, the horizontal components of the 
ainson the cables and on the back-stays, will be equal, 
d will thus counteract each other, aud there will conse- 
ently be no horizontal or oblique strain on the pier, 
iat is, the strain on the pier will be vertical. 



a 

d 

tV 

o l 


... 

-’V-A 

s. J 

s 

-At 

n 



To find the amount of the ten¬ 
on on I he back-stay, and of the pres- 

n*e on the pier; on dg in either Fig. 4 
1 C or 4 D, lay off ds, equal, by scale, to the tension 
the cable at d. Draw d v perpendicular to the surface 
r» on which the rollers rest. We assume that ran is 
rizontal, as is generally, but not necessarily, the case : 
d d v, therefore, vertical. Draw s v horizontal, or par- 
si to m n.* 

Then s v will give the horizontal pull of the main cable 
the wagon, and d v will give the vertical pressure of 
i wheel d on the tower (to which that of the wheel d'has 
to be added). From d r lay off d' o horizontal, and equal 
sv; and draw r o vertically. Then rf'r will give the 
louutof the pull on the back-stay ; and ro will give the 
tical pressure of the wheel d' on the pier; which 
ist be added to d v for the fofal vertical pressure. 


Fitf.4 D 



9 



..j n 

rS 

V 


Fi«\4 C 


a d' 7 

ci _ 


9 


)r the various strains may be calculated, thus: 

[orizontal pull 
iUT or do at the 
“ fop of the pier 

h. trainri'rln back- 



_ Horizontal pull at _ Tension ds i 
— middle of span cable at d 


X Cosine of a dg. 


stay 
4res on pier, perp 


l = Horizontal pull s » or d o at top of . „ 
j pier, or at middle of span . Cosine off d u. 


o surf on 
!l# ers rest 


which the rol 


3 l / 

- $=dv + ro=( 


Tension ds _ 

on main X _l_ /Tension ** ' x 

cable at d ad 9 ) Von back-stay * 


?ine of\ 

Idu) 




Fi«\4 F 

Art. 7. When, ns Is sometimes the ease in light 
bridges, the piers are posts, P Fig 4 E, of w'ood or iron, hinged at 
the bottom, and having the cables and back-stays firmly fixed to 
ir tops; from d draw d s, equal, by scale, to the tension on the main cable at d ; and d w toward 
foot of the post. From * draw a r parallel to the back-stay, and meeting d tv in r. Then will 
give the strain in the back-stay, aud d r will give the amount and direction of the pressure upon 
post. 

Art. 8. As in the Niagara bridge, the cables often merely rest upon 

vable trucks, or saddles, T Fig 4 F. curved on top to avoid sudden bends in the cables, and resting 
>n loose rollers which lie upon a thick horizontal iron plate bolted to the top of the pier, and are 
e to move horizontally. In such cases the angles ad g and l d'u are made equal; so that the pulls 


The tines a l and s v must be drawn parallel to the surface m n on which tlietcagon rest*,whether 
d surface be horizontal or inclined. 









































620 


SUSPENSION BRIDGES, 


d * and d' r are equal, as are also their horizontal components p d and d’o • and the pressures on th 
pier are vertical; and if changes of temperature or of loading produce slight changes in the angle 
a d g and t d' to. the truck will (by reason of the inequality thus brought about between the hon 
zoutal components) move far enough to restore the equality between the angles, and between th 
horizontal components, and consequently the pressure upon the pier will at all times be vertical. 


Art. 9. To find, approximately, tlie length of a mainchaii 

C b d\ Fig. 1 ; having the span c d, and the middle deli a 0. See preceding table, Art l. 


Half length of main chain — ]/ 1 % (defl 2 ) -J* chord;*. 

In Menai bridge the chord cd is 579.874 ft; and the dell is 43 ft. 

According to the above formula, the entire leugth is 588.3 feet. By actual measurement the chaii 
Is precisely 590 feet. The approximate rule below gives 589.764 ft. 

Noth. The leugths obtained by this rule are only approximate, because the calculation is 
upon the supposition that the chains form a parabolic curve; whereas, in fact, the curve of a finis 
bridge is neither precisely a parabola, nor a catenary, but intermediate of the two. 

The following simple rule by the writer is quite as approximate as the foregoing tedious out 
when, as is generally the case, the deli is not greater than ^ of the chord, or span. 

Length of main chaiu when dell does not exceed one-twelfth of the span = chord -J- .23 defl 


Art. lO. To find, approximately, the length of the ver 
suspending rods jc i/, Ac, Fig 1; assuming the curve t< 
be a parabola. 

Let at, Fig 1. be auy poiut whatever in the curve ; aud let x w be drawn perp to the chord cd; tit 
x f perp to ah; then iu any parabola, as at -2 ; a >c- : : a b : b f. And b f thus found, added to 6 
(which is supposed to be already kuowu, being the length decided on for the middle suspending rod 
gives x s, the length of rod reqd at the poiut x; and so at any other point. 


If' h f thus found l>c taken from the middle deflection a h 

it leaves tr ur; and thus any deflection tv x of the main chaiu or cable, may 
found when we know its hor dist, a to, from the center, a, of the spau. 

Iu the foregoiug rule, the floor of the bridge is supposed to be straight ; but generally it is raise 
toward the center; aud in that case, the rods must first be calculated as if the floor were straigh 
and the requisite deductions be made afterward. When it rises in two straight lines meeting iu tl 
center, the method of doing this is obvious. When an arc of a circle is used, its ordinates may t 
calculated by the rule given on page 141a, and deducted from the lengths obtained by this rub 
Or, having drawn the curve by the rule for drawiug a parabola. the dimensions can be approi 
imated to by a scale. The adjustments to the precise lengths must be made duriug the actual cot 
struction of the bridge, by means of nuts on their lower scrcw-ends. The rods require, therefor 
ouly to be made long enough at first. 

The towers, piers, or pillars, which nphold the chains 
cables, admit of an endless variety in design. According to ci 

cumstances, they may consist each of a single vertical piece of timber, or a pillar of cast or wroug 
iron; or of two or more such, placed obliquely, either with or without connecting pieces; like t 
bents of a trestle, Or they may be made (with any degree of * 

namentation) of cast-iron plates; as in iron house-fronts. Or they may be of masonry, brick, 
concrete; or of any of these combined. 


.bo 


no 

ffli 


Each of the snspending-rods, through which the floor of the bridge 

upheld by the main cliaius, requires merely strength sufficient to support safely the greatestlo; 
that can come upon the interval between it and half-way to the nearest rod on each aide of it; i 
eluding the wt of the platform, &c, along the same interval. 


In anchoring the backstays into the gronml, it is necessary 

secure for them a sufficiently safe resistance against a pull equal to the strain, upon the backsta ' 


l 


As to the anchorage of the cables below the surface of the groun 

natural rock of firm character is the most favorable material that can preseut itself. When it is 
present, serious expense in masonry must be incurred in large spans, in order to secure the nec« 
weight to resist the pull of the cables. Our Figs 4)$ give ideas of the modes most frequently adi_ r 
For a very small bridge, such as a short foot-bridge, for instance, the backstavs may simply be 
chored to large stones, t. Fig A, buried to a sufficient depth. Or, if the pull is too great for so sim 
a precaution, the block of masonry, mm, may be added, enclosing the backstay. A close coveri 
of the mortar or cement of the masonry has a protecting effect upon the iron. 

To avoid the necessity for extending the backstays to so great a dist under ground, they are usua r 
curved near where they descend below the surface, as shown at B, D, and E; so as sooner to rea 
the reqd depth. This curving, however, gives rise to a new strain, in the direction shown by t 
arrows in Figs B and I). The nature of this strain, and the mode of finding its amount, (knowi 
the pull on the backstay,) are very simple; and fully explained under the head of Funicular Hi 
chine. _ The masonry must be disposed with reference to resisting this strain, as well 
that of the direct pull of the backstay. With this view, the blocks of stone on which the bend re 
should be laid in the position shown in Fig D ; or by the single block in Fig R. Sometimes the be 
Is made over a cast-iron chair or standard, as at x, Fig F, firmly bolted to the masonry. 

Fig E shows the arrangement at the Niagara railway bridge of 821 % ft span. The wire backsta 
end at cc; aud from there down to their anchors, they consist of heavy chains; each link of whi 
is composed of (alternately) 7 or 8 parallel bars of flat iron, with eye ends, through which pass bo 
* Each of the 7 bars of each link is 1.4 ins thick, by 7 ins wide, near I 




■fe 


* When chains of iron bars are used instead of wire cables, they are usually made as at p 6 
Since bar iron has but about half the tensile strength of wire, the chains must have a sectional ai 
twice as great as that of a cable. 















SUSPENSION BRIDGES, 


621 


>west part of the chain ; but they gradually increase from thence upward, until at c, c, where they 
uito with the wire cable, the sectioual area of each link is lilt sq ins. These chain backstays pass in 
' curve through the massive approach walls, (28 ft high,) and descend vertically down shahs s, *, 25 
deep in the solid rock. Here they pass through the cast-iron anchor-plates, to which they are con- 
Lned below by a bolt 3)4 ins diam. The anchor-plates are 6)4 feet square, and 2)4 ins thick ; except 
ir a space of about 20 ins by 26 ins, at the center where the chains pass through, where they are 1 



nt thick. Through this thick part is a separate opening for each bar composing the lowest link, 
rum this part also radiute to the outer edges of the lower face of the plate, eight ribs, 2)4 ius thick. 
Tie shafts s, s, have rough sides, as they were blasted ; and average 3 ft by 7 ft across; except at the 
ittom, where they are 8 ft square. They are completely tilled w ith cement masonry, with dressed 
ds, well in contact with the sides of the shafts; and thoroughly grouted, thus tightly enveloping 
e chains at every point; as does also the masonry of the approach wall ww; which extends 28 ft 
>ove ground; and is 6 ft thick at top, and 10)4 ft thick at its base on the natural rock. 

D, Figs 4)4, shows a mode that may be used in most cases, for bridges of any span. The depth 
id the area of transverse section of the shaft, and consequently the quantity of masonry in it, will 
'■pend chiefly upon whether it is sunk through rock, or through earth. If through firm rock, then 
its sides be made irregular, and the masonry made to fit securely into the irregularities, much re- 
nee may be placed upon it to assist the weight of the masonry in resisting the pull on the back- 
ys. Earth also assists materially in this respect. 

F is the arrangement in the Chelsea bridge of 333 feet span, across the Thames, at London ; Tlios. 
_ge, eng. The space from one wall 6 6, to the opposite one, is 45 feet; and is built up solid with 
ickwork aud concrete; except a passage-way 4 ft wide, and 5 ft high, along the backstay ; and a 
sail chamber behind the anchor-plates. It rests chiefly on piles. 

The arrangement by Mr Brunei, in the Charing Cross bridge,London,*isverv similar. In it also 
e entire abutment rests on piles; and is 40 ft high, 30 ft thick, and solid, except a narrow passage- 
v along the chains. The backstays extend into it 60 ft. Span 676 feet. Dell 50 feet. 

(j is intended merely as a general hint, which, variously modified, may find its application in tho 
se of a small temporary, or even permanent bridge; for the number of pieces, i , t, Ac, may be in- 
eased to any necessary extent; aud they may be made of iron or stone, instead of wood. 

j is* estimating 1 the action of the backstays upon the ma- 
j»siry, «*rc, to which they are anchored, it is safest to consider tho 

Easton along them to continue undiminished to their very ends ; although, when they are embedded 
masonry, friction causes it to diminish; especially when they are curved, as in E, Figs 4)4, in 
uich cuse the friction is greatly increased, and the tension thereby materially reduced as the ends 
e approached. Frequently, however, they are not so embedded; for, although embedding preserves 
e iron, many engineers prefer to leave an open space around the entire length of the anchor-chains ; 
well as around the anchor-plates; in order that they may be examined from time to time. To this 
d, the masses mwiof masonry in Figs 4)4, may he made not solid, but to consist of two parallel 
tlis, between which the backstav may pass; ntid the anchor-stones, or anchor-plates, will extend 
rnss the space between the walls, and have their bearing against the ends of tho walls. In F, the 
ble may be supposed either to be tightly surrounded by the masonry, and gn.utcd to it; or else to 
surrounded by a cylindrical passage-way like a culvert, so ns to be at all times accessible. Ana 
e same with regard' to the cable in the vertical shafts at D or E. 

Art 35 Art 43, Ac, of Force in Rigid Bodies, will assist in calculating L he resistance 

lich the masses mm of anchorage masonry oppose to the pull of the backstays. Soft friable stone 
jst be carefullv excluded from such parts of these masses as are most directly opposed to this pull. 
If blocks of stone large enoueh for securing good bond are not procurable, heavy bars of iron, or 
■earns, may life advantageously Introduced for that purpose. ...» 

The masses must be founded at such a depth as not to slide by the yielding of vac earth in front 

Experience shows that with due attention to periodical painting, and renewal of woodwork, a 
no -lv designed suspension bridge will be very durable, The transverse floor joists should be of 


jht iron ; to prevent interruption to travel" while putting in new wooden ones. 


I'ni'tioiilnr fftre slionltl bft b^stowod up©** Ilk© sti*©ngtli ©f 
li© joint* of tl»© si«l© psirapets; for the undulations and lateral motions 
the bridge expose them to violent, deranging forces in every direction. Tho parapets should be 
irh and stout; aud not restricted to mere service as hand-rails or Kurils. , 

\ -. a ru'e of thumb, one-half the sq rt of the span will be about a good height for them in ordinary 
ses, provided it is not less than a hand-rail requires. __ 


j# Removed to Clifton, England, in 1863, and replaced by an iron truss railway and foot bridge. 



































622 


SUSPEMSION BRIDGES. 


We do not think that diagonal horizontal bracing should, as is usual, be omitted under the floor. 
It may readily be effected by iron rods. , 

All the cables need not be at the sides of the bridge. One or more of them may be over its axis, 
especially in a wide bridge. One wide footpath in the center may be used, instead of two narrow 


ones at the sides. ... . - . 

The platform or roadway should be slightly cambered, or curved upward, to the extent say or abo 

of the span. 

Art. 11. The Niagara suspension bridge, built in 1852-3,*John 

Roebling, engineer, consists of a siugle span of 821 % It measured straight from center to center i 
towers ; and 800 ft of clear suspended length of roadway. It has two floors or roadways : the upp 
one for a single-track railway, is 25% It; and the lower one, for common travel, 24% ft wide, out to oul 
of everything. The lower one is 11* ft wide in the clear of everything. They are 17 ft apart verti 
•ally. The trusses are 18 ft total height, throughout. They are on the Pratt arrangement; 
with verticals 5 ft apart from cen to cen; and single oblique iron rods, 1 inch square, running i 
each direction across four of the 5 ft panels. Where these rods pass each other, they are tied togethe 

by 10 or 12 turns of inch wire. Each vertical consists of two pieces of 4% by 6% timber, placet 
4% ins apart, to allow the oblique rods to pass between them. Both upper and lower floor girders ar 
iu two pieces, of 4 by 16 ins each. Pairs 5 ft apart. The tops and bottoms of the verticals pass be 
tween the two pieces which form each floor girder. No tenons or mortises are used in the framing. 


There are four cables of iron wire ; two on each side of the bridge. Each 

cable is 10 ins diam. The wire is scant No. 9 of the Birmingham wire gauge, or scant .148 inch diam , 
Sixty wires have a united transverse section equal to one square inch of solid iron. Each of the fou j 
cables contains 3610 wires, with a united cross-section of 60.4 sq ins of solid metal. Therefore, th 
area of solid metal in a section of all the four cables together is 241.6 sq ins, or 1.678 sq ft; weighin 1 
814 lbs per ft of span. The wires of each cable are first made up, iu place, into 7 small strands; an ; 
these are firmly bound together throughout by a continuous close wrapping of wire. The strength o > 
each individual wire is 1640 lbs, or .73214 of a'ton. This is equal to 98400 lbs, or 43.93 tons per sq inc j 
of solid metal; or to 5943360 lbs, or 2653.3 tons per cable ; or to 10613.2 tons ultimate strength of th t 
four cables together. One cable on each side of the bridge deflects 54 ft; and the other 64 ft; averag ; 

deflection 59 ft, or about of the span. With this av defl the tension on the cables at the tops of th 
towers averages 1.82 times the total suspended wt of the span and its load. See table, Art 1. The w 
of the suspended span itself is about 900 tons; and if the greatest extraneous load on the two flooi I, 
together be taken at 1% tons per ft run, we have the total suspended wt 900-f- (800 X 1 %) — 1900 ton , 
And 1900 X 1.82 =3458 tons tension at towers; or very nearly' % of the ultimate strength of the cable J 
without any allowance for momentum, or wind. But such loads, although possible, are not permitte - 
to come upon the bridge. 

The wires were perfectly oiled before being made into strands; and when the strands were bein 
bound together to form a cable, the whole was again thoroughly saturated with oil and paint. 

The cables do not hang vertically; but the two upper ones are about 37 ft apart from center to cei > 
ter, where they rest upon the towers, (where all four are on the same level;) and are drawn to wlthi j 
13 ft of each other at the center of the span; and at the level of the railway track on top of ti , 
bridge; while the two lower ones are about 39 ft apart at the towers, and 25 ft at the center of the spai j 
and at the level of about halfway between the two floors. 

This drawing-in of the cables contributes much to lateral stability; as do also the upper and low 
floor of stout plank. There is no horizontal diagonal floor bracing. jj 

There are 624 suspenders of wire rope, 1% ins diam, and 5 ft apart, or corresponding with the flr 
girders, which they uphold; and with the wooden verticals of the trusses. They do not hang vert ? 
cally ; but incline inward. 

The masonry towers are all founded on rock. They are 78% ft high above the bottom of the bridgi , 
and 60% ft above the upper floor. The two at each end of the spau are 39 ft apart from center to cei j 1 
ter. At the level of the lower floor they are 19 X 20 ft; and 21 ft apart in the clear. At the level < J 
the upper floor they are 15 ft square; and 24 ft apart in the clear. Front there they taper regular ; ! 
to the top, where they are 8 ft square. They are built of limestone, in heavy dressed hor course 
laid in cement; vertical joints grouted. The upper courses are dowellcd. On top of each tower is ! 
cast-iron plate, 8 ft sq, and 2% ins thick, bedded in cement. Part of the top of this plate is plane ! 
as upon it move the rollers which support the cast-iron saddles on which the cables rest. At ea< 
tower, each cable has its separate saddle and rollers. Each saddle rests on 10 cast iron rollers 25 
ins long, and 5 ins diam, carefully planed. See Fig 4 F. They lie loosely, and close togethei !' 

and are kept in place by side flanges on the hed-plate. 

The cast saddles are each 5 ft long, by 25% ins wide. Their bottoms, which rest on the rollers, a 
flat, and planed. Their tops are curved to a rad of 6% ft; to suit the bend of the cables over the pier 
and each saddle has a longitudinal groove, in which the cable lies. The passage of the heaviest trai 
produces less than % an inch of movement in a saddle. 

The floors have a camber of 5 feet. 

A change of 100° Fah of temperature causes an average variation of about 2% ft in the deflecti 5 
of the cables, or in the camber of the roadways; and one of 150°, (about the extreme to which t 1 
bridge is exposed,) about 3% ft. The passage of a train weighing 291 tons, and covering the ent * 
length of the span, caused a deflection of 10 ins; and an ordinary train deflects it only from 3 t< p 
inches. 

This bridge has, since the year 1853, demonstrated the applicability of the suspension principle " 
large span railway bridges. Its entire cost was not quite $400,000. 


Art. 12. The wire suspension bridge near Freybnrg, Swi 
zerland, finished in 1834, Mr. Chaley, engineer, and still in full service. i 3 ’ 

very simple construction, and has served as the prototvpe for several in this country. It is t i 
common travel only : and is narrow: its entire width of platform being but 21% ft; and its cU ' 
available width but 19 ft. The dist from cen to cen of its towers is 889 feet; and its clear span 1 
tween abutments 800 ft; or the same as the Niagara. There are 4 cables, each 5 ins diam. Each , 
them consists of 1056 wires of No. 10, or full % inch diam, (or 71 wires to the sq inch of solid mrta j, 
arranged in 20 strauds of about 53 wires each. The four cables, therefore, have a united area of I , 
60 sq ins of solid metal; weighing 202 lbs or .09 of a ton, per ft run of span. All its suspenders i . 

• In 1886 the wooden piers and trusses here described were replaced by iron ones. 













SUSPENSION BRIDGES, 


623 


vertical; about 5 ft apart; and each upholds one end of a transverse floor girder. It has no side 
trussing except the slight one of the wooden hand-railing, which is about 6 feet high; and conse¬ 
quently, with its great span it is quite flexible. The deflection of the cables is yU 0 f the span ; hence 
the strain upon them at the top of the towers at either end. is 1.82 times (see table p 615) the wt of 
the suspended span itself, and its extraneous load; and supposing the wire to be as good as that of 
the Niagara, the breaking strain of the four cables would be 60 X 41 — 2640 tons; and their safe 
strain cannot be taken at more than as much, or 880 tons. The suspended weight reqd to produce 

880 

this safe strain would of course be —- = 484 tons. The suspended weight of the span itself cannot 

1.82 

well be less than .3 of a ton per ft run ; or 240 tons in all; * thus, leaving 484 — 240 =: 244 tons for 
the maximum safe extraneous load. This amounts to .305 of a ton per ft run of span ; or 36 lbs par 
sq ft of its platform, 19 ft wide in the clear. The French allowance is 41 lbs per sq ft; t and since no 
allowance is here made for momentum or wind, it is plain that this celebrated bridge, on account of 
its slight cables, and its flexibility, is by no means a strong one. In that respect, as well as steadi¬ 
ness, it is much inferior to the one next spoken of. It is said, however, to have withstood very severe 
tempests; and also to have been occasionally completely covered by crowds of people. If so, their 
lives were not very secure. 

Art. 13. The wire suspension bridg-e across the Schuylkill 

at Philada, finished in 1842,JCIias Ellet, Jr, engineer, is somewhat similar in character, and in the 
dimensions of its details, to the preceding; but being of much less span, is much stronger. Its span 
from cen to cen of towers is 358 ft; suspended platform between abuts 342 ft. It has ten cables of 3 
ins diam ; five on each side. Their united sections present 55 sq ins of solid iron ; or nearly as much 
as the preceding bridge of 800 ft clear span. The five cables on either side have differeut deflections, 
ranging between the yV and the yL 0 f the span from tower to tower. The dist from cen to cen of 
towers at either end of the span is 35)^ ft; and on top of each tower the cables (considerably flattened 
at that point)-lie side bv side on a single roller about 30 ins long, and 6 ins smallest diam, which has 5 
grooves, for their reception. Each cable is drawn-in about 3t£ ft at the center of the span. At in¬ 
tervals of 20 ins the parallel wires of the cable have a close wrapping of finer wire for a distance of 
8 ins. 

The suspenders are of wire; and are % inch diam; and 4 ft apart. On any one cable they are 20 
ft apart. They all incline slightly inward. 

The width of the platform from out to out is 27 ft; and in clear of hand-rails 25 ft. Inside of the 
hand-rail is a Tootwav, 4 ft 4 ins wide, on each side of the bridge. The remaining 16 ft 4 ins serves 
for a double carriage way, or double-track street railway. Figs 5 show' the arrangement of the wood¬ 
work, on a scale of ^ 
inch to a ft. The trussing 
of the parapets is on the 
Howe system, 
which does not appear to 
be as well adapted as the 
Pratt, to suspension 
bridges. The diagonals 
in the Fairmount bridge 
work themselves out of 
[{place laterally, by the 
’ vibrations of the bridge; 

and we have occasional- 
[I Jy seen several of them 
I almost on the point of 
(falling out entirely. 

Being under municipal 
charge, it is of course 
neglected. The upper 
chords u, are 12 ins 

wide by 6 ins deep; the lower ones l. and the stringer c, below them, are each 12 wide, by 7 deep. 
The diagonals fare all i ins wide, by 5 deep. The angle-blocks at their ends are of cast iron, hollow, 
and about % inch thick. The vert iron rods v, (in pairs,) are % inch diam near the center of the 
span • and 1 % at its ends. The top chords are spliced on each vert face by an iron bar. of 5 ft by 3 
ins by H inch : with 4 bolts passing through them. The splice of the bottom chord has merely i 
1 bolts, side by side ; (see Figs 5 ;) which (exceptS) are to a scale of inch to a ft. The floor girders g, 
4 ft apart from cen to cen, are 6 by 14 ins at their ends; and 6 by 16 at center. 

The floor is of two thicknesses of 2-inch plank; except the footpaths, which are single thickness. 

The wires were well oiled when the cables were made; and afterward painted. 

At S is shown the mode of uniting a suspender with a cable, o, by means of a small cast-iron yoke 
a w hich straddles the cable; and on the back of which is a groove % of an inch wide, in which the 
susnender rests. The metal of the yoke is about H inch thick. Since the lower ends of the wires 
which compose a suspender cannot themselves be formed into a screw-bolt, for upholding the floor 
girders, they are passed through the eye of a screw-bolt of bar iron ; then doubled on themselves, and 
held bv a wrapping of wire. It is well to introduce a yoke here also, to prevent the wear of the wires 
by friction. The small fig on the right of S is an edge view of the yoke g. 



* This is probably nearly its actual weight, as obtained by comparing it with the Fairmount bridge! 
vhich, by a careful estimate by the writer, weighs .375 of a ton per ft run t i>nt is considerably w;der 
,han the Freyburg; and carries four lines of light street-rails. But if the Frey burg has longitudinal 
[{vfa+q it will weich about .03 ton more per ft run. , ... 

t The greatest load that can come upou an ordinary bridge, is a dense crowd of people; and this 
he French engineers estimate at 41 fi>s per sq ft of platform. This is certainly as great as can well 
incur under ordinary circumstances; but it may be considerably exceeded. The French estimate, 
Moreover includes no allowance for wind, or for the crowd being in motion. Including these, the 
nr iter thinks that no suspension bridge should have a less safety than 3 against 100 tbs per sq ft, 
rdded to the weight of the bridge itself. A less coeff of safety is admissible in a wire bridge than 
a wi iron trussed one. on account of the greater reliability of the material. See foot-note, p 606. 

< Removed, 1873, and replaced by a truss bridge of 348 ft span, by Keystone 1.ridge Co. 




































024 


SUSPENSION BRIDGES, 


I 


There is no transverse bracing tinder the floor; nor are there longitudinal floor joists resting on the 
girders. Owing to the want of the distributing effect of these ; and to the use of so niauv small cables 
instead of but 2 or 4 larger ones ; as well as to the inefficient trussing of the hand railing or para¬ 
pets, the bridge is much less steady than it would otherwise be. 

With wire of the same quality as the Niagara, (or 44 tons per sq inch breaking strength,) the Fair- 
mount bridge would, with a safety of 3, (omitting momentum and wind,) sustain an extraneous load 
of 346 tons; which is equal to 1.01 ton per ft run of span ; or 90 fts per sq ft of its clear platform. 
This last is 2.5 times as great as the strength of the Freyburg, with the same quality of wire. The 
Fairmount is, however, we believe, built with wire of but 36 tons per sq inch ultimate strength. If 
so, its greatest extraneous load becomes reduced to 260 tons; or .76 ton per ft run; or 68 ftps per sq ft 
of platform, or nearly twice that of the Freyburg. 

The towers are of cut granite, in heavy courses. They are 8% ft square at the ground line, or level 
of the floor; about 5 ft sq at the top; and about 30 ft high. The backstays have the same angle of 
direction as the main cables. 

Art. 14. The Wheeling- bridge across the Ohio at Wheeling, Vir¬ 
ginia, also by Mr Ellet. had a span of 1010 ft between the towers; and 960 feet clear span between the 
abuts ; and was 26 ft wide from out to out. Its mode of construction was much the same even in de¬ 
tail as that of the Fairmount bridge; except in having 12 cables instead of 10. The 12cablescou- 
sisted of 6600 wires of No. 10 Birmingham gauge, presenting a sectional area of 93 sq ins of solid metal, 
weighing 313 lbs, or .14 of a ton, per foot of span. The weight of the woodwork was about the same 
per foot run of span as in the Fairmount- Altbongh its clear span was 2.8 times as great as the 
Fairmount, yet its cables had but 1.7 times as great area of solid metal. The entire suspended wt 
between towers, is stated at but 440 tons; therefore, with an average deflection of of the span, 
for a safety of 3 against 100 lbs per sq ft of platform of 24 ftclear width ; or 1.07 tons per ft run of span, 
the area of solid metal in the cables should have been 175 sq ins, with 44 ton wire like that of the Ni¬ 
agara ; or 214 sq ins, with 36 ton wire, which we believe was the quality actually used.. 

. The suspension canal aqueduct at Pittsburg. Penn, 

built in^l84o, John A Roebiing, Esq, engineer, has seven spans of 160ft each. Deflection 14)4 ft; oi 
about yy of the span. It has but two cables, each 7 ins diam. The two together con tain 3800 No. 10 
wires, making 53 sq ins of solid metal section. Ultimate strength of each wire 1100 fts ; equal to 35.2 
tons per sq inch of solid metal; and making the ultimate strength of the two cables together 1866 tons. 

The prism of water in the wooden aqueduct is 4 ft deep ; by 14% ft average width; and weighs 265 
tons per span. The wt of one span of the structure itself is about 111 tons; making the total sus¬ 
pended wt at each span 376 tons. The tension on the two cables at either end of a span, with a deli 

of y^, is 1.46 times the total suspended weight; see table, p 615. Hence it is in this case 376 X 1.46 

1 

~ 549 tons; and the strength of the cables is ^ =3.4 times the constant strain upon them. 

-tr-oK S * de water 1® a towpath for horses; and on the other a footpath ; each 7 ft clear width. 

, , occupied by horses and people, the foregoing safety would be reduced to about 3 The 

n, a jr d „! 0 „ , T. a fr, n ,V lt< T.f Iy ^ 0 \ he Y ight ’ inasmuoh as they displace a bulk of water equal to 
thur own wt, and but little of the displaced water remaius on a span at the same time with the boat. 

The great wt of the water prevents undulations; and the aqueduct is therefore very steady. On this 
account a less coefT of safety is admissible than on a common bridge. 

The aqueduct leaked badly along its lower corners. 

Art. 16. In 1796, Mr James Finley, of Fayette Connty, Penn, 
suspension bridges in ihe V. S.x and built several with 

spans of 200 feet and less. Many of them were very primitive structures; but answered sufficiently 
well for the times. They had usually either two or four chains, composed of links from 7 to 10 feet 
long formed by bending about 1%-inch square bars of iron, and welding their ends together 
each link-end, was a vertical suspender rod of 2 ins bv H inch iron • which at its lower eno’ 
bent and welded into a stirrup for upholding one end of a transverse floor beam. On these beams 

rested longitudinal joists supporting the floor plank. Finley used deflections as great as l or even 

8 P? n : a “ 0 d Hi 8 piers frec I uen ‘>y single wooden posts; the two at each end being braced 
together at top. Such were used in a span of 151% ft clear, across Will’s Creek, Alleghan v Co Penn 

. n h . d Tho^n'^r'r l he dC K W?S ^ ^ th !l span ’ The double links of 1 % inch sq iron, were’lO feet 
long. The center link was horizontal, and at the level of the floor; and at its ends were stirruned 

atrJrt H« t rL t I^ S |?” e r g ii: der - 3 - , Frora the endsof this central’link, the chains were clrried^ 

( aight lines to the tops of the single posts, ‘25 ft high, which served as piers or towers The hack* 

“• *»»« *»8le *• ins cables; and eaEh was Jd te 

K55 A" 5rS r -asaas were 

wWwV i an ' 1 a “ r h ® erv ?nt engineer friend, who in 1838 took the sketch and ineasnrements nnon 
description is based, informed the writer that the iron was as perfect, and as sharp on all 
iL-h d nUi’ as on the da y lt J! as built. The iron was the old fashioned charcoal of full 30 tons ner -n 
l‘ ch nlt'uiate strength. The united cross-section of the two double links was 7 56 so ins • which at 
30 tons per sq inch, gives 227 tons for their ultimate strength ; or say 76 tons with a safetv’of 3 Now 

of span; equal to .7 ton for a bridge wide enough for two vehicles to pass. This primitive 








SUSPENSION BRIDGES, 


625 


idg’e would therefore safely sustain a greater load per foot 
n of span, than the Frey burg’. 

hese old bridges frequently failed by the rotting of the end posts; or were carried away by fresh- 
but we have never heard of a failure from the breaking of the chains. Many of them were built 
i much more perfect style than the one just described ; and on the most used roads in the Union. 



He - • ■ “■ ■ -'i : : - - *»' • 

rt 1 f. . '■ ' > - 

■ 

**.. •> i • • . > si ■> 

1. tfSfttsfc- W 


Ml M».M I' '• if>« 1 < < *• ft *4 OH f • - 





626 


TEST BOEINGS. 


Pierce’s Well-borer, made by the Pierce Artesian and Oil Well Supply C< 
80 Beaver Street, New York, is an excellent tool for boring into soil 
clay, sand, or gravel, even when quite indurated, or when frozen. It 



Fig 2. 


dr 

I 


This consists of a hollow iron cylinder, about 5 ins dianv X 30 ins long, with a va 
at its foot, opening upward. It is lowered to the bottom of the hole; covered w 

U’fltpr tn u ilnntli a! O t/-v /I ft ___ .i ,_: .i.i . i i , . . 


water to a depth of 2 to 4 ft, and churned quickly up and down 4 to 6 ins, by li« 
20 or 30 times, during which the sand fills the pump, which is then drawn up ; 
emptied. From 10 to 20 ft in depth of sand, mud, &c, per hour can thus be tal 
from a 6 to 18-inch hole. This pump is also used for removing broken earth 
from a hole bored in compact earth by the Pierce borer first described. 

Tile cost of a Pierce auger, with derrick, boring-rods, rope, sand-pump, <fce. 
complete,is (1888) about $175. The auger weighs from 150 to 200 lbs accc 
ing to size. Boring-rod V/ 2 ins sq, 3^ lbs per ft. Derrick, 150 lbs. 

Sand-borer, ligs 3 and 4, like the sand-pump just describee 
used inside of tubing, and for the same purpose. The hollow iron cylinder C. 10 
ivm'- 1 ! ^ ins * 0Ufr ’ sI 'des vertically on the rod, but the screw is fast to the 
While boring, the sand below and around the cyl keeps it in the position show 


not bore through hard rock, or through large boulders 
consists of two sheet-iron cylindrical segments S S, call 
“ pods,” having their lower or cutting edges shod with ste 
These edges project (as show n in Fig 1) beyond the sides 
the auger, and thus make the hole larger than it, so that 
cannot bind or stick. The two cutting edges are equidista 
from the vert cen line of the tool, and this insures a straig 
and vert hole. At a the auger is attached to the lower e 
of a vert boring rod composed of a number of 1%-inch squa 
iron bars, or 2)/£-inch iron tubes, about 10 to 15 ft lor 
jointed together at their ends by means of square socki 
joints. At the top of this boring-rod is a swivel-hook, 
means of which the entire apparatus is hung to the end ot 
rope, which passes over a pulley at the top of a derrick 
tripod, and down to a drum worked by a windlass and ge; 
ing. By means of this drum and rope, the auger and boring-rod (which at first cc 
sists of only one bar) are lifted, and suspended over the intended hole. The auj: 
is then lowered, and rotated hor by two men or one horse, working at the ends 
levers which grip the boring-rod a few ft above the ground. The swivel at the t 
of the boring-rod permits this rotation to take place without twisting the ro' 
The shape of the auger is such that its rotation feeds or screws it into the groui 
and the man at the windlass has, during the boring, merely to keep the rope tig 
so as to prevent the auger from boring too fast, and becoming clogged. In abou 
revolutions the auger fills with earth. By means of the windlass it is then rai 
to about 2 ft above the ground; and by unkeying and removing the band b the an 
is opened like a pair of tongs, and the earth emptied into a wooden box which 
in the meantime been placed over the hole. The box is then removed and empt 
and the boring proceeds as before. When the boring has reached a depth of a 
10 ft, a second bar must be added to the top of the rod. For this purpose the 
and auger are raised a few inches; a slight frame-work of hoards is placed on t 
ground, close to the boring-rod and surrounding it; and a flange is clasped tigh 
to the rod just above, and close to, the framework. The framework and flange, n 
support the rod and auger; the swivel-hook and rope are removed, and attached 
the upper end of the second bar, which is then raised, and its lower end is fastei 
into the socket-joint upou the top of the first one. The rope is then drawn tigl 
the flange removed; the auger lowered t6 the bottom of the hole; and the hor 
resumed. Additional lengths ot boring-rod are attached in the same way from t!i 
to time, as required by the descent of the auger. 

The borers are made from 6 to 18 ins diain, or larger to special order. If desi 
the boring may be made from 24 to 36 ins diam by attaching a reamer to the au 
This auger will bore to a depth of 100 ft or more at the rate of from 5 to 20 ft 
hour. It removes stones as large as half the diam of the hole. In dry soils a hue 
f»l of water is poured into the hole each time the auger is raised. 

The Pierce well-borer may be advantageously used in boring the holes for sa 
piles, p 650, and at times, instead of driving wooden piles, it n 
be better to plant them (butt down if preferred) in holes bored by this auger* 
ming the earth well around them afterwards. This will save adjacent buildi 
from the jarring and injury done by a pile driver. 

If sand, mud, or loose gravel is reached in boring with this t< 
the hole is reamed out 4 ins larger, and a tubing of inch boards is inserted i 
the hole, and driven into and through the sand or gravel, w hich is then remo' 
front within the tubing by means of the sand-pump furnished with each maclii 


I; 

























ARTESIAN WELL BORING. 


627 



Fig 3. Six revolutions of the rod and screw fill the cyl with sand. The rod is then 
Illitte J. This first draws the screw up into the cyl, as in Fig 4; and a valve at tho 
i[foot of the screw closes the bottom of the cyl, and prevents the sand 
from falling out when the borer is lifted from the hole. The rod is 
hollow, and open at top and bottom. This allows passage of the air, 

- and thus prevents resistance from suction in withdrawing the borer. 

This tool is rotated and withdrawn in the same way as the earth borer 
1 first described. Price (1888), $32. 

The Pierce Co also furnish a steel prospecting auger, from 
l to 4 ins diam, and 2 ft long, for boring holes from 2% to 6 ins diam, 
ind to depths of 10 to 50 ft, into day, sand, or fine gravel, of 
ill of which it brings up samples. It is turned by wrenches, and by 
t nan or horse power, as is the well-borer; but requires no derrick, as 
Ht can be withdrawn by hand. Price (1888), of auger alone, about 
115 to $30. 

The boring tool shown in vert section by Fig: 6, 

i and in hor cross section by Fig 5. is very useful for boring slial- 
e ilow holes by liaml through surface soils, clay, and gravel, and 
: winging up samples. The borer proper consists of a cylinder of 
i. spring steel, 3 or 4 ins diam. and 4 or 5 ins high, with sides ^ 
i nch thick, having a vert slit (see cross section) throughout its 
i.ieigbt, and beveled to a cutting edge all around its foot, as 
shown in the vert section. At its top it is riveted, as show n, or 
i! Ivelded, to the inverted-u-shaped forging, which, by means of tho 
|t ocket at its top, is screwed to a length of gas-pipe which serves 
i is a handle, and to wjiich other pieces are joined by sockets as 
»oring proceeds. 

i The boring is done by two men, who grasp the handle, and, 

(iiolding the tool vert, drive it into the ground by repeatedly 
i! lifting it and forcibly bringing it dow n upon the same spot. As 
, he tool strikes the ground, the beveled shape of its cutting edge 
pauses it to open slightly, and when the downward pres is re- 
ieved in lifting it, it springs back and grasps the earth which 
!.-'iiHS entered it. It soon fills; and the men, finding that it ceases 
t | o penetrate readily, lift it to the surface aud empty it. The 
,ji haracter of its contents from different depths, measured along 
,j| he handle, is noted from time to time. 

( In six days of 8 hours each, three men (one 
eating at intervals) using one such auger 
>etween them, bored 20 holes,averaging 9% 
t each, m loam, gravel, clay, and decom- 
I iosed mica schist, at a cost of 22 cts per foot. 

Vages of each man, $2 per day. 

For work in loam, clay, or non-rnnning 
\ and, an effective screw-auger can 
>e made by any good blacksmith, by merely 
s (ironing a one-inch sq bar of iron or steel 
P ,nto corkscrew shape about 2 ft long, with 

complete turns 6 ins in diam ; its lower end sharpened to form a vertical cutting 
idge, which should project say .5 of an inch beyond the spiral of the screw, in order 

i n diminish friction. It will bring up full samples. Requires a derrick, or some 
tlier simple mode of lifting, when the screw is full. 

Artesian Well drilling. Deep vert holes in earth and rock, 6 and 8 ins 
a diam, such as are reqd for artesian vvells for water and oil, and for mining exphua- 
ions, are drilled by repeatedly lifting and dropping, in the same vert line, a heavy 
jron hit. Fig l,p 629, with a steel cutting-edge. The hit Is partly revolved horizon- 
, illy after each blow, to insure roundness of hole. The length of the cutting-edge 
f the bit is a little greater than the diam of the bit, and the hole is thus made suf- 
, ciently large to prevent the bit from binding in it. See also diamond drill, p 852. 

I The bit is the low'ost one of a series of iron and steel liars, Ac, Figs p 629, screwed 
igether at their ends, and called a “string of tools.” The string of tools 
aries in length from 25 to 60 ft, according to the size and depth of the hole, and the 
ardness of the rock; and its diam throughout (above the cutting-edge) is an inch 
v two less than that of the hole. Its weight is from 8(>0 to 4o00 lbs. Its upper 

I iember is always a “rope-socket,” Fig 4 (without a swivel), to which tho lower end 
Of the sup|>orting rope cable is attached. This cable pusses up out of the hole to 
i bur lever, which, by means of a horse-power or steam-engine, is kept eon- 
tantly moving up and down with a see-saw motion. The string <»f tools, with the 
utting edge of the bit at its lower end, is thus alternately lifted from 2 to 4 ft, aud 




Fig 5. 


Fig 6. 


44 


























628 


ARTESIAN WELL BORING. 


let fall, from 30 to 50 times per minute, and so drills its way into the rock or eartl 
From 4 to 10 ft in depth of water are kept in the hole, to facilitate the drilling an 
the removal of dehris. Alter water is reached, the drilling may be continued, evei 


if the hole is full of water; but a great depth of water of course diminishes the for 
of the blows of the bit. A suitable arrangement must be provided for paying 
out the rope as the boring tool descends. A clamp is attached to the cable 


and the man in charge, by turning the clamp, twists the rope, and thus turn? 


the bit horizontally about one-fifth of a revolution after each stroke, unti 
six or eight complete revolutions have been made in one direction, lie then re 
verses the motion, and makes an equal number of turns, at the same rate, in th 
opposite direction. 

After drilling a few feet, the string of tools is lifted out of the hole by means ol 
the cable, to allow the removal of the debris which has accumulated in tli 
hole. This is done by means of a saml-pump, which is a sheet-iron cylinde 
say 4 ins diam, and 4 to 6 ft long, provided, at its foot, with a valve opening upwar 
The pump is lowered to the bottom of the hole, and filled with the mixed water an 
debris by churning it up and down a number of times. Sometimes, in addition t 
the valve, the pump is fitted with a plunger, which is at the foot of the pump whe 
the latter is let down to the bottom of the hole. The plunger is then drawn up int 
the pump, and the debris follows it. In either case, the pump, when filled, is liftei 
out of the hole and emptied; the string of tools is again lowered into the hole,am 
the drilling resumed. The debris must be removed after every 3 to 5 ft of drilling 
Otherwise it would interfere too greatly with the action of the bit. 

Wells are usually drilled from <> to 8 ins diam. For diams les 
than 6 ins, the tools are so slender that they are liable to be broken in a deep hole 

The same apparatus is used for drilling' through the earth abov 
the rock, before the latter is reached. This is called “spudding.” In this c 
the sides of the hole must be prevented from caving in. For this purpose a wrougli 
iron pipe of such diam as to fit the hole closely, and inch thick, is inserted int 
the hole, and is driven down from time to time as the drilling proceeds. The pi 
is driven by means of a heavy maul of oak, or other hard wood, 14 to 18 ins squar 
and It) to 16 ft long. This maul is attached, by one end, to the lower end of tl 
same cable which, during drilling, supports the string of tools. It is thus repea 


tally lifted, and dropped upon the head of the tube, which is protected by a cast-in; 
‘•driving-cap.” The foot of the tube is shod with a steel cutting-edge ring .or “ste 


ro 

rap.” The foot of the tube is shod with a steel cutting-edge ring,or “ste 
shoe.” When the tube has been driven as far as it will readily go, the maul is i 
moved from the end of the rope; the string of tools substituted; and the drillin 
resumed within the pipe. 

The pipe is put together in lengths of from 8 to 18 ft, and the drilling and pip 
driving proceed alternately until the rock is reached, and the foot of the pipe fore* 
into it to a depth of a few ins, or far enough to shut off quicksand or surface watt * 

If quicksand is encountered, the string of tools is removed, and th 
sand-pump is used inside of the pipe. 

For reaming out, or enlarging', holes, or for straightcuin 

crooked ones, &c, special tools, such as reamers, <fec, are substituted in place of tl 
boring hit. 

Special care must be taken to have nil the rubbing surfaces thoi 
oughly lubricated. The pulley in the mast-head, and the pinion-whee 
ot the horse power (if such be used) should be well oiled every two or three horn 

In very cold or wet weather, a shed of rough hoards, or a cove 
ing of canvas, about 8 ft high, should be erected, to protect the men; and, if stea 


Pierce 

make 


is used,2 or 3 hoards should be used as a covering for the belt, which will slip if we 
^Tho following description is based upon the improved machines made by tl 

York, wl 
and hors , 
•nth,in ni 

dril 

ines, t 


F 


improved 

Artesian and Oil Well Supply Co., 80 Beaver Street, New Y 
a specialty of artesian well machinery, and of steam engines 
powers for its operation. They also sink Wells to any required dentil 
country 

For holes from 200 to lOOO ft deep, this Co furnish portable 
ing machines, to he worked by horse or steam power. In these nmchi 
drill-rope, extending from the string of tools up out of the bole, passes over a shea 
at tho top ol a wooden mast; down to, and around, a pulley fast to the workii 
lever; and thence, by way of a pulley fixed at the foot of the mast, to a drum up 
which it is wound. To this drum a friction and ratchet wheel is attached, for p: 
ing out the cable as the tools descend. 

The mast is hinged six feet above its foot, so that its upper part may 
laid hor when the machine is to 1#> moved. When at work, it is held in position 
two timber struts or braces, bolted to it near its top, and having their lower on 
fastened to the “drill-jack,” which is a light and strong framework. 9 ft lot 
8 ft wide, and 4 tt high, at- the foot of the mast, containing the working lever whi 


I 1 * 










ARTESIAN WELL BORING. 


629 


:fi 


uid 

and, 


Fig 2. 


Fig 1. 


Fig 4. 


isoe the rope and lets it fall, the drum on which the rope is wound, Ihe shaft 
f , 111 which work the lever, &e. The operator stands at the foot of the mast, 
means ot foot- and hand-levers within his reach, regulates 
the movements of the machine. One of these governs 
l’ e pawl and ratchet wheel regulating the paying out of the 
hie. By letting the ratchet-wheel of the drum move one 
! tell, the hit is fed down quarter of an inch. 
re r he operator, by moving a slide with his foot, holds the 
irking lever down, out of reach of the cam, thus stopping 
? up-and-down motion of the rope and tools. By means 
another lever he can now put the rope-drum in gear with 
3 main driving-shaft, so that the rope is wound up on the 
am, and the tools drawn up out of the hole. Another 
‘ t ® er controls the separate reel on which the light rope, car- 
ng the sand-pump, is wound. All these operations are 
" rformed by the same power (horse or steam, as the case 
y be), which works on without stopping; the various 
“ inges being made by merely throwing the different parts 
o, or out of, gear with the main driving-shaft. 

[>ne of these portable machines requires 
l " o horses or a small steam-engine, a man to attend the 
c : lie, and another man to operate the machine, empty the 
id-pump, change the tools, &c. It can be transported on 
irm wagon over any common road. Two men can unload 
set it up, and commence drilling, in two hours; and, un- 
i S-steam is preferred, the two horses used for its transpor- 
‘tion furnish the motive power. The machine can be taken 
vn and reloaded in the wagon in two hours. 

'’igs 1 to 4 show the tools us<m 1 wills those ma- 
lines. For the different sizes of machine they differ 
efly in their dimensions and weights. Fig 1 is the 
illing hit, called a “ Z ” bit from the shape of its cut- 
g-edge. This edge is 6 ins long. The bit is 80 to 36 ins 
g, and weighs about 100 lbs. Its top is screwed into the 
t. of the “auger-stem,” Fig 2, which is of 3-inch 
nd iron, 12 ft long, and weighs 350 lbs. Its use is that 
i weight, giving additional force to the blows of the bit. 
top is screwed into the foot of the 44 tlrill-fars,” Fig 
I* and to the top of these is screwed the 44 rope-socket,” 

4, to which the drilling cable is attached. If the hit, 
aug-er-stem, becomes wedged in the hole 
’any means, the operator stops the'churning motion 
the tools, and the rope is let out about 12 ins. This per¬ 
il s the upper link U of the drill-jars, Fig 3, to slide down 
t! iut 12 ins in the slot S in their lower link. The churn- 
motion is then started again, and the upward jerk of 
hi link U against the upper end of the slot loosens the 
ef Is. 

!« These machines are made in a number of sizes, to drill 
ft es from 200 to 1000 feet deep. The string of tools weighs 
i m 800 to 1800 lbs : and the machine complete including 
ft .Is, rope, mast, etc., but exclusive of power, from 1 00 to 
if >0 lbs. They cost from $70° to $1500 exclusive of power. 

(k e smaller sizes may be worked by horse power. A horse 
i8|«rer weighs about 800 lbs., and costs about $'75. Steam 
ifjine, 1GO0 to 3600 lbs., $150 to $300. 

For well* from lOOO to 3000 feet deep, a 
tionary machine, with a walking-beam, is used, similar 
those employed in the oil regions of Pennsylvania. A 
s^iare pyramidal derrick is erected, 74 feet high, 20 feet 
lare at base, 4 feet square at top. Each of its 4 corner legs 
of 2 inch X 8 inch and 2 inch X 10 inch planks, spiked 
;ether so as to form a 10 inch X 10 inch angle-piece, 2 
j;hes thick. The legs are braced together by horizontal 
a diagonal timbers. The walking-beam is of timber, 26 
t long, 12 inches wide, and 26 inches deep at the middle 
o its length, where it is pivoted to the top of a weoden post 
i inches square and 12 feet high, call si a “Samson post.” 
tils post, at Us foot, is dovetailed into the main sill of the machine, which is 18 
' .hes wide X 24 inches deep. 


Fig 3. 


















































































































630 


ARTESIAN WELL BORING. 


The motive power is a 15-hp steam-engine, which, by means of a belt and pulle 
crank and pitman, working at one end of the walking-beam, gives to the latter i 
see-saw motion. To the other end of the beam, and immediately over the well, 
suspended, by means of a hook, a “ temper-screw.” This last is composed of t\ 
bars of iron, about % X 2 ins, 5 ft long, hung 2 ins apart, fastened together at tin 
top ends, at which point there is an eye, which is suspended on the walking-bea 
hook. At the bottom of the two bars there is a sleeve-nut, and between the t\ 
bars and passing through the nut, is a screw 5 ft long, at the bottom of which th 
is a head, which carries a swivel, set-screw, and a pair of clamps. These grasp 
cable, 2 or 2% his diam, which carries at its lower end the string' of too 
This, for a 2000-ft hole, consists of a steel bit, 3 or 4 ft long, weighing 200 to 400 1 
an auger-stem of 4 or 5-inch round iron, from 24 to 30 ft long, and weighing fi 
1200 to 2100 lbs; steel-lined drill-jars 8 ft long, weighing 600 to700 lbs; a sinker 
of round iron of same diam as the auger-stem, 12 to 15 ft long, and weighing from 
to 1100 lbs ; and a rope-socket, '2% ft long, weighing 200 lbs. Total 'length of stri 
of tools, 50 to 00 ft, total weight, 3000 lbs ; or, for an 8-inch hole in the hardest roc 
4000 lbs. The sinlter-bar is added to give additional wt, and thus to assist 
pulling the cable down through the water, either in lowering the string of tools 
in working she drill-jars. The. shapes of the other tools are given by Figs 1 to 
Special tools are used for recovering articles that may be accidentally dropp 
into the hole. 

Tile drilling cable is wound on a drum, called a bull-wheel shaft, at t 
foot of, and inside of, the derrick. While drilling is going on, it passes from t 
bull-wheel shaft loosely over the sheave at the top of the derrick, and down to t 
clamps at the lower end of the temper-screw on the end of the walking-beam, 
the drilling progresses, the temper screw is turned or fed out by the man in charj 
who also, by means of a clamp, twists the rope, so a6 to change the position of t 
bit after each stroke. 

When the tools are to be lifted out of the hole, the cable is disengaged from t 
clamps on the temper-screw, and is wound upon the bull-wheel shaft, which, for t 
purpose, is thrown into gear with the steam-engine; the pitman being at the sai 
time removed from the. crank-pin, so that the walking-beam is at rest. As in t 
portable machines, the saud-pump is also raised by the same power which does 
drilling. 

About lOOOO ft b m of rough lumber are reqd for the derrick, w 
ing-beam, sills. Ac, and about 3000 ft more for sheds over the boiler, engine,and 

In ordinary hard limestone rock, such a machine will drill about \\ 
per hour under the most favorable circumstances. Two men are require* 
one to attend to the boiler, sharpen the bits, Ac, and one to operate the machi 
The cost of the apparatus, in 1888, is from $1800 to $2500. I.umb 
for the derrick. $350 to $450 more. The cost of drilling, in 1888, in lit 
stone, is about $6 per ft ruu for 6-iuch holes; $8 for 8-incli. In granite and tt 
rock, $12. 

Tor quantity of masonry in walls of wells, see p 158. 







COST OF DREDGING, 


631 


COST OF DREDGING. 

Dredging is generally done by skilled contractors, who own the requisite machines, 
ows or lighters, &c ; and who make it a specialty. It is necessary to specify whether 
le dredged material is to be measured in place before it is loosened: or after being 
^posited in the scow: because it occupies more bulk after being dredged. It was 
Mind, in the extensive dredgings for deepening the River St Lawrence through the 
ake of St Peter, that on an average a cub yd of tolerably stiff mud in place, makes 
4 yds in the scow; or 1 in the scow, makes .715 in place. Also stipulate whether the 
‘inoval of bowlders, sunken trees, &c, is to constitute an extra. These often require 
wing and blasting under water. The cost per cub yd for dredging varies much 
ith the depth of water: the quantity and character of the material: thedist to which 
has to be removed; whether it can be at once discharged from the machine by 
leans of projecting side-shoots or slides; oT must be discharged into scows, to be re- 
loved to a short dist by poling, or to a greater dist by steam tugs; w hether it can be 
ropped, or dumped into deep water by means of flap or trap doors in the bottom of 
ic hoppers of the scows ; or must be shovelled from the scow s into shallow water, (at 
iy 4 to S cts per yd ;) or upon land, (at say from 6 to 10 or 20 cts for the shovelling 
one, or shovelling and wheeling, as the case may be;) whether much time must*be 
msumed in moving the machine forward frequently, as when the excavation is 
arrow, and of but little depth; as in deepening a canal, &c; whether many bowl¬ 
ers and sunken trees are to be lifted ; whether interruptions may occur from waves 
i storms; whether fuel can be readily obtained, &c, &c. These considerations may 
lake the cost per cub yd in one case from 2 to 4 times as great as in another. The 
ztual cost of deepening a ship-channel through Lake St Peter, to 18 ft, from its orig- 
ial depth of 11 ft, for several miles through moderately stiff mud, was 14 cts per 
ib yd in place, or 10 cts in the scows; including removing the material by steam 
igs to a dist of about a mile, and dropping it into deep water. This includes re- 
airs of plant of all kinds, but no profit. It was a favorable case. When the buckets 
ork in deep water they do not become so w r ell filled as wdien the water is shallower, 
ecanse they have a more vertical movement, and, therefore, do not scrape along as 
reat a distance of the bottom. Hence one reason why deep dredging costs more 
er yard; in addition to having to be lifted through a greater height. Perhaps the 
Allowing table is tolerably approximate for large works in ordinary mud, sand, or 
ravel; assuming the plant to have been paid for by the company ; and that common 
ibor costs $1 per day. 







632 


COoT OF DREDGING. 


Table of actual cost of (Ircd^iii^ on a large scale; includ¬ 
ing- dropping the material into scows, alongside: or into 
side-slioots, on board. Common labor 81 per day. Repairs 
of plant are included ; but no protit to contractor. (Original.) 


Depth 
in Ft. 

Cts per Yard, 
iu place. 

Cts per Yard, 
ia scow. 

Less than 10 

8.4 

6 

10 to 15 

9.8 

7 

15 to 20 

11.2 

8 

20 to 25 

14.0 

11 ) 


Depth 
in Ft. 


Cts per Yard, 
in place. 


I Cts per Yard, 
iu scow. 


25 to 30 
30 to 35 
35 to to 


18.2 

25.2 

35.0 


13 

18 

25 


For towing of the scows by steam tugs to adist of 34 mile, and dropping the mud into deep water.^38 
4 cts per yard iu the scow ; for 34 mile, 6 cts; for % mile, 8 cts ; for 1 mile, 10 cts Add profit to con- 


* 


ill I! 

sti 


tractor. On a small scale work is done to a less advantage; and a corresponding increase must be made 
in these prices. Also, if the contractor himself furnishes the dredgers and plant, a still further addi¬ 
tion must be made. It is evident that the subject admits of no great precision. Small jobs, even in 
favorable material, but iu inconvenient positions, may readily cost two or three times as much per yd 
as the above: and in very hard material, as in cemented gravel aud clay, four or five times as much 
for the dredging. The cost of towing, however, will remain as before, if wages are the same. 

The cost of dredgers, tugs, &c, will vary of course with their capabilities, strength of construction, 
style of finish, whether having accommodations for the men to live on board or uot. &c. When for 
use iu salt water, the bottoms of both dredgers and scows should be coppered, to protect them from sea- 
worms; and if occasionally exposed to high waves, both should be extra strong. The. most powerful 
machines on the St I.awrence cost about $15000 each ; and removed in 10 working hours on an average 
about 1800 cub yds in place, or 2520 in the scows. Good machiues, capable, under similar circumstances, 
of doing as much, may, however, be built for about $25000 to $30000. To remove this quantity to a 
dist of >4 to 1 mile, would require two steam tugs, costing about $0000 to $10000 each ; aud 4 to (i scows, 
(some to be loading while others are away,) holding from 30 to 60 cub yds each ; aud costing from $800 
to $1500 each at the shop. Scows with two hoppers are best. Such a dredger would require at least j 
8 or 10 men, iucluding captain, engineer, fireman, and cook. Each tug 4 or 5 men ; and each scow 2 
men. The engineer should be a blacksmith ; or a blacksmith should be added. In certain cases a 
physician, clerk, assistant engineer, &c, may be needed. 

Dredgers are often built on the principle of the Yankee Excavator, with but a single bucket or dip-til 
per, of from 1 to 2 cub yds capacity. Hull about 25 by 60 ft Draft 3 ft. Cylinder about 7 or 8 insj|Y ] 
diam; 15 to 18 inch stroke; ordinary working pressure nib to 80 Bis per sq inch, according to hardness 
of material. Cost $8000 to $12000. Will raise as an average days' work ^10 hours) from 200 to 500 
yds iu place, or 280 to 700 in the scow, according to the depth, nature of the material. <tc. Require 
5 or 7 men in all aboard, including cook. Burn 34 to 1 ton of coal daily. Tolerably large bowlders, 
and sunken logs, can be raised by the dipper.* -i 

When the material is hard and compacted, the buckets of dredgers should be armed with strong ■. 
steel teeth projecting from their cutting edge. On arriving at such material, every alternate bucket 
is sometimes unshipped. By arranging the bnckets so as to dredge a few feet in advance of the bull. h 
low tongues of dry laud may be cut away ; the machine thus digging its own channel. The daily U! 
work iu such cases will uot average half as much as in wet soil. 

. On small operations, dredgers worked by Iwo or more horses. 

instead of by steam, will answer very well in soft material; or.even in moderately hard, by reducing 
the size and number of the buckets. A two-horse machine will raise from 50 to 100 yards of ordiuary 
mud iu place, or 70 to 140 in the scow, per day, at from 12 to 15 ft depth. 

Soft material in small quantity, ami at moderate depth, may be removed by the 
slow and expensive mode of the bag-scoop, or bag-spoou. 

This is simply a bag B, made of canvas or leather, and having iti ptl 1 
mouth surrounded by an oval iron ring, the lower part of which is l, 
sharpened to form a cutting edge. It has a fixed handled. nnd’iU 
swivel handle i. One man pushes the bag down into the mud by A, 
w hile another pulls it along by the rope g; and when filled, another lt i 
raises it by the rope c, and empties it. If the bag is large, a wtud- B( 
lass may be used f ir raising it. The men may work from a scow ing¬ 
raft property anchored. Or a long-handled metal spoon, shaped !ike 
a deeply-dished hoe. mav be used by only one man ; or a larger spur o j, 
may be guided by a man, and dragged forward and backward by a 
horse walking in a circle on the scow, &c, &c. 



The weigh! of a, cub y(! of wet dredged mud, pure sand, or gravel, averages T , 
about 1% tons; say 111 lbs per cub ft: muddy gravel, full \]/ 2 tons; say 125 tbs per i| r 
culi ft. Pure sand or gravel dredges easily: also beds of shells. Wet dredged clay p, 
will slide down a shoot inclined at from 5 to 1, to 3 to 1, according to its freedom 
from sand, Ac; but wet sand or gravel will not slide down even 3 to 1, without a free ][ 
flow of water to aid it: Otherwise it requires much pushing. 


* The writer has seen cases iu which a circular saw for logs in deep water, would have been 
very useful addition to a dredger. It should be worked by steam; and be adjustable to differet 
depths. It wool i cost but about $560. 

The American Dredging Co, No. 234 Wgluut St, Philada, make dredgers 

many patterns ; aud contract for dredging on any scale. 






































FOUNDATIONS. 


63a 


FOUNDATIONS. 

4. volume might be occupied by this important subject alone. We have space for 
, ly a few general hints ; leaving it to the student to determine how far they may 
' applicable in any given case. In ordinary cases, as in culverts, retaining walls, 
, if excavations, or wells, <tc, in tike vicinity, have not already proved that the soil 
reliable to a considerable depth, it will usually be a sufficient precaution, after 
ving dug and levelled off the foundation pits or trenches to a depth of 3 to 5 ft, to 
t it by an iron rod, or a pump-auger; or to sink holes, in a few spots, to the depth 
4 to 8 ft farther; (depending upon the weight of the intended structure;) to ascer- 
n if the soil continues firm to that distance. If it does, there will rarely be any 
k in proceeding at once with the masonry; because a stratum of firm soil, from 4 
8 ft thick, will be safe for almost any ordinary structure; even though it should 
underlaid by a much softer stratum. If, however, the firm upper stratum is ex- 
sed to running water, as in the case of a bridge-pier in a river, care must be taken 
! preserve it from gradually washing away; or from becoming loosened and broken 
by violent freshets ; especially if they bring down heavy masses of ice, trees, and 
ter floating matter. These are sometimes arrested by piers, and accumulate so as 
form dams extending to the bottom of the stream ; thus creating an increase of 
oeity, and of scouring action, that is very dangerous to the stability both of the 
ttom and of the structure. When the testing has to be made to a considerable 
)th, it may be necessary to drive down a tube-of either wrought or cast iron, to 
went the soil from falling into the unfinished hole. If necessary, this tube may 
in short lengths, connected by screw joints, for convenience of driving; and the 
th inside of it may be removed by a small scoop with a long handle.f 
Boring'S in common soils or clay may be made 100 ft deep in a day or two by a 
imon wood auger 114 ins diam, turned by two to four men with 3 ft levers. This will bring up 
iples. For this and other earth-boring tools see p 626. 

n starting the masonry, the largest stones should of course be placed at the bot- 
n of the pit so as to equalize the pressure as much as possible; and care should 
taken to bed them solidly in the soil, so as to have no rocking tendency. The 
ct few courses at least should be of large stones, so laid as to break joint thoroughly 
h those below. The trenches should be refilled with earth as soon as the masonry 
1 permit; so as to exclude rain, which would injure the mortar, and soften the 
ndation. It is well to ram or tread the earth to some extent as it is being deposited, 
f the tests show that the soil (not exposed to running water) is too soft to support 
. masonry then the pits should he made considerably wider and deeper; and after- 
rd be filled to their entire width, and to a depth of from. 3 to G or more It, (de¬ 
fine, nn the weight to he sustained,) with rammed or rolled layers of sand, gravel, 
stone broken to turnpike size; or with concrete in which there is a good propor- 
n of cement On this deposit the masonry may be started. The common practice 
mch eases, of laying planks or wooden platforms in the foundations, for building 


inhierranenn caverns in limestone regions are a frequent source of trouble, against which it in 
uIt to adopt precautions. 












634 


FOUNDATIONS. 


upon, is a very bad one. For if the planks are not constantly kept thoroughly wet. 
they will decay in a few years; causing cracks and settlements in the masonry. 

Some portions of the circular brick aqueduct for supplying Boston with water 
gave a great deal of trouble where its trenches passed through running quicksands 
and other treacherous soils. Concrete was tried, but the wet quicksand mixed itself 
■\\ith it, and killed it. Wooden cradles, 4c, also failed; and the difficulty was finally 
overcome by simply depositing in the trenches about two feet in depth of strong 
gravel.* Sand or gravel, when prevented from spreading sideways, forms one of tin 
best of foundations. To prevent this spreading, the area to be built on may be sur 
rounded by a wall; or by squared piles driven so close as to touch each other; or ii I 
less important cases, by short sheet piles only. But generally it is sufficient simph ' 
to give the trenches a good width; and to ram the sand or gravel (which are all tin 
better if w-et) in layers; taking care to compact it well agaiDSt the sides of the trend 
also. Under heavy loads, some settlement will of course take place, as is the cast 
in all foundations except rock. If very heavy, adopt piling, Ac. See Grillage, p 641. 


When nn unreliable soil overlies a firm one, but at such a depth 

that the excavation of the trencher (which then must evidently be made wider, as well as deeper 
becomes too troublesome, and expensive; especially when (as generally happens in that case) water 
percolates rapidly into the trenches from the adjacent strata, we may resort to piles. See p 641 

When making deep foundation pits in damp clay, we must remember that 

this material, being soft, has, to a certain degree, a tendency to press in everv direction, like water 
1 lira causes it U. bulge inward at the sides; and upward at the bottom The excavations for tuuuels 
or lor vertical shafts, often close in all around, and become much contracted thereby before they c-ir 
be lined; therefore they should be dug larger than would otherwise be necessary. The botton'i.s of 
canal and railroad excavations in inoistclay are frequently pressed upward by the weightof the sides 
Wr y c,s *y rapidly absorbs moisture from the air, and swells, producing effect 

similar to the foregoing. Its expansion is attended by great pressure : so that retaiu.ng-walls hacked 
with dry rammed clay will be in dunger of bulging if the clay should become wet. It is a treacherous 

material to work in. For concrete foundations, see ps 680 &c. 

.• *? th< i greatest load that may safely be trusted on an earth founda- 

„r" k ! ne a, l vlscs , «><>t to exceed 1 to 1.5 tons per sq ft. But experience proves that on good com- 
pact gravel, sand, or loam, at a depth beyond atmospheric influences, 2 to 3 tons are safe, or even 4 
to b tons if a few ins of settlement may be allowed, as is often the case iu isolated structures without 
tremors, tears may elapse before this settlement ceases eutirelv. Pure clay, especially ir damn is 
more compressible, and should not be trusted with more than 1 to'2.5 tons, according to the case. All 

earth foundations must yield somewhat. Equality of pressure is a main 
point to aim at. Tremor increases settlements, and causes them to continue 

even iu . weak soils, great care must be taken not to overload in such cases, 

even if piled. Foundations in silty soils will probably settle, in years, at the rate of from 3 to 
12 in* per ton (up to 2 tons) per sq ft of quiet load, ir not on piles. * 

1 1 g _ shows an easy mode of obtaining a foundation in certain cases It is tht 
stollc V^orV'ip' ra ” (1 ° St St ° ae) ° f the French 5 iu English, “random 

It is merely a deposit of rough angular quarry stone thrown into the water; the largest ones beinc 
at the outside, to resist disturbance from freshets, ice, floating trees, Ac. A part of tlfe inteHor may 
be of small quarry chips, with some gravel, sand, clay, Ac. When the bottom is irregular rock tbit 
process saves the expense of levelling it off to receive the masonry. For 2 or 3 feet below the surface 
of the water, the stones may generally be disposed by hand so as to lie close and firmly Small sr awh 
packed between the larger ones will make the work smoother, and less liable to be displaced bv violence 
Ci ‘i»- S " lay at tlm £* be useful for connecting several of the large stones together for great ei 

stability. Kip-rap, however, is apt to settle. 

f the bottom Is so yielding as to he liable to wash away in 

freshets, it may, m addition, be protected, as in Fig 2, by a covering of the same kind 

of stones, as at c: extend¬ 
ing all around the struc¬ 
ture. Or the main pile 
of stones may be extend¬ 
ed as per dotted line at d; 
so that if the bottom 
should wash away, as pei 
dotted line at o, the 
stones d will fall inte 
the cavity, and thus pre¬ 
vent further damage 

driven as an additional precaution. For greater security, the'bed^the river maJ 

L , d e W, or 8C ?°P e . d » nder the entire space to be covered by the main deposit^ 
per dotted lines in Fig 3, to as great a depth tis any scouring would he apt to reach 

* Smeaton mentions a stone bridge built upon a natural bed of gravel onlv^W 9 f. 




























FOUNDATIONS, 


635 


this excavation also to he filled with stone. Such foundations are evidently best 
adapted to quiet water. The masonry should rest on a strong plattorm. 




Large deposits of stone, as in these 
two figs, greatlv increase the velocity, 
tnii the scouring action of the stream 
iround them, especially in freshets ; un¬ 
less the bottom on each side from the de¬ 
posit he dredged out to such an extent 
that the original area of w r ater shall not 
he reduced. If the bottom is treacherous, 
this should be done before depositing the 
covering stones c, Fig 2. Judgment and 
experience are necessary in such matters, 
us in all others connected with engineer¬ 
ing. Mere study will not guard against 
constant failures. Theory and practice 
must guide each other. 

Fig 3 is another simple method; and 

or to tb, e.cop. of U. ».W l» time of Mgh fr..b.t. I. . 
verv effective one Here the piles are first driven into the river bottom for the support of the pier 
then the deposit of stone is thrown in. for the support and protection of the piles; preventing them 
from bending under their loads; and shielding them from blows from floating bodies. The tops of 
the niles beine cut off to a level, a strong platform of timber is laid on top of them, as a base for the 
masonrv The top of the platform should not be less than about 12 or 18 ins below ordinary low 
water, to prevent decay. Mitchell's iron screw pile; or hollow piles of cast iron, may be used instead 
of wooden ones. See pp 641 to 646. 

Fi^s 4 represent a convenient method of establishing a foundation in water by 
means of a timber crib, A A, without a bottom. It should be built of 
squared timbers, notch¬ 


ed together at their 
crossings, as shown at 
Fig5; each notch being 
of the depth of the 
stick. By this means 
each timber is support¬ 
ed throughout its entire 
length by the one below 
it; and resists pulling 
in both directions. Bolts 
also are driven at the 
intersections; at least 
in the sides of the crib, 
to prevent one portion 
from being floated off 
from the other. The 
crib is thus divided into 
square or rectangular 
cells, from 2 to 4 or 5 ft 
on a side, according to 
the requirements of the 
case. The partitions 
between the cells are 
put together in the 
same manner as those 
at the sides of the cribs; 
and consequently, like 
the latter, form solid 
wooden walls. 


.. . , , ot onv convenient spot; and when finished, may be towed to its 

The crib may be framed ^ = nosiiion ‘and then sunk by throwiug stone into a few 

final place, where it is carefully n I Q ’ se These platforms should be placed a little 

cells provided with platforms, as at c c, f P P , ., { 1 settling slightly into the soil, and 

above the lower edge of the cells, so as * P rev «“i p lh 1 ® c ™ f he /ells are filled with 

thus coming to a full bearing upon the bottom £ f teyt ba^been n ” ybe;also, a protection, t f, of 
rough stone. A stoiut top pla.tf° r m “‘’‘y t ’ ”jf t p e s j,j C s are exposed to abrasion Pom 

random stone, to Prevent unde™,nngby.thecurrent. ^ the Rngle3 length- 

ice, Ac, they may be covered m wlij • foundation may be made partly of random stone, as m 
FiKndT and on top of thfs may be sunk a crib, with its top about 2 tt under low water, as a base 

for'the masonrv. This is much safer than random stone alone. 

A rock bottom it may be necessary to scribe the bottom of the 

On uneven rOCM. «, lln k hv means of a loaded platform on its top, or by 

crib to tit the rock; or thecrib timbers are Within a short distance above the bottom. Being 

filling some of its oelU, unUl i' l » may be thrown into the cells, and allowed to find 

there kept in a horizontal position, “ rm , ; a leve i sup port for it. Tlie cells may then b* 

flUed ra^dTi^ap deposited outside around the crib to prevent the small stones from being displaced. 





































































































636 


FOUNDATIONS. 


A crib with only an ontsirte row of colls for sinking it may be 

built; and the interior chamber ma}' be filled, with concrete underwater. The masonry may then 
rest on the concrete alone. If the crib rests upon a foundation of broken stone, the upper interstices 
of this stone should first be levelled off by small stone or coarse gravel to receive the concrete of the 
inner chamber. 

Or crib like Fig". 4 may be sunk, and piles be driven in the cells, which 

may afterward be filled with broken scone or concrete. The masonry may then rest on the piles only, 
which in turn will be defended by the crib. If the bottom is liable to scour, place sheet-piles or 
rip-rap arouud the base of the crib. 

Ky all means avoid a crib like e, Fig 514, much higher at one part 

than at another, if the superstructure s is to rest on the timber of the crib instead of on 
piles, or on concrete independent of the timber; for the high part of the crib will compress more uuder 
its load than the low part, and will thus cause the superstructure to lean or to crack. 

A crib either straight sided or circular, with only an outer row of cells for |>ml> 
dling may l»e used as a cofferdam (see cofferdams, p 637). The.joints 

betweeu the outer timbers should be well caulked; and care be taken, by means of outside pile-planks 
gravel, &c, to prevent water from eutering beneath it. 


The cast-iron Bridge across the Schuylkill at Chestnut St, 

Pliala, Mr. Strickland Kneass, Engineer, affords a striking example of crib 
foundation. The center pier stands on a crib, an oblong octagon in plan ; 81 by 87 feet at base • 2t 
by 80 ft at top; and (with its platform) 29 ft high. Its timbers are of yellow pine, hewn 12 ins 
square : and framed as at Fig 5. The lower timbers were carefully cut or scrihed to conform to the 
irregularities of the tolerably level rock upon which it rests. These were ascertained (after the 8 ft 
depth of gravel bad been dredged off) in the usual manner of mooring above the site a large floating 
wooden platform, composed of timbers corresponding in position with all those of the lower course 
of the intended crib, both lougitudinal and transverse. Soundings were then takeu close together 
aloug all these lines of timber. Most of the cells are about 3 by 4 ft on a side, in the clear. A fen 
of them had platforms at the level of the second course from the bottom, for receiving stone for sink¬ 
ing the crib; the others are open to the bottom. 

The crib was built in the water; and was kept floating, during its construction, with its unfinished 
top continually just above water, by gradually loading it with more stone as new timbers were added 
The stoue required for this purpose alone was 300 tons. When the crib was towed into position and 
moored, 150 tons more were added for sinking it. All the cells were afterward filled with rough drv 
stone, and coarse gravel screenings; making a total of 1060 tons. A platform of It bv 12 inch squared 
timber covered the whole; its top beiug 2($ ft below low water. The pier alone, which stands on this 
crib, weighs 3255 tons; and during its construction it compressed the crib ins The weight of 
superstructure resting on the pier, may be roughly taken at 1000 tons more. 


iu icm < • |m* ii it louuu.'iuoti previously 

merely levelling off the natural surface, &c. 


All ordinary caisson is merely a strong scow, or a box with¬ 
out a lid ; and with sides which may at pleasure be readilv detached from its bottom. It is built on 
land, and then launched. The masonry may first be built in it, either in whole or iu part, while 
afloat; and the whole being then towed into place, and moored, mav be sunk to the bottom of the 
river, to rest upon a foundation previously prepared for it, either by piling, if necessary; or b\ 

The bottom of the caisson constitutes a strong tinibe’ 
platform, upon which the masoury rests; auc 
is so arranged, that after it is sunk, the sides 
may be detached from it, and removed to be 
rebottomed for use at another pier, if needed. 
This detaching may Vie effected bv some su 
contrivance as that shown in Fig 6, wht 
P P w is the bottom of the caisson, to whi 
are firmly attached at intervals strong ii UI 
eyes t; which are taken hold or by hooks d, a 
the lower end of long bolts E n,’ reaching ti 
the top timbers S t>f the crib, where they art 
confined by screw nuts n. By loosening the 
nuts v, the hooks d can be detached front the 
eyes t; and the sides cau then he removed 
fi oni the bottom; there Vicing do other connec 
tion between the two. These hooks aud eve* 
are usually placed outside of the caisson : Uit 
screw nuts n being sustained hv the projecting 
ends of cross pieces, as 11, Fig 9. ^ he ini 

proper position given them in our Fig wat 
merely for convenience or illustrating theprin 

one side detachable from the others, in order to float ThfcaJson‘aw^^r^ 
unle,s it be floated away before the masonry has been built so high ns to render the prec-iution^ise 
less, h tg 6 shows one ot many ways of constructing a caisson) with sides consisting of unwjh, 
coi net-posts, I; cap pieces S, on top ; and. sills g at bottom, resting on the bottom platform P P«^ 
intermediate uprights T, framed into the cans and siUs; the whole heing ^ P f 

two thtekuesses of planking H, which, as well as the platform, should'he well calked to^reven' 
leakiug. Tarpaulin also may be nailed outside to assist in this The greatest trouble trem 

It 8 ' e . i he , pla * fo . nu - 0n l "P of the platform is firmly spiked a timber o o extend m 

all around it just inside of the inner lower edee of thf* nf thn mnee r* . ’ extend.nj 

the sides from being forced inward by thepfessufeofthewater „i %" S V* t0 prev ? n 

will of course vary with the requirements of the case In deep caissons inside I;” 1 s . of coimructl ° l 

AS the rnU n ™ tl L e vessel gradually sinks while the masonry is befog built wH in if 



























FOUNDATIONS. 


637 


>)ete the interior after sinking the caisson. Indeed, masonry or brickwork, in cement, may thus be 
milt hollow at first, resting on the platform; the masonry itself forming the sides of the caisson. 
)r the sides may consist of a water-tight casing of iron, or wood, of the shape of the intended pier, 
ic. This casing being confined to the platform, becomes, in fact, a mould, in which the pier may be 
ormed, and sunk at the same time by filling it with hydraulic concrete. For 
concrete foundations, see p 680 &c. 




i 

it 

s 

■i 

it 

I! 


On rock bottom the under timbers of the platform may be cut to suit the irregularities 
is already stated under “ Cribs.” Or the bottom may be levelled up by first depositing large stones 
iround the area upon which the caisson is to rest; and then filling between these with smaller stones 
ind gravel; testing the depth by soundiug. Or a level bed of cement concrete may, with care, be 
leposited in the water. If there are deep narrow crevices in the rock, through which the concrete 
nay escape, they may be first covered with tarpaulin. Diving bells may often be used to advantage, 
u all such operations. But in the case of very irregular rock, it will often be better to resort to cof- 
dr-dams. The draft of a caisson (the depth of water which it draws) whether empty or loaded, can 
>e found by page 236. Valves for the admission of water for sinking the caisson are 

isuaily introduced. If, after sinking, it should be necessary to again raise the whole, it is only 
lecessary to close the valves, and pump out the water. Guide piles may be driven and braced along- 
fide of the caisson, to insure its sinking vertically, and at the proper spot. Or it may be lowered by 
;crews supported by strong temporary framework. 

Assuming the uprights I, T, &c. Fig 6, to be sufficiently braced, as at ee, Fig 7, the following table 
urill show the thickness of planking necessary for different distances apart of the uprights, (in the 
dear,) to insure a safety of six against the pressure of the water at different depths; and at the 
lame time not to bend inward under said pressure, more than part of the distance to which 

hey stretch from upright to upright; or at the rate of *4 inch in 10 ft stretch; inch in 5 ft, &c. 
such a table may be of use in other matters. 


: Fable of thickness of white pine plank required not to bend 
* more, than part of its clear horizontal stretch, under 
different heads of water. (Original.) 


■ 1 


HEADS IN FEET. 


Stretch 






iu Ft. 


40 

30 

20 

10 

5 




Thickness in Inches. 


3 


3^4 

3 

2% 

234 


4 


4 % 

4 

334 

2?4 

*34 

6 


6% 

6 

5 34 

4M 

334 

6 


9 

8 

7 

5/4 

134 

10 


UH 

10 

8 % 

7 

5 >4 

12 


13 J4 

12*4 

mi 

854 

6% 

15 


16% 

15 

13 

1034 

»34 

20 


**34 

20 

17/4 

14 

11 


t'oder-dams are enclosures from which the water may be pumped out, so as 
o allow the work to be done in the open air. Their construction of course vanes 
neatly. In still shallow water, a mere well-built bank of clay and gravel; or of 
’ags nartlv filled with those materials when there is much current, will answer 
jvery purpose; or (depending on Ihedepth) a single ordouble row of sheet-piles; or of 
if mured piles of larger dimensions, driven touching each other; their lower ends a 
evv feet in the soil; and their upper ones a little above high water, and protected 
>utside by heaps of gravelly soil or puddle, (as at P in Fig J,) to prevent leaking, 
[’he sheet-piles may be of wood; or of cast iron, of a strong form. 

The sufficiency of a more bank of well-packed earth in still 
vater is shown by the embankments or levees, thrown up m all countries, to pre¬ 
sent rivers from overflowing adjacent low lands. The general average ot the levees 
don" 700 miles of the Mississippi, is about 6 f t high; only o It wide on top, side- 
ilones 1 Vcy to 1. In floods the river rises to within a foot or less of then tops, and 
'reouefitly hursts through them,doing immense damage. They are entirely toashght. 

The method of a single row of 12 by 12 inch squared piles, driven in contact with 
>ach other, (close piles, ) and simply hacked by an outer deposit of lmpervioiis soil, 
s very effective; and with the addition of interior cross-braces or struts, like cc,.I i 0 
r t0 p re vent crushing inward by the outside pressure of the water and puddle when 
iSSKS. l.«™ successfully employed in from 21.to 25 ft dep h of m 

v 1 1 i(1 1 there was not sufficient -Current to wash away the puddle. 1 he cross mutes 
in- inserted successively, as the water is being pumped out; beginning, of course, 
.vith the upper ones. The ends of these braces may abut oif longitudinal timbers, 
silted to the piles for the purpose. Another method is ft strong co ™“ 
uised of uprights framed into caps and sills; and covered outside with squared 
imhers or plank laid touching each other, and well calked ; as in the caisson, I ig 
i'-butwi thout a bottom^Between the opposite pairs of uprights are strong interior 
itrute, i c“ Hg 7,Caching from side to ride, to prevent crushing inward. The 


















638 


FOUNDATIONS. 


upper series of these usually supports a platform for the workmen, windlasses, Ac. 
Ihe crib having been built on land, is launched, taken to its final place, and sunk by 
piling stones on a temporary platform resting on the cross-struts ; the bottom of the 
stream having been previously levelled off, if necessary, for its reception. 

To prevent leaking under the bottom of the crib, sheet-piles may be driven around it, their beads 
extending a few feet above its bottom; or a small deposit of outside puddle may be placed around it, 
as shown at the stone deposits 11, Fig t. Or a broad flap of tarpaulin may be closely nailed arouud 
and a little above the lower edge or the crib; so arranged that it may be spread out loosely ou the 
river bottom, to a width of a few feet all arouud the outside of the crib; and the puddle may be placed 
upon it. Such a tarpaulin is also very useful in case the river bottom is somewhat irregular, and 
cannot be levelled off without too great expense; in which case the crib cannot come to a full beariug 
upon it; and consequently the water would leak or flow beneath freely. It is especially adapted to 
uneven rock ; where sheet-piles cannot be driven. An artificial stratum of impervious soil may, how- 
ever. be deposited on bare rock ; in which case the siuking of the crib, and the subsequent operation! 
will be the same as on a natural stratum. These expedients are evidently more or less applicable in 
other cases, where, to avoid repetition, they are not specially mentioned. 




iS C J ril * coffer-dam : in which the sides, instead of being 

? 1 drivpn 1 AftS S? 1, ir 1 -» y i“ ni 1 the ]f t ins,anoe ' » re sheathed with vertical sheet-piles 
i, : , u y en after the crib is sunk. It is much inferior to the last, owing to its greater 

twi lty t0 - 6ak , f , In °.” e of th ‘ s description, Fig 7, successfully used in 16 ftwater 
l a ( f ?T mens ir S oft ie crl ) dd ft by 80 ft. Along each long side were 7 uprights l t 

Sd h&J UH ? “MX*'* r£r of these ^flnotched, 

.and held by dog-irons, 6 cross-braces c c, of 12 ins square. The distance between the 

two ,” pper ones was ° ft in th® c,ear 1 gradually diminishing to 18 ins between the 

the mSe Vh" aCC< ? l V\ t of tl ! e lncre «wed pressure of the water in descending On 
the outside of the uprights, and opposite the ends of the braces, were bolted longi- 


SECTION. 


PLAN. 




tudinal timbers to support the outside pressure against the 3-inch sheet-piling ss 
0t ^ er I° nS A tn(Iina l P l . eces 0 con fine the heads of tiie sheet-piles to the ton of the" 
crib after they are driven. The feet of the sheet piles were cut to an angle as at m • 
to make them draw close to each other at bottom in driving. S ’ 

The sheet-piles will drive in a far more regularand satisfactorv manner, with the 
arrangement shown in Figs 8. Here o o are the uprights; c c are pairs of longitudinal 
































































FOUNDATIONS. 


6o9 


pieces, notched and bolted to the uprights, near both their tops and their feet; and 
at as many intermediate points as may be desired. The sheet-piles I, are inserted 
between these; and of course are guided during their descent much more perfectly 
than in Fig 7. The crib at top of p 636 may be used as a cofferdam. 

When the current is too strong to permit the use of outside puddle, P, Fig 7, tha 
principle of coffer-dam shown in Fig 9, is generally used; in which both sides of the 
puddle are protected from washing away. The space to be enclosed by the dam is sur¬ 
rounded by two rows of firmly-driven main piles p p, on which the strength chiefly 
iepends. They may be round. In deciding upon their number, it must be remem¬ 
bered that they may have to resist floating ice, or accidental blows from vessels, &c. 
With reference to this, extra fender- piles may be driven. A little below the tops of 
the main piles are bolted two outside longitudinal pieces w w, called wales; and oppo¬ 
site to them two inner ones, as in the fig. The outer ones serve to support cross- 
timbers 11 , which unite each pair of opposite piles, and steady them ; and prevent 
their spreading apart by the pressure of the puddle P. The inner ones act as guides 
for the sheet piles s s , while being driven ; after which the heads of the sheet-piles 
ire spiked to them. In deep water these sheet-piles must be very stout, say 12 ins 
i square ; to resist the pressure of the compacted puddle. 

A gangway m, is often laid on top of the cross-pieces 11, for the use of the 
workmen in wheeling materials, &c. The puddle P is deposited in the water in the 
space, or boxing, between the sheet-piles. It should be put in in layers, and com¬ 
pacted as w ell as can be done without causing the sheet-piles to bulge, and thus open 
their joints. The bottom of the puddle-ditch should be deepened, as in the fig, in 
lease it consists, as it often does, of loose porous material which would allow water to 
leak in beneath it and the sheet-piles. This leaking under the dam is frequently a 
source of much trouble and expense. Water w ill find its w ay- readily through almost 
any depth and distance of clean coarse gravelly and pebbly bottom, unmixed with 
earth. Sand is also troublesome; and if a stratum of either should present itself ex¬ 
tending to a great depth, it will generally be expedient to resort to either simple 
cribs, Fig 4; or to caissons; with or without piles in either case, according to cir¬ 
cumstances. But if such open gravel, or any other permeable or shifting material, 
as soft mud, quicksand, &c, is present in a stratum but a few feet in thickness, and 
underlaid by stiff clay, or other safe material, leaking may be prevented, or at least 
i much reduced, by driving the sheeting-piles 2 or 3 ft into this last; and by deepening 
the puddle-trench to the same extent. It may sometimes be better, and more con¬ 
venient, to dredge away the bad material entirely from all the space to be enclosed 
by the dam, and for a short distance beyond, before commencing the construction of 
(the latter. If the dam, Fig 9, is (as it should be) well provided with cross-braces, 
like c c, Fig 7, extending across the enclosed area, the thickness or width o o of the 
puddle, need not be more than 4 or 5 feet for shallow depths ; or than 5 to 10 ft for great 
ones; because its use is then merely to prevent leaking. But if there are no braces, 
it must be made wider, so as to resist 'upsetting bodily; and then,with good puddle, 
o n may, as a rule of thumb, be % of the vertical depth o 1 below high water; except 
when this gives less than 4 ft; in which case make it 4 ft; unless more should be 
required for the use of the workmen, for depositing materials, &c. Or if the excavation 
for the masonry is sunk deeper than the puddle, the dam must be wider; else it may 
be upset into the excavated pit. 

The excavated soil may be 
raised in buckets by windlasses, or by hand, in 
■uccessive stages. The pumps may be worked 
by hand, or by steam, as the case may require; 
as also the windlasses generally needed for 
lowering mortar, stone, &c. More or less leak¬ 
ing may always be anticipated, notwithstanding 
every precaution. 

Where a coffer-dam is exposed to a violent 
current, and great danger from ice, &c, the ex¬ 
pensive mode shown in Figs 10 may become 
necessary. The two black rectangles c c, repre¬ 
sent two lines of rough cribs filled with stone, 
and sunk in position; one row being enclosed 
by the other; with a space several feet wide be¬ 
tween them. Sheet-piles p p are then driven 
around the opposite faces of the two rows of 
cribs: and the puddle is deposited within the boxing thus provided for it. as shown in the fig. 

Where the current is not strong enough to wash away gravel backing, we may. on rock especially, 
enclose the space to he built on, by a single quadrangle of orlhs sunk by stone; and after adopting 
precautions to prevent the gravel from being pressed in beneath the cribs, apply the backing.* 

Figs U)]/ z show the plan, outside view, and transverse section, to a scale of 20 ft to 
an inch, of a coffer dam on rock, in 8 to 9 ft water, used successfully on the Schuylkill 
Navigation. 

* A pure clean coarse gravel is entirely unfit for such purposes. A considerable proportion of 
earth is essential for preventing leaks. 



























0 TU 


FOUNDATIONS. 


°| l rock. Uprights b, about 1 ft square, and 10 ft apart from center to cenl 
alofig the sides of the dam; and 10 ft m the clear, transversely of the dam, support two Hues of ho 
zontil stringers, i 1 ; inside of which are the two lines or sheeting piles, * s, enclosing between tliei 
a width of 7 ft o gravel puddle. Two fiat iron bars (t t, of the transverse Section, tie together eac 

fir irnn “Th g | S 6 h lh ;'" e bar " are * »> ch thick, by 2% ins deep, and 9 ft long. Their hooked en 
fit li to eye-bolts c, which i>ass through the uprights 6; outside of which they are fastened by keys, 

w^ d «ss l iti S ! e | C M Between the keys and b, were washers. At the corners of the dam (see jdai 
were additional tie-bars, as shown. A small band of straw, as seen at v wranned around the tie 
bars just inside of the sheet-piles ; and kept in place by the puddle ; effectually prevented the leakin 
wh oh generally proves so troublesome in such cases. The stout oblique braces, oo were merel 
spiked to the outside faces of the uprights b. They are not shown iu the transverse section. This dai 
Uit.ioM 0 ' 1 shore; ip sections 30 to 40 ft long. These were floated into place, and weighteddowr 
et-pilnd, and puddled with gravel. The dam had sluices by which water was admitted whe 

first fouiid^by ^ fr ° m eXceediu 8 9 ft ’ The ^rights 6 6 wer 


1 TR.SEC 

° 1? 


OUTSIDE, 

b b 

i JL 


PLAN 


///*?' 








E^siol 



hin ^. f T c 2 ff( T f1, ‘ ims ma y be token from what is said under “ Dams ” nr 
water. 0 ' ° S affords U8e ™ suggestions forcoffer-dams also, on rock in shallow 


ipSip 

rows of piles from the dam to tlu* uimm r 13 °* ten ex P e d>ent to drive twe 


corning the friction ofthesoil against the outfit •or 1 ™l u,mi for over 
to dredge away some of the outer material also ’ «V.ti!«»fc e "* beeo,ne neeessar y 
expedient to drill holes in deen v t 1 ? X n rwek may at times he 

Ac. This may be done JmS iS'T the f"' 1 * of piles, or of iron rod,* 

P "“!f «*• *° *»<*- » *•*»•»» to Lend, .ndt u *rk?u?l» o^n a w" ^ 

prove,,, iu o»n e „h.„ „ r TgtZSgEtfSSX 3^ 


yellow P i„, Its, f«.'^™ritopSo,-l 1 0« ! . Uemlocl1 ' 5 t0 6cts per foot Hue.,, Bay 
















































FOUNDATIONS 


041 


Generally these are thinner than 

t 


n 



V , 


in .round them, &c, they are called sheet-piles. 

« hey are wide; but fre- 
|uently they are square; 

,,ml as large as bearing 
dies; and are then called 
‘lose piles. To make 
hem drive tight to¬ 
other at foot, they 
,re cut obliquely as at 
f. Occasionally, when 
Iriven down to rock 
hrougli soft soil, their 
eet are in addition cut 
o an edge, as at i, so as 
o becqjne somewhat 
truised when they reach 

he rock, and thus fit closer to its surface. Their heads are kept in line while driv- 
ng, by means of either one or two longitudinal pieces a and a, called wales or 
tringers. These wales are supported by gauge-piles, or guide-piles, previously driven 
n the required line of the work, and several ft apart, for this purpose. See Figs8. 

A dog-iron d, or round iron, may also be used for keeping 
he edges of the piles close at top to those previously driven, both 
uring and after the driving. Its sharp ends, c c, being driven into 
lie tops of the wales ww, (shown in plan,) it holds the descending 
die o firmly in place. At?», d,p. Fig 11, are other modes occasionally 
sed for keeping the piles in proper line. Aty>. the letters ss denote 
mall pieces of iron well screwed to the piles, a little above their feet, 
o act as guides; very rarely used. At m are shown wooden tongues 
f, sometimes driven down between the piles after they themselves 
lave been driven; to assist in preventing leaks. In some cases 
heet-piles are employed without being driven. A trench is first 
ug to their full depth for receiving them; and the piles are simplv 

ilaced in these, which are then refilled. Closer joints can be secured in this manner than by driving. 

When piles are intended to sustain loads $n their tops, whether driven all their 
ength into the ground, or only partly so, as in Fig 3, they are called bearing 
[tiles. They are generally round; from 9 to 18 ins diam at top; and should he 
traight, hut the bark need not be removed. White pine, spruce, or even hem- 
ock, answer very well in soft soils ; good yellow pine for firmer ones; and hard 
>aks, elm, beach, &c, for the more compact ones. They are usually driven from about 
to 4 ft apart each way, from center to center, depending on the character of the 
[on, and the weight to be sustained. A trend-wheel is more economical than 

nt, 

it, 

M 




w 


w 


1 — 

.... 1 

rri 

1 

I 

£ 

d 

- 1 

1 { 


w 


w 


Fi(jl2 


lie winch for raising the hammer, when this is done by men. Morin found that 
lie work performed by men working 8 hours per day, was 3900 foot-pounds per man, 
ier minute by the tread-wheel; and only 2600 by a winch. 



After piles have been driven, and their heads carefully sawed off to 

i level, if not under water, the spaces between them are in important cases filled up level with their 

tops with well rammed gravel, stone spawls, or concrete, in order 
to impart some sustaining power to the soil between the piles. Two 
courses of stout timbers, ("from 8 to 12 ins square, according to the 
weight to be carried) are theu bolted or treenailed to the tops of the 
( = == j3£ = te^5r If—*— piles and to each other, as shown in the Fig. forming what is called 

ip -13 1 3 „— rw... a grillage. On top of these is bolted a floor or plat- 

H form of thick plank for the support of the masonry ; or the timbers 

ml of the upper course of the grillage may be laid close together to form the floor. The space below the 
e; loor should also, in important cases, be well packed with gravel, spawls, or concrete, 
lie If milder water, the piles are sawed off by a diver, or by a circular saw driven 
>y the engine of the pile-driver, and the grillage is omitted. Instead of it the masonry or concrete 
a ay be built in the open air in a caisson, which gradually sinks as it becomes filled ; or on a 

trong platform which is lowered upon the piles by sorews as the work progresses. Or a strong 

taissou limy first be sunk entirely uuder water, and theu he tilled with concrete, up to near 

os water; the caisson being allowed to remain. Or the caisson may form a cofferdam, to be first 
mnk, and then pumped out. If the ground is liable to wash away from around the piles, as iu the 
lase of bridge piers, <tc, defend it by sheet-piles, or rip-rap, or both. 

The cost of a floating steam pile driver, in Phiiada, scow 24 ft by 50 ft, 

raft 18 ins, with one engine for driving, and one (to save time) for getting another pile ready ; with 
ue toil hammer, is about $8000; and *500 more will add a circular saw, Ac. for sawing off piles at any 
eqd depth. Requires engineman, oook, and 4 or 5 others. Will burn about half a ton of coal per day. 
)nviug 20 ft into gravel, and sawing off. will average from 15 to 20 piles per day of 10 hours 
and about twioe as many. On laud about half as many as iu water. 


In 


Tile gunpowder pile driver Invented by Mr. Thomas Shaw, the well-known 
neohanical engineer of Phiiada, is a very meritorious machine. The hammer is worked by small 
artridges of powder, placed one by one in a receptacle on top of the pile; and exploded by the ham- 
ner itseir It can readily make 30 to 40 blows of 5 to 10 ft. per minute ; and, since the hammer does 
ict come into actual contact with the piles, it does not injure their heads stall; thus dispensing 

lith iron hoops Ac, for preserving them. When only a slight How is required, a smaller caitndbe 







































D4Z 


FOUNDATIONS. 




is used. To drive a pile 20 ft into mud averages about one-third of a pound o 
powder; into gravel, 4 times as much. This machine does not assist in raising th 
pile, and placing it in position, as is done by ordinary steam pile drivers; the latter 
however, average but front 6 to 14 Mows per minute. 

Piles have been driven by exploding small charges of dynamitt 
laid upon their heads, which are protected by iron plates. 

Steam-hammer pile <lrivers, operating on the principle of that devise 
by Nasmyth about I860, are economical in driving to great depths in diliicu 
soils where there are say 200 or more piles in clusters or rows, so that the niachiu 
can readily be moved from pile to pile. 

The steam cylinder is upright, and is confined between the upper ends of tw 
vertical and parallel I or channel beams about 6 to 12 ft long and 18 ins apar 
the lower ends of which confine between them a hollow conical bonnet cast 
ingv” which fits over the head of the pile. This casting is open at top, and throng 
it the hammer, which is fastened to the foot of the piston-rod,* strikes th§ head u 
the pile. Each of the vertical beams encloses one of the two upright guide-timb 
or “leaders,” of the pile driver, between which the driving apparatus, above 
scribed, is free to slide up or down as a whole. 

When a pile has been placed in position, ready for driving, the bonnet casting 
placed upon its head, thus bringing the weight of the beams, cylinder, hammer,an 
casting upon the pile. This weight rests upon the pile throughout the driving, th 
apparatus sliding down between the leaders as the pile descends. 

The steam is conveyed from the boiler to the cyl by a flexible pipe. When it 
admitted to the cyl, the hammer is lifted about 30 or 40 ins, and upon its escape tl 
hammer falls, striking the head of the pile. About 60 blows are delivered per mil 
ute. The hammer is provided with a trip-piece which automatically admits steal 1 
to the cylinder after each blow, and opens a valve for its escape at the end of tl 
up-stroke. By altering the adjustment of this trip-piece, the length of stroke (an 
thus the force of the blows) can be increased or diminished. The admission an 
escape of steam, to and from the cyl, can also be controlled directly by the attendan 
The number of blows per minute is increased or diminished by regulating the sui 
ply of steam. 

In making the up-stroke, the steanV, pressing against the lower cyl head,of cours 
presses downward on the pile and aids its descent. 

The chief' advantage of the.se machines lies in the great rapidit 
with which the blows follow one another, allowing no time ior the disturbed eartl 
sand, Ac, to recompact itself around the sides, and under the foot, of the pile. Th 
enables the machines to do work which cannot be done with ordinary pile driver 
They have driven Norway pine piles 42 ft into sand. They are less liable tha 


others to split and broom the pile, so that these may be of softer and cheaper w< 
The bonnet casting keeps the head of the pile constantly in place, so that the t 

fin nnf ^ ilnil ^ nrn f mif /if lino Tlioi>. 1. .... .1 r. 1 -. .... l 


do not “ dodge ” or get out of line. Their heads have, in some' cases, been set on fir 
by the rapidly succeeding blows. 

These machines consume from 1 to 2 tons of coal in 10 hours ar 
require a crew of 5 men. They work with a boiler pressure of fro; 
60 to 75 lbs per sq inch. 


They are made by R. J. & A. B. Oram, 80 Griswold St, Detroit, Mich, an 
by Vulcan Iron Works, 86 N. Clinton St, Chicago, Ill. 



about $1075 on cars at works. 

The Vulcan Works make a railroad ear, furnished with a pilc-drivinj 
machine, and 2 hor engines, with boiler, for raising the hammers. Either th 
Nasmyth, or the ordinary, hammer may he used. The car can be made self-prc 
pelling when desired. It is larger than an ordinary platform car, and is provide 
with lateral supports tp enable it to drive piles to the right or left of the road-bet 
The leaders are hinged, so that they can be laid horizontally upon the car when nc 
in use. 


!tti' 

ill* 


jc 


* In the Cram machine the hammer is fastened to the lower end of the cylinder 
instead of to that of the piston-rod. The cylinder rises and falls with the hammei 
and its weight is thus utilized in increasing the force of the blows Steam is sup 
plied to the cylinder through the piston-rod, which is made hollow for this purpos* 
The piston-rod is fixed in place between the vertical beams by means of a cross-hea 
at its top. The piston is fastened to the foot of the piston-rod 















FOUNDATIONS. 


643 


Rules for the Sustaining' Power of Piles. 


, 'ey differ very much. No rule can apply correctly to all conditions. The ground itself between 
e piles, m most cases, supports a part of the load; although the whole of it is usually assigned to 
ne piles. Again, in very clayey soils, there is greater liability to sink somewhat with the lapse of 
* l “ consequence of the admission of water between the pile and the clay ; thus diminishing the 
riction between them. The less firm the soil, the more will the piles he affected by tremors ; which 
Iso tend in time to cause sinking. lu some cases this sinkiug will not be (hat of the piles settling 
iu ! P t r .u earoun d them ; but that of the entire compacted mass of piles and earth into 

Inch they were driven; settling down into the less dense mass below them. Piles are sometimes 
.lined for settlements which are really due to the crushing (flatways) of the timbers which rest 
n mediately upon their heads. See p 436. 


a 11 ^ Ie London bridge across the Thames, each pile tinder some of the 
|'ers sustains the very heavy load of 80 tons. They are driven hut 20 feet into the 
tiff, bine London clay ; and are placed nearly 4 ft apart from center to center; which 
fl too much for such piers and arches. At 3 ft apart scant, they would have had but 
Id tons to sustain. They are 1 ft in diam at the middle of their length. Uglyset- 
t' ements, some of them to the extent of about a ft, have occurred under these piers. 
* llwckfriars bridge, in the same vicinity, exhibits the same defect. By some 
■ ns is ascribed in both cases to the gradual admission of water between thecfav and 
le piles, perhaps by capillary action of the piles themselves; or perhaps by direct 
■aktng It may, however, be owing in part to the crushing of the platforms on 
>p of the piles; or to a bodily settlement of the entire mass of piled clay, into 
le unpiled clay beneath, under the immense load that rests upon it. This’here 
mounts to 5 V 2 tons per sq foot of area covered by a pier; and is probably too much 
) trust upon damp clay, when even the slightest sinking is prejudicial. 

Maj J. Sanders, U. S. Engs, experimented largely at Fort Delaware in river 
tud; and gave the following in the Jour. Franklin Inst, Nov 1851. For the safe 
•ad for a common wooden pile, driven until it sinks through only small and 
early equal distances, under successive blows, divide the height of the fall in ins, 
y the small sinking at each blow in ins. Mult the quot bv the weight of the 
ammer, ram, or monkey, in tons or pounds, as the case may be. Divide the 
rod by 8. He does not state any specific coefficient of safety. 


( Example. At the Chestnut St Bridge, Philada, the greatest weight on any pile is 18 tons, 
r Kueass had the piles driven until they sauk %, or .75 of an inch under each blow from a 1200 8 > 
unmer, falling 20 ft. Was he safe in doing so? Here we have the fall in ins 20 X 12 = 240 And 
40 „ 384000 

175 - = 320; and 320 X 1200 = 384000 B>s ; and —— 


— 48000 lbs, — 21.4 tons safe load by Maj San- 


firs’ rule. The soil was river mud. 

Our own rule is as follows. Mult together the cube rt of the fall in ft; the wt of hammer in fts; 

P nrnrl hv fhn last elnVimr in ino 1 1 Tho nimh'nni 11 . 1 


- - ~ • ”•"~ vuv uuov IV 01 uuv iau 1U 1 

i id the decimal .023. Divide the prod by the last sinking in ins. -f- 1 . The quotient will be the 
ctreme load that will be just at the point of causing more siuking. For the safe load take from 
»te twelfth to one half of this, according to circumstances. Or, as a formula, 

Cube rtof v Wt of hammer 


Extreme load __ fail <n feet * in pounds 
iu tons ~ 


X .023. 


Last sinking in inches -f- 1 


Example. The same as the foregoing at Chestnut St Bridge. 
714 ft. Hence we have 

Extreme load _ 2 714 X 1200 X '° 23 = *LI 

in tons .75 1 1.75 


Here the cube re of 20 ft fall is 


= 42.8 tons. 


: Or say half of this, or 21.4 tons, the load for a safety of 2. Major Sanders’ rule makes the safe 
ad 21.4. The actual one is 18 tons. 

A safety of 2 is not enough for river mud. See “ Proper load for safety," p 644. 

But although Major Sanders’ rule and our own agree very well in this instance if a safety of 2 be 

I ken for each , they differ widely in some others. Thus at Neuilly Bridge, France, Perronet’s 
;aviest hammer weighed 2000 Bis, fall 5 ft, sinkage .25 of an inch in the last 16 blows; or say .016 
ch per blow. The piles sustain 47 tons each. Our rule gives 38.8 tons for a safety of 2; while San- 
lirs’ rule gives 515 tons safe load I If, as we think probable, there was no actual sinking at the last 
I ow, then our rule gives 39.3 tons for a safety of 2 ; while Sanders’ gives infinity. 

At (file Hull Docks, England, piles 10 ins square, driven 16 ft into alluvial mud. by a 1500 lb ham- 
er, falling 24 ft, sauk 2 ins per blow at the end of the driving. Thev sustain at least 20 tons each, 
• according to some statements 25 tons. Our rule gives 33.2 tons for the extreme load; or 16.6 fora 
trety of only 2. Sanders gives for safety 12.06 tons. As before remarked, 2 is not safety enough for 
ud. In mud, it is not primarily the piles, but the piled soil that settles, bodily, for years. 

At the Koyal Border Bridge, England, piles were very firmly driven from 30 to 40 ft in sand 
id gravel, in some cases wet. Pine was first tried, but it split and'broomed so badly under the hard 
-iving, that American elm was substituted, with success. They were driven until they sank but .05 
ich per blow, under a 1700 fl) monkey, falling 16 ft. They support 70 tons each. Our rule gives 47 
iris for a safety of 2 ; while Sanders gives 364 tons safe load ! 

Itisthe writer's opiuion, however, that the piles did not actually sink, as was (and always is, in 
'■ch cases) taken for granted by the observers; but that they were merely compressed or partially 
•nshed by overdriving. Most of the piles were driven until they sank (?) only an inch under 150 
llows; but we doubt whether they were any safer, or farther in the ground, than when they had re¬ 
vived only one of them ; and consider such extreme precaution worse than useless. 

In gome experiments (1873) at Philada, a trial pile was driven 15 ft into soft river mud. by a 
;>00 lb hammer ; its last sinking being 18 ins under a fail of 36 ft. Only 5 hours after it was driven 
was loaded with 6.5 tons ; which caused a sinking of but a very small fractiou of an inch. Our rule 









644 


FOUNDATIONS, 




gives 6.4 tons as the extreme load. Under 9 tons it sank .75 of an inch ; and under 15 tons, 5 ft. B 
Maj Sanders’ rule its safe load would be 2.14 tons. 

A U. 8 . Govt trial pile, about 12 ins sq, driven 29 ft through lavers of silt, sand, and elav, ham 
mer 910 lbs, fall 5 It, last sinking .375 of an iuch, bore 26.6 tons; but sank slowly under 27*9 tout 
Our rule gives 26 tons extreme load. 

French engineers consider a pile safe for a load of 25 tons, when it is driven to the refusal o 
1344 fts, falling 4 ft; our rule gives 24.2 tons for safety 2. They estimate the refusal by its not sink 
ing more than .4 of an inch under 30 blows. In many important bridges Ac they drive until there i 
no sinking under an 800 R> hammer, falling 5 ft. Our rule here gives 31.5 tons extreme load ; or 15. 
for safety 2. 

As to the proper load for safety, we think that not more than one-half the extreme load give 

by our rule should be taken for piles thoroughly driven in firm soils; nor more than one-sixth whe 
in liver mud or marsh ; assuming, as we have hitherto done, that their feet do not rest upon rock 

If liable to tremors, take only half these loads. 


Piles may be made of any required size as regards either length or cross section, by bol 

ing and fishing together sidewise and leugthwise, a number of squared timbers. 

Piles with blunt ends. At South Street Bridge, Phila, 1200 stout piles of Nova Scotia sprue 
with blunt ends were driven 15 to 35 ft, partly in strong eravel, by a common steam pile driver, at 
total cost (piles and driving) of $1 to $8 each. At Wilmington Harbor, Cal, Mr. C. B. Sear 
U. S. Army. (Jour. Am. Soc. C. E., Dec 1876) found that iu firm compact wet sand, after the first fi 
blows the piles would not penetrate more than .5 to 1.5 ins at a blow, no matter how far the 2400 
hammer fell. The unpointed ones of which there were many thousands, drove quite as readily to ave 
age depths of 15 ft in this sand as the pointed ones, and with much less tendency to cant. As a hig 
fall had no farther effect than to batter the heads he reduced it to 10 ft. which drove an average o 
about .72 inch to a hlow. To insure straight driving, the ends must be at right angles to the lengt 
Instead of driving piles to moderate depths it mat at times be better to merely plant them bu 
down in holes bored by an auger like Pierce’s Well Borer. See p 626. This will avoid shakit 
adjacent buildings. See “ In Mobile Bay,” p 646. 

The ultimate friction of piles even with the bark on, and driven about 3 ft apart from ce 
to cen probably never much exceeds about 1 ton per sq ft even when well driven into dense moi 
sand or loamy gravel; nor more than .5 to .75 of a ton in common soils and clays; or than .1 to 
of a ton in silt or wet river mud depending on the depth and density. 

The friction of east iron cylinders seems to be about .3 that of piles. 


f 




There is a great difference In the penetrability of differer 

sands. Thus, in the Lary bridge, uo special ditficulty was fouuct in driving piles 35 ft into deep w 
sand ; while, in other wet localities, piles of very tough wood, well shod with iron, cannot be drivf 
6 ft into saud, without being battered to pieces. The same difference has been found in the case < 
screw-piles. At the Brandywine light house these could not be forced more than 10 ft into the clea 
wet sand. Stiff wet clay (and clean gravels) also differ very much in this respect. Generally tht 
are penetrable to any required depth with comparative ease ; but we have seen stout hemlock pil 
battered to pieces in driving 6 ft through wet gravel; and Mr. Rendel found that at Plymouth 
•‘could not by any force drive screw-piles more than about 5 ft into the clay, which is not as stiff 
the London clay,” on which the forementioued new London and Blackfriars bridges were foundet !' 
and iuto which even ordinary woodeu piles were driven 20 ft without special difficulty. 

A mixture of mud with the sand or gravel facilitates driving very much ; but before beginning 
extensive system of piling, a few experimental ones should be driven, to remove doubt as to t 
trouble and expense that may be anticipated. Mere boring will often be but a poor substitute for thi 


the same time with a low fall; aud this gives less time for the soil to compact itself around the pil 
between the blows. At times a pile may resist the hammer after sinking some distance; but str 
again after a short rest; or it may refuse a heavy hammer, and start uuder a lighter one. It n> 
drive slowly at first, and more rapidly afterward, from causes that may be difficult to discover. Ti 
driving of one sometimes causes adjacent ones previously driven, to spring upw ard several feet, 
pile is in the most favorable position when its foot rests upon rock, after its entire length has b 
driven through a firm soil, which affords perfect protection against its bending like an overload 
column; and at the same time creates great friction against its sides; thus assisting much in si 
taining the load, and thereby relieving the pressure upon the foot. A pile may rest upon rock, at 
yet be very weak; for if driven through very soft soil, all the pressure is borne by the sharp poin 
and the pile becomes merely a column in a worse condition than a pillar with oue rounded end. S 
Fig 1, page 439, Strengt h of Iron Pillars. In such soils the piles need very little sharpening; iudee 
had better l>e driven without any; or even butt end down. 

The driving of a pile in soft ground or mud wIN generally canse an adjacent one previously drive 
to lean outwards unless means be taken to prevent it. 

Iu piling an area of firm soil it is best to begin at its center and work outwards ; otherwise the st 
may become so consolidated that the central ones can scarcely be driven at all. 

Elastic reaction of the soil has been known to cause entire piled*ar 

to rise, together with the piles, before they were built upon. 

In very firm soil, especially if stony; or 
even in soft soil, if the piles are pointed, and 
are to be driven to rook; their feet should 
tie protected i>y shoes of either wrought 
iron, as at a, ,s,and b, Figs 13; spiked to the 
pile by means of the iron straps »?, forged 
to them; or of cast iron, as at c, where the 
shoe is a solid inverted cone, the wide flat 
base of which affords a good bearing for the 
flat bottom of the pile-point. The dotted 
line is a stout wrought-irou spike, well se¬ 
cured in the cone, which is cast around it; 
this holds the shoe to the pile. Regular 
















































FOUNDATIONS. 


645 


Moderate!v°oom 1 ^ct W 'r> I Ur^ n fm^!*»>r e * 8h 18 t0 30 ,b ® : but sheet iron ma ? h « n.sed when the roil is but 
It t helmtt » nil 4 fn a !' a , EU m °, re 80 : aud solid iron or sleel Points, from 2 to 4 ins square 

lrill%£* Iong ’ whei \ ver ^ Co,ll ^ t ;lIld stony. Holes .nay be 
Vom hv? H k - ^° r H recelvin f tlie points of piles, and thus preventing them 

i he m tnr ,Qg ^ d ° #1 a tU . b6 ' aS ? guide t0 the dri11 - after the earth is cleaned out of 

. * pi^os^rvo tli^ lu^iKis to some extent from splitting under the 

li > 5Tn i n t r w i 1 de raD The t 3 e e are re ho Snall - V SUrr0Unded b ^ a hoo P h ’ Fi S d ■' from .4 to 1 inch thick ; and 

vill crush8Dlit‘ a nd h S.1^nt hl u ,r ’. a . ome y me9 but imperfect aids; for in hard driving the head 

often snlitonenThehZlt .h^r’ fre< l u ® nU . v v for nian y feet below the hoop: moreover, the 
P.^ °P^n. The heads, therefore, often have to be sawed, or pared off several time* 
>efore the pile is completely driven; and allowance must be made for this loss in ordering piles for 
oy given work; especially in hard soil. Capt Turnbull, U S Top Eng, states that at Jhf Potomac 

eoth of^ahoufanHnoh 3 "T P reserv c d fr onnnjury by the simple expedient of dishing them out to a 
epthof about aninch.and covering them by a loose plate of sheet iron ; as shown in section ate, 

f the rim Phes mal hfT b [? omi "g or crushing of the head, materially diminishes the force 
Lo .t S I ? ay . be dr, . ven through small loose rubble without much labor. Shaw's driver 

om not injure the heads. Piles which foot on sloping rock may slide when loaded. 

. T ive a I >i,e **ea<l below water a wooden punch, or follower, as 

tp, Figs 13, may be used. The foot of this punch fits into the upper part of a casting f f, round or 
quare, according to the shape of the pile; and having a transverse partition o o. The lower part 
f the casting is fitted to the bead of the pile t; and the hammer falls on top of the punch. When 
riving piles vertically in very soft soil, to support retaining-walls, or other structures exposed to 
onzonfcal or inclined forces, care must be taken that these forces do not push over the piles them- 
elves ; for in such soils piles are adapted to resist vertical forces only, unless they be driven at an 
aclinatton corresponding to the oblique force. 

A broken pile m;iy be drown out, or at least be started, if not very 

rmly driven, bv attaching scows to it at low water, depending on the rising tide to loosen it Or a 
>ng timber may be -used as a lever, with the head of an adjacent pile for its fulcrum. Or a crab 
•orked by the engine of the pile driver. In very difficult cases the method devised by Mr J. Monroe 
E, may he used. A 4 inch gas pipe 15 ft long, shod with a solid steel point, and having an outer 
boulder for sustaining a circular punch, was thereby driven close to and 2 or 3 ft deeper than two 
ties driven 12 ft, in 37 ft water, and broken off by ice. Four pounds of powder were then deposited 
i the lower end of the pipe, and exploded, lifting the piles completely out or place. It will often be 
est to let a broken pile remain, and to drive another close to it. May be drawn by hydraulic press. 
Ice adheres to piles with a force of about 30 to 40 lbs per sq inch, and in 

sing water may lift them out of place if not sufficiently driven. 

Iron piles and cylinders. Cast iron in various shapes has been much 
sed in Europe for sheet piles: especially when intended to remain as a facing for the protection of 
tncrete work, filled in behind and against them.* Cast iron cylinders, open at both ends, mav be 
‘ sed as bearing piles; and may be cleaned out, and filled with concrete, if required. The friction in 
riving is greater than in solid piles, inasmuch as it takes place along both the inner and the outer 
irfaces. This may be diminished by gradually extracting the inside soil as they go down. Thev 
•quire much care, and a lighter hammer, or less fall than wooden ones, to prevent breaking: to 
hich end a piece of wood should be interposed between the hammer and the pile; or the ram may be 

f wood. But it is better to use them in the shape of screw cylinders, which, 

oreover, gives them the advantage of a broad base, as in the following. 

Brunei's process. He experimented with an open cast-iron cylinder, 3 ft 
;• lter diam; 1% ins thick; in lengths of 10 ft, connected together by internal socket and joggle joints, 
•cured by pins, and run with lead. It had a sharp-edged hoop or cutter at bottom: and a little 
love this, one turn of a screw, with a pitch of 7 ins, and projecting one foot all around the outside 
'the cylinder. By means of capstan bars and winches, he screwed this down through stiff'clay and 
tnd, 58 feet to rock, on the bank of a river. In descending this distance the evlinder made 142 
Evolutions ; sinking on an average about 5 ins at each. The time occupied in actually screwing was 

hours; or about 1 ft per hour. There were, however, many long intervals of rest for clean- 
>g away the soil in the inside. After resting, there was no great difficulty in restarting. The next 
; will give an idea of the arrangement of the screw. 


The screw-pile of Alex. Mitchell, Belfast, consists usually of a rolled iron 
laft A, Figs 14, from 3 to 8 ins diam; and having at its foot a cast-iron screw 
S S. with a blade of from 18 ins to 5 ft diam. The screws used for light houses, 
* <posed to moderate seas, or heavy ice-fields, are ordinarily about 3 ft diam, have 
M turns or threads, and weigh about 600 lbs. The round rolled shafts are from 
to 8 ins diam. They are screwed down from 10 to 20 ft into clay, sand, or coral, by 
: »out 30 to 40 men, pushing with 6 to 8 capstan bars, the ends of which describe a 
rule of about 30 to 40 ft diam. For this purpose a platform on piles has frequently 
be prepared. In quiet water, this may be supported on scows; or a raft well 
looreti may lie used when the driving is easy; or the deck of a large scow with a 
ell-hole in the center for the pile to pass through. Roughly made temporary 
*ihs, filled with stone and sunk, might support a platform in some positions. The 
latform must evidently be able to resist revolving horizontally under the great 
ushing force of the men at the capstan bars; and on this account it is difficult 
i drive screws to a sufficient depth, in clean compact sand, by means of a floating 
latform. The feet of the piles must be firmly secured to the screws, to prevent 


* Cast iron, intended to resist sea-water, should be close-grained, 

ird. white metal. In such, the small quantity of contained carbon is chemically combined with the 
etal; but in the darker or mottled irons it is mechanically combined, and such iron soon becomes 
ft, (somewhat like plumbago,) when exposed to sea-water. Hard white iron has been proved to 
sfst for at least 40 years without any deteriotation : whether constantly under water, or alternately 
3t and dry. Copper aud bronze are but slightly and superficially affected by sea-water; but destruc- 
/e galvanic action takes place if diff metals are in contact. See p 218. 







646 


FOUNDATIONS 


their being lifted out of them by the upward force of waves against the super¬ 
structure. At y p, Figs 14, is shown a mode of splicing or uniting the different 
lengths or sections of a pile. The point of junction is at t; r r is a stout iron ring 
forged on to the lower pile p, y 

about a foot or 18 ins below its 
top v. A strong cylindrical cast¬ 
ing n n, enclosing the ends of 
the sections, rests on this ring, 
and is pinned through the piles, 
as at tt. On this casting are 
also cast projections cc c, for at¬ 
taching rods</$r, and beams ?',&c, 
necessary for bracing the struc¬ 
ture from pile to pile. The time 
actually required for driving a 
screw is from 2 to 10 hours, in 
favorable circumstances. 

At the Brandywine lighthouse, on 
a sand bank of very pure sand, cov¬ 
ered 6 nr 8 ft at low water, and from 
11 to 13 ft at high, they could not be 
forced down, from a fixed platform, 
for more than 10 ft. At other places 20 ft in sand is reached without much trouble, where the sand 
contains a good deal of mud, but its bearing power is then less. This (ultimate) ranges between 
about 1 and 6 tons per sq ft according to purity, depth, compactness, &c, of the sand. In important 
cases the bearing power should be tested. 

Mitchell's piles have been screwed about 40 feet into a mixture of clay and sand, with screws 
4 ft diam. Thev pass through small broken stone and coral rock w ithout much difficulty ; and will 
push aside bowlders of moderate size. Ordinarily, clay or sand will present no great obstruction ; 
but occasionallv either of them will do so. Perfectly pure clean sand, as a general rule, gives most 
difficulty. At the Braudywiue shoal the driving was aided by a spur and pinion placed as low as the 
water permitted ; and the levers were worked by 30 men. The dauger of twisting off the shaft is 
the limit for screwing them. Thev are much used for the anchoring of chains for mooring buoys, &c. 
On land, small screws, with short hollow shafts, make good durable supports for depot pillars, cranes, 
wooden telegraph poles, station signals in marine surveying, &c, &c. They can readily be unscrewed 
for removal. Horses or oxen may be used in driving large screws. The Brandywine light house 
stands on 9 screw-piles, which are surrounded by 30 others of 5 ius diam, as fenders. They have to 
resist not only moderate seas, but immense fields of Boating ice, miles in extent. An unfinished 
structure was destroyed by ice, which at times injures the bracing of the standing one. 



Fig 15 


Test boeiiijjs should be made to ensure that the screws do not stop just 
above a very weak stratum which may endanger their bearing power. So with any piles. 

By means of a jet of water forcibly impelled through a tube by a forct 

pump, the most obstinate sands (but not stiff clay or cemented gravel) w ill be loosened, and the sink 
iug of screw piles, or wooden cues, or even the largest cylinders, be greatly facilitated. In a govern 

ment pier at Cape Ilenlopen in very compact sand, in which 6 out of" 
screws previouslv broke before reaching 10 ft, the use of the jet was found to remove more thai 
three-fourths of the resistance.* The pileyj to be sunk having first been placed in position as in Ft 
15, the lower open ends 11 of a bent iron tube 1st of one and a 
quarter ins bore were stood upon the upper face of the screw disk, and 
there held firmly by 3 or 4 men while the pile was being screwed down 
by the capstan c, which was worked by a leading rope r. From the 
bend s of the pipe, a hose h. 2 ins diam. led to the force pump, the 
cylinder of which was 5 ins bore, and 9 ins stroke, and worked about 
80 full strokes per minute, by a mule walking on a tread wheel on a 
floating platform/. There was now tio trouble in screwing the piles to 
any required depth. Previous trials by playing the jet beneath thedisk 
gave unsatisfactory results. 

In Mobile B ay several thousands of wooden piles, 
from 18 to 48 ins diam, were sunk from 10 to 20 ft into obstinate sand, 
at the average sinking rate of about 1 ft per second, entirely by means 
of jets. The jet was propelled by a city steam fire engiue. on a steam¬ 
boat, through its own hose, with a one and a quarter inch nozzle. 

During the descent the nozzle n n was held loosely in its place near 
the foot of the pile, by two staples » s and by a string t reaching to the 
surface. The piles were suspended by their heads from shears, by the 
tackle of which their descent was regulated. The sand settled flrmly 
around the piles in a few miuutes after they were sunk.f 

At Tensas River. Alabama, for iron cylinders 6 ft diam 

(enclosing piles, see p 651), in deep light shiftiug sand, the jet was forced by a small 
rotarv pump of 200 to 300 revolutions per minute, through a canvas hose 3 ins diam, 
into a central conical cast iron vessel 10 ins diam. from which radiated 12 gas pipes 1 
inch diam, and about 30 ius long. At the outer end of each of these radii was an 
elbow to which was attached a long vertical pipe reaching down into the cylinder, 
and made in 10 ft lengths with screw ends for prolonging them as the cylinder went 
down. This apparatus was raised and lowered by a light block and line; and by it 
alone each cylinder was sunk about 16 ft into the light sand in a few hours.J 



71 


* Report Sec of War 1872. t John W. Glenn. C E, Van Nostrand, June 1874, 

J Gabriel Jordan, C E; Trans Am Soc C E, Feb 1874. 









































FOUNDATIONS. 


647 


At the I.ovan Viaduct. Mr James Brnnlee, England, in a light 

sandy marl of great depth, sunk hollow cast iron cylinders of 10 ins outer diant, to a depth of 20 ft, 
by means of a jet pipe 2 ins diam passing down inside of the cylinder, and through a hole in its base, 

I which was a cast iron disk 30 ins diam, and 1 inch thick, strengthened by outside flanges. The con¬ 
necting flanges of the cylinder sections are outside, thus impeding the descent, as did also the broad 
bottom disk ; still 3 or 4 hours usually sufficed for the sinking of each, to 20 ft depth. Actual trial 
showed that their safe sustaining power was about 5 tons per sq ft of bottom disk. 

At l<ock Ken viaduct each pier consists of two cylinders, open at both 
ends; of cast iron, 8 ft in diam; l^ins thick; in lengths of 6 ft, weighing 4 tons 
each ; and bolted together by inside flanges, with iron cement between them. The 
cylinders stand 8 ft apart in the clear; and are in 36 ft water. “ A strong staging 
was erected; and 4 guide-piles driven for each cylinder. The several lengths being 
previously bolted together, these were lowered into their places. Each cylinder sank 
ny its own weight one or two ft through the top mud, and then settled upon the sand 
m l gravel which form tin* substratum for a great depth. Into this last they were 
sunk about 8 or 9 ft farther, by excavating the inside earth under water, by means 
d f an inverted conical $crew-|mn, or dredger, of J4 inch plate iron. This was 
|i ft greatest diam, and 1 ft deep; and to its bottom was attached a screw about I ft 
long, for assisting in screwing it down into the soil. Its sides had openings for the 
entrance of the soil; and leather flaps, opening inward, to prevent its escape. From 
opposite sides of the pan, 3 rods of % inch diam projected upward 4 feet, and were 
there forged together, and connected by an eye-and-bolt joint to a long rod or shaft, 
it the upper eud of which was a four-armed cross-handle, by which the pan Avas 
screwed down by 4 men on the staging.” 

‘•When a pan was full, a slide which passed over the joint at the bottom was lifted; and the pan 
was raised by a tackle. This pan raised about 1 cub ft at a time. A smaller one of only 1 ft diam, 
uid 1 ft deep, raising about J4 cub ft, was used when the material was very hard. By this means 
.he cylinders were sunk at the rate of from 2 to IS ins per day. The slow rate of 2 ins was caused 
jy stones, some of them of 50 lbs. These were first loosened by* a screw-pick, which was a bar of 
ron 3 ft long, with circular arms 12 ins long projecting from the sides. After being loosened by this, 
he stones were raised by the pan. The expense of all this apparatus was very trifling; and the ex- 
iavation was done easily and cheaply. After the excavation was finished, and the cylinder sunk, 
lefore pumping out the water, concrete (gravel 2, hydraulic cement 1 measure) was fiiled in to the 
lepth of 12 feet, by meaus of a large pan with a movable bottom; and about 12 days were left it to 
oarden. The water was then pumped out, and the masonry built in open air. In some of the cylin- 
lers. however, the water rose so fast, notwithstanding the 12 ft of concrete, that the pumps could not 
teep them clear; and 6 ft more of concrete had to be added in those. Finally random-stone, or rough 
try rubble, was thrown iu around the outsides of the cylinders, to preserve them from blows and 
tndermining." * The masonry extends 20 ft above the cylinders, and above water. 

Tlie vacuum and plenum processes. We can barely allude to 
the general principles of these two modes of sinking large hollow iron cylinders. In 
,the vacuum process of Dr. Lawrence Holker Potts, of London, the cylinder 

c, Fig 16, while being sunk, is closed air-tight at top, by a 
trap door, opening upward. A flexible pipe p, of India- 
rubber, long enough to adapt itself to the sinking of the 
cylinder, and provided with a stopcock s, leads from the 
cylinder to a vessel v; which may be placed on a raft, or a 
scow, or on land, as may suit circumstances. The cylinder 
being first stood up in position, as in the fig, the water is 
pumped out, and the interior soil removed if the cylinder 
has sunk some distance by its own weight. The cock 
s is then closed, and the air is drawn out from the vessel v 
>y an air pump. The cock is then opened, and most of the air in the cylinder rushes 
nto the void vessel v: ; thus leaving the cylinder comparatively empty, and therefore 
ess capable of resisting the downward pressure of the external air upon its top. 
.'his pressure, as is well known, amounts to nearly 15 lbs on every sq inch ; or nearly 
> ton per sq ft of area of the top. Consequently the cylinder is forced downward in 
he bed of the river, by this amount of pressure, in addition to its own weight. At 
he same time, the pressure of the air upon the surface of the water is transmitted 
hrough the water to the soil around the open foot of the cylinder; so that if this 
oil be soft or semi-fluid, it will be pressed up into the nearly void cylinder, in which 
s no downward pressure to resist it. The descent varies from a few inches, to 4 or 5 
, t each time. The process is then repeated, by admitting air again into the cylin- 
j ler, opening the trap-door, removing the water and soil, as before, &c. Additional 
I engths of cylinder may be bolted on, by means of interior flanges, 
f It is adapted only to soft soils, and to wet sandy ones ; but is not sufficient- 

y v powerful in very oompaotones; nor doea it answer where obstructions from bowlders, logs, &o, occur ; 

•Hollow Iron Piles either cast or wrought with solid pointed feet, to be driven by the hammer 
ailing inside of them and striking against the top of the solid foot, are a recent devioe of great use in 
- n-inv cases. They are made in seotions of which enough oan be gradually united to reaoh any 
equired depth. They avoid the danger of bending which attends striking the top. The iron feet are 
welled outwardly a little to ditniuUh earth-friction against the pile above them. 


t 



IxijlG 




















613 


FOUNDATIONS. 


The pipe p 


the removal of which requires men to enter the cylinder to its foot; which they cannot do in the rarefied 
air. The pipep should be of sufficient diam to allow the air to leave the cyliuder rapidly, so that the 
outer pressure may act upon the top as suddenly as possible. 

At the Goodwin Sands light-house, England, hollow cylinders ft in diam, were sunk 34 ft into 
sand by this process, iu about 6 hours; where a steel bar could be driveu only 8 ft by a sledge-ham¬ 
mer Others, 12 ins in diam, have been sunk 16 ft into sand within less than an hour. In this last 
iustance the air-pump had two barrels, 434 ius diam, 16 iuch stroke, worked by 4 men. 
was of lead, and only inch diam. 

The plenum process, invented by Mr Triper, 
of France, consists in forcing air into the cylinder 
C C, Fig 17, to such an extent as to force out the 
water, compelling it to escape beneath the open foot, 
into the surrounding water. The interior of the cylin¬ 
der being thus left dry to the bottom, men pass down it 
to loosen and remove the soil at and below its base. When 
this is done, they leave; the compressed air is allowed to 
escape; and the cylinder, being no longer sustained by 
the upward pressure of the compressed air beneath its 
top, sinks into the cavity, or the loosened material at its 
foot. Fig 17 shows the simple arrangement by which 
workmen are enabled to enter or leave the cylinder, 
without allowing the compressed air to escape; as well 
as the general principle of the entire process. 

L b is a separate small chamber, the air-lock, which is 
removed when a new length of pipe is to be added ; and afterward 
replaced and firmly bolted on. This chamber has a small air-tight 
door d, by which it can be entered from without; and another, o, 
opening into the cylinder. The flaps, (, h, of both doors, open in¬ 
ward, or toward the cylinder. This chamber also has two stopcocks; one, a, in its floor, communi 



eating with the cylinder; and one e. above, communicating with the open air. At s is a bent tube 
also with acock, which passes air-tight through the side aud the bottom of the air lock. Throng! 


it the compressed air is forced into the cylinder, by an air force pump or condenser: aud through i 
the same air is allowed to escape at a later period. A siphon is shown at n 7171. A drum w is usee 
for hoisting the excavated material from the bottom, to the air-lock ; its axle ii passes air-tight through 
stuffing-boxes iu the sides of the lock ; the hoisting being done by men outside. This is the genera 


k 


_ „_ip_ _,_j genera 

arrangement employed by Mr W. J. McAlpine, CE, of New York, at Harlem bridge; and from his 
description of it, ours has been condensed. The cylinders were there 6 ft diam, 1 ins thick, and it 
lengths of 9 ft, bolted together through inside flanges/, as the sinking went on. The air-lcck is 6 f 
diam, by nearly 6 ft high ; with sides of boiler iron; and top and bottom of cast iron. 

Now suppose the cylinder C C to be let dow n, and steadied in position, as in the fig; and the air 
lock b I, to be adjusted ou top of it. The next process is to force in air through the curved tube s 
the flap t of the lower door o t and the cock a. being previously closed. As the compressed air accu 
mulates in the cylinder, it forces out the water; which escapes partly heueath the bottom of the cvl " 
inder, aud partly by ri.siug through the siphon tin, and flowing out at q. The door o being alrea’dj '' 
closed, and that at d open, the air in the air-lock is in the same condition as that outside; so tha 
workmen can enter it readily. Having done so, they close the door d, and the cock e ; and open tht Cl 
cock a , through which condensed air from the cylinder rushes upward, soon filling the air-lock 
n hen this is done, the flap t is opened, and the men descend through the door oby a ladder, or by ! 
bucket lowered by the drum to, to the bottom. Here they loosen and excavate the material as dee 
as they can; and, filling it into a bucket or bag, they signal to those outside, w r ho raise it to the ait 
lock. M hen done, they ascend to the air-lock, close the door o, and the cock a ; and open the cock e 
through which the condensed air in the lock soon escapes, leaving the internal air the same as tha 
outside. The door d is then opened, the buckets of earth are removed, and the men go out. Finalb 
the cock at * is opened, the condensed air in the cylinder escapes through it to the outside air, an! 
the cylinder sinks by its own weight into the cavity and loosened soil prepared for it at its base. an< 
which is now forced up into the cylinder by the rush of the returning water. The process is lliei 
repeated. The sinking will often vary from 0 to 10 or more feet at oue operation. Until depths o 
40 or 50 ft, most meu can endure the pressure of the condensed air ; but as the depth iucreases thi 
becomes more difficult, and positively dangerous to life. Cast-iron cylinders 15 ft diam; and greu 
caissons, Eig 18, have been thus sunk; but at times at great expense and trouble. 

Tl»e cylinder should be guided in its descent by a strong frame, whicl 

may be supported by piles. Otherwise it will be apt to tilt, and thus give great trouble to settle 1 
upon Us exact place. Have been sunk in deep water by divers undermining inside. 

The plenum process as applied at the South St bridge, Philada 
by Mr. John W. Murphy, contracting engiueer, differs materially from that described above • am 
moreover deserves notice ou account of the great simplicity aud efficacy of his plant. This cousiste. 
partly of two canal boats, decked, each 100 ft long, by 17*^ ft wide, and 8 ft depth of hold The' 


were anchored parallel to each other. 15 ft apart. Supported by the boats, aud over the space betweei 

1 SO ft high ; at tiie top of which was attached tackle fo ™ 


them, was a strong four-legged shears about; 


handling the cast iron cylinders. In the hold of one of the boats was a lSuriei'> li 
Coni pressor having two pistons of 10 ins diam, and 9 ins stroke; together witl 

1 to h /11 1 A I* » 4- 1 1 < 1 lr , , ^ 4 1 . . 1 . 1. .. .. X ± . .. — A- * ww * ^ _ 


its boiler. On the deck of the same boat stood a vertical air-tank or regulator 
>y 2 ft diam. made of quarter inch boiler iron. This served to maintain a supplv of com 


22 ft long, by * ii, uiam, rnaae or quarter men boiler iron. This served to maintain a supply of com 
pressed air in the submerged cylinder in case of an accidental stopping of the compressor - whic 
otherwise would probably be fatal to the laborers in the cylinder. The condensed air flowed fror 
this air-tank to the air-lock of the cylinder through a hose 4 ins diam, made of gum elastic and can 
vas, and so long, and so placed, as to extend itself as the cylinder went down, thus maintaining th 
communication at all times. Entirely across both boats, and across the interval between them ex 
tended two heavy wooden clamps, each 3 ft wide by 18 ins high; each compose. 






















•f three pieces of 12 X 18 Inch timber strongly bolted together. At the centers of these clamps the 
wo inner vertical sides which faced each other were hollowed out to the depth of a foot by concavi- 
les corresponding to the curve of the cylinders. The distance apart of the clamps was regulated by 
wo strong iron rods, having screws and nuts at their ends for that purpose. Thus when a section 
fa cylinder was hoisted by means of the shears into its position over the space between the two 
oats, the two concavities of the clamps were brought into coutact with it. and the nuts being then 
crewed up, the cylinder was firmly held in place by the clamps. The shears could then be used to 
aise another section of the cylinder to its place upon the first one, that the two might be bolted to¬ 
other. By repeating this process the height of the cylinder would soon become too great to allow 
le shears to place another section upon it; in which case the nuts of the screws were slightly 
oosened, and the cylinder was allowed to slip down slowly into the water until its top was but a 
ttle above the surface. The screws were then again tightened, and the cylinder again held fast 
util other sections were added and bolted to it. When there was danger that the upward pressure 
the condensed air might lift a cylinder, the clamps were raised by the shears clear of the boats: 
len tightened to the cylinder, and a platform of planks laid upon them, and loaded with stone. 

1 lie air-lock was so arranged as not to require to be removed when a new sec- 
j on was to be bolted on. This was effected as follows. Sections of the cylinder were bolted together 
the manner just described, until its foot rested on the bottom, with its top a few feet above high 
| ater. A heavy cast iron diaphragm \% inches thick, to form the floor of the 
.f r-lock, was then placed on top Then was added another 10 ft high section of the cylinder, to form 
1 e , oh f m , ber of^e air-lock. These were bolted together ; and then another diaphragm was added 
I top to form the roof of the air-lock. These diaphragms were furnished with openings, and with 
|,.ors and valves corresponding with those shown in Fig 17. and remained permanentlv in tha 

f 1 nders when the work was finished. If the depth of soil to be passed through before reaching 
ck is so great as to require other sections of cylinder to be bolted on above the top of the air-lock 
\ is may be done to any extent, inasmuch as it is immaterial whether the air-lock is under water or 

)t. To keep the cylinder both air- and water-tight the faces of 

e flanges before being bolted together were smeared with a mixture of red and white lead and cot- 
a fiber. 

At the South St bridge the cylinders were 4, 6, and 8 ft diam ; in lengths 

sections 10 ft long. They were all 1^ inch thick. Inside flanges 2% ins wide, IV thick, with bolt- 
les 1 % inch dtam, by 5 ins apart from center to center. The bottom edge has no flange. A 10 ft 
:tion of an 8 ft cylinder weighs 14600 tbs; of a 6 ft one, 10800; of a 4 ft one, 6800. An 8 ft dia- 
jf ra S r ?' 2800 tts ; 6 ft, 1600 ; 4 ft, 783. The rock under the soil was quite uneven in places ; but was 
, £ lled off a * the cylinders went down. These were then bolted to it by cast iron brackets, 
he work went on. day and night, summer and winter: with no inter¬ 
s' 11011 from the tides, floods, or floating ice; and the thirteen columns were sunk, filled with con- 
,te L a u com P‘ eted ln I* months ; much of which was consumed in levelling off the rock, and bolt- 
£ the brackets. The want of guides caused much tilting, trouble, and delay. 

tise and fall of tide about 7 ft. Greatest depth of soil, gravel, &c, passed through, 30 ft: least 6 
Depth of water about 25 ft. The work was under charge of John Anderson, a very skillful and 
ergytic superintendent of such matters. The entire neat cost of the cvlin- 

it ’ 8 > n Pl ace . and filled with hydraulic concrete, was approximately $92 per foot of total length 
the 8 ft ones , $64 for the 6 ft; and $40 for the 4 ft diums. There were three gangs of workmen * 
1 each gang worked 4 hours at a time. See a full and very instructive description with eiigrav- 

•<} hr n VC Clnn ffh- QainAnintAfi/tln^ 1 !*»«*>... 0 .1 . 1 T ... . . . . n . .. _ 

V 

tiding ^ _ _ _ _ _ _ ^ ^ 

>ught-iron, 8 ft diam, 66 ft long, at an angle of 45° with the hor, intended as struts to prevent the 
vement of one of the abut piers of Chestuut St bridge, Phila. 

}ast iron cylinders have cracked through, around their entire circum- 
:nce, in many parts of the U. S. in very cold weather; owing to the diff of contraction of the 
a, and of the concrete filling. Ignorant use of them may be attended by great danger. 



The shaded part of Fig 18 shows a transverse section of the caisson of* yellow- 
: ne timber and cement, for the Brooklyn tower of Bast River (N Y) 
" ipension bridge, of 1600 ft clear span. It is 168 ft long at bottom, and 102 ft wide, 
ongitudinal section resembles the transverse one, except in being longer, and in 
•wing more shafts J. Of these there are 6, arranged in pairs, for expedition and as 
recaution against accident. Namely, two water-shafts J, each 7 ft by 6^£ ft across, 
H removing by buckets and hoisting apparatus, the material excavated beneath the 

caisson; together with such 
water as may accumulate at 
o o; two air-shafts of 21 ins 
diam, through which air is 
forced from above, to expel 
the water from the chamber 
C S S D below the caisson, so 
as to allow the laborers to 
work there at undermining; 
the expelled water escaping 
under the foot C D of the cais¬ 
son, into the river; and two 
supply shafts of 42 ins diam, 
for admitting laborers, tools, 

, The several shafts of course have air-chambers on top, on the same principle as 
1 17, to prevent the escape of the compressed air in s s, 

« 

a 























riOU 


i‘UL J.1JA 11UHJ, 


Tbe shafts are of *4 inch boiler iron. Tbe foot C D, nine timbers high, is continuous, extend 
entirely around the caisson ; its bottom is shod with cast iron ; its four corners are strengthened 
wooden knees 20 ft long. 

From the bottom, up to the line N, N, 14 ft. the caisson is built of horizontal layers of timbers 
foot square; the layers crossing each other at right angles; and the timbers of each layer touch' 
each other well forced and bolted together; and all the joints tilled with pitch. To aid in prevent 
leakage, the nuts and heads of the screws have India-rubber washers; also all outside seams, as 
as all the seams of the layer of timbers N, N, are thoroughly calked: and a layer of tin, encbi; 
between two layers of felt, is placed outside of each outer joint; and over the entire top of the h 1 
next below N, if. 

When the.caisson was built up to N, N. on land, it was launched, floated into position, and anchoi^ 
after which were added for sinking it. fifleeu courses of timbers one ft square; and laid one ft a;] 
In the clear; with the intervals tilled with concrete. The top course A B is of solid timber, to s< 1 
as a floor for supporting machinery, &c. It was sunk some feet beiow the very bottom ot ' 
river, in order to avoid the teredo. 

Cribs are sunk outside of the eaisson. to form temporary wharves for boats carrying away excav 
material ; and for vessels bringing stone, Ac. 

When the eaisson was smnk, and the water forced out from the chamber or space CSS D.work | 
began to excavate uniformly the enclosed area of river bottom, so as to allow the cnisson to des< 
slowly unti>it reached a Arm substratum. The space C 8 S D, as well as the shafts, was then fillei 
solid with concrete masonry. A coffer-dam was built on top of the caisson ; and in it the rep 
masonry of the tower was started. The total height of this tower including the caisson, is about 
ft. For full details see report. 1873, of W. A. Roebliug the chief engineer. 


Hollow cylinders, or other forms of brickwork or n 
sonry, with a strong curb or open ring of timber or iron beneath them, may 
gradually sunk by uuderminiug and excavating from the inside; and form very stable foundati] 
Under water this may be done by properly shaped scoops, with or without the aid of the diviug 
according to the depth. &e. On land it will often be the most economical and satisfactory m 
especially in firm soils. The descent may be assisted by loading them, if, as sometimes happuus 
friction of their sides against the earth outside prevents their siukiug by their own weight. A t 
cylinder, 46 ft outer diam, walls 3 ft thick, has been sunk 40 ft in dry s’aud and gravel, without 
difficulty. It was built 18 ft high, (on a wooden curb 21 ins thick,) and weighed 300 tons befort 
siukiug was begun. Tbe interior earth was excavated slowly, so that the siukiug was about 1 fi| 
day ; the walks being built up as it sank. Tunnel shafts are at times so sunk. 

On the Rhine for a coal shaft, a brick cylinder 25b£ feet diam was first tj 

sunk by its own weight 76 ft through sand and gravel; then an interior one, 15 ft diam, was suti 
the same way to the depth of 256 ft below the surface; of which depth all the 180 ft below the i 
cylinder was a running quicksand. At 256 ft friction rendered the cylinder immovable. The qi il 
sand was removed by boriug; no pumping was done; but the water was permitted to keep the cy 1 ■* 
The entire fbandatioo for a Targe pier of masonry has been sunk in this manner, in a single m 1 
a sufficient number of vertical openings being left in it for the workmeu to desceud, or for tools ik 
inserted for undermining. This is generally a very slow and tedious operation, especially u L 
water. It may often be expedited by diving-bells or by diving-dresses. It will generally be bett j 
make the mass wider at bottom than above it, so as to diminish friction against the outside err! 
On land, water may at times be used for softening the bottom earth. By keeping the interior or ' 
hollow masonry dry, it may even be built downward from the surface ; by undermining ouly a u 
tion of its circumference at a time, filling said portion with masonry, and then removing and ti 
the other portion ; and so on in successive stages of 2 or 3 ft downward at a time. This mode m:, 
adopted also when friction has stopped the sinking of a mass by Us own weight when undermiu 

Tile Hand pump as used at the St Louis bridge will often be of service iu if 
ing sand from cylinders while being sunk in water. With a pump pipe of 3.5 ins bore, and a v 0 
jet under a pressure of 150 lbs per sq inch, 20 cub yds of sand per hour were raised 125 feet. A ’ | 
air has als® been successfully used in the same way, as at the East River, N Y, suspension bridg' | 

Faseine**. On marshy or wet quicksand bottoms, foundations may be lai o 
first depositing large areas of layers of fascines, or stout twigs and small branc n 
strongly tied together in bundles from C to 12 ft long, and from 6 ins to 2 ft in d j 

The layers or strata of bundles should cross each other. A kind or floating raft or large maid 
is first made of these, and then sunk to the bottom by being loaded with earth, gravel, stone.-\\ 
In this mauner the abutments and piers of the great suspension bridge at Kieff. in Russia, with > „ 
of 440 ft. were founded in 1862. on a shifting quicksand. There the fascine mattresses extend 
beyond the bases of the masonry which rests upon them. 

Fascines may be used in the same way for sustaining railway embankments. Stc, over mi s 
ground, but they will settlte considerably. 

Sand-piles. We have already alluded to the use of sand well rammed in li. 
Into trenches or foundation pits; but it may also be used in soft soils, in the 61 '# 
of piles. A short stout wooden pile is first driven 5 to 10 feet or more, accordii “ 
the case. It is then drawn out, and the hole is filled with wet sand well ram ; 
The pile is then again driven in another place, and the process repeated. The in«J 
Yals may be from 1 to 3 ft iu the clear. Platforms may he used on these piles f t 
wooden ones. If the sand is not put in wet, it will he in danger of afterward : 
ing from rain or spring water. In this case, as with fascines, it is well to tes '! 
foundation by means of tidal loads. Some settlement must inevitably take ].„ 
until all the parts come to a full bearing: but it will be comparatively trifling. 
same occurs in every large work to some extent; as in a roof or arch of great s ^ 
whether of wood, iron, or masonry ; so also with all tall piers, walls, Ac, &c. S i " 
foundations under water should be surrounded by stout well-driven sheet-pilin ’! 
prevent the enclosed sand from running out in case the outer sand is washed a 
and should also be defended by a deposit of random-stone. See Sand-piles, p ( t 


■n 








JVUUBAimi.LLIA It. 


TTTT 


l On bad bottoms under water, small artificial islands of good soil have 

en deposited; and the masonry founded upon them. Canal locks and other structures may at 
0 nes be advantageously founded in this way in marshy soils. If necessary, a depth of several feet 
ij the bad soil may be dredged out before the firmer soil is deposited; and the latter may be weighted 
a trial load to test its stability. 

#■ The mode of layiug a foundation under water, by building the masonry upon a timber platform 
>sK>ve water, upheld by strung screws, and lowered into the water as the work 
n finished in the open air. a course or two at a time, has of late been much employed with entire 
jcess, in large bridge-piers in deep water. It however is not new. It was suggested more than 
•() years ago by Belidor. 

P Piles are driven 6 to 10 ft apart around the space to be occupied by the pier; having their tops con- 
?i cted by heavy timber cap-pieces. These last uphold the screws, which work through them. The 
{tole is braced against lateral motion. 

a A clump OF piles well driven ; and then enclosed by an iron cylinder sunk to a 
nt bearing, and filled with concrete, is an excellent foundation. The piles may 
n tend to the top of the cylinder, and thus be enclosed in the concrete. Sucli an 
( rangement has been patented by S. B. Cushing, C. K., Providence, R. I. The cyl- 
der and concrete serve to protect the piles from sea-worms, and from decay above 
w water; and are not intended to support the load above them. 

Cost of a diving 1 outfit, with two dresses, air-pump and tubes, about $750, 
1 • $000 with cheaper pump. Alfred Hale & Co, 30 School St, Boston, Mass. 

Two men can work the air-pump to 50 ft depth. 

b 

0 


I[ 

STONEWORK. 


I Where work is done on a large scale, blasting can sometimes be done at from 10 
1 20 per cent less cost per cubic yard by means of machine drills and 
|ynamite, than by hand drills and gunpowder. Ordinarily, how- 
j-er, the cost is about the same, and the advantage of the newer 
< ethods consists rather in economy of time, convenience, and having the work more 
J itirely under control. In ordinary railroad work in average hard rock, and when 
; >mmon labor costs $1 per day of ten hours, the cost per cubic yard, for loosening, 
'ill ordinarily range between 30 and 60 cts, including tools, drilling, powder, &c; 
1 1 erage 45 cts. 

: Holes for blasting, drilled by hand, are generally from 2 y 2 to 4 ft 

,;ep; and from 134 to 2 ins diam. Churn-drilling is much more expeditious 
id economical than that by jumping, mentioned below. The churn-drill is merely 
round iron bar, usually about 1J4 ins diam, and 6 to 8 ft long; with a steel cutting 
Ige, or bit, (weighing about a lb, and a little wider than the diam of the bar,) welded 
ii its lower end. A man lifts it a few inches; or rather catches it as it rebounds, 
irns it partially around; and lets it fall again. By this means he drills from 5 to 
> feet of hole, nearly 2 ins diam, in a day of 10 working hours, depending on the 
laracter of the rock. Prom 7 to 8 ft of holes ins diam, is about a fair day’s 
ork in hard gneiss, granite, or compact siliceous limestone; 5 to7 ft in tough cmn- 
ict hornblende; 3 to 5 in solid quartz; 8 to 9 in ordinary marble or limestone; 9 to 
) in sandstone; which, however, may vary within all these limifs. When the hole 
more than about 4 ft deep, two men are put to the drill. Artesian, and oil wells, 
i rock, are bored on the principle of the churn-drill. See also diamond drill, p 652. 


*The jumper, as now used, is much shorter than the churn-drill. One man (the holder} sitting 
iwn, lifts it slightly, and turns it partly around, during the intervals between the blows from about 
to 12 lb hammers, wielded by two other laborers, the strikers. It can be used for holes of smaller 
ameters than can be made by the churn-drill; because the bolder can more readily keep the cutting 
td at the exact spot required to be drilled. It is also better in conglomerate rock ; the hard siliceous 
,-bbies of which deflect the churn-drill from its vertical direction, so that the hole becomes crooked, 
id the tool becomes bound in it. The coal conglomerates are by no means hard to drill with a 
in per. The jumper was formerly used for large deep holes also, before the superiority of the churn- 
dll became established. . . m . , . . .. 

Either tool requires resharpening at about each 6 to 18 inches depth of hole; and the wear of the 
I eel edge requires a new one to be put on every 2 to 4 days. With iron jumpers, the top also be- 
.mes battered away rapidly. As the hole becomes deeper, longer drills are frequently used than at 
le beginning. The smaller the diameter of the bole, the greater depth can be drilled in a given 
me ; and the depth will be greater in proportion than the decrease of diam. Under similar circuw- 
anc’es, three laborers with a jumper will about average as much depth as one with a churn-drill. 
The hand-drill, in which the same man uses botli the hammer and the short drill, is chiefly used 
, r shallow holes of small diam. With it a fair workman will drill about as many feet of hole from 
to 12 ins deep, and about inch diam, as oue with a churn-drill can do in holes about3 ft deep, and 
ius diam. in the same time, Only the jumper or the hand-drill can be used for boring holes which 
e horizontal, or much inclined. 







MACHINE ROCK-DRILLS. 


Art. 1. Machine Rock-drills bore much more rapidly than hand dril 

and more economically, provided the work is so great as to justify the prelimim 
outfit. They drill in any direction, and can often be used in boring holes so loca 
that they could not be bored by hand. They are worked either by steam direct 
or by air, compressed by steam or water power into a tank called a “ receiver,” fL 
thence led to the drills through iron pipes. The air is best for tunnels and sha 
because, after leaving the drills, it aids ventilation. 

Art. 2. Such drills are of Iwo kinds: rotating drills hi 
percussion drills. In the former, the drill-rod is a long tube, revolving ab< 
its axis. The end of this tube, hardened so as to form an annular cutting-edge 
kept in contact with the rock, and, by its rotation, cuts in it a cylindrical hole, g „ 
erally with a solid core in the center. The core occupies the core-barrel. Art 
The drill-rod is fed forward, or into the hole, as the drilling proceeds. The del u 
is removed from the hole by a constant stream of water, which is led to the bott L 
of the hole through the hollow drill-rod, and which carries the debris up tlirouL 
the narrow' space between the outside of the drill-rod and the sides of the hole, f 

In percussion drills, the drill-rod is solid, and its action is that of iL 
churn drill, p 651. 

Art. 3. In the Brandt (European) rotary drill, the cutting-edge at 1, 
end ot the tubular drill-rod is armed with hardened steel teeth. It is pressed u^ai |i 
the rock under enormous hydraulic pressure, and makes but from 5 to 8 revoluti.L 
per minute. F 

Art. 4. The Diamond drill is the only form of rotary rock-drill ext ! ‘ 
si vely used in America. In it, the boring-rod consists of a number of tubes join | 
rigidly together at their ends by hollow interior sleeves 


b is „ ca,led a “core-bit.” Its cutting-eiL 


v lit -, » * uuitf-uiu cuiting-e< 

has imbedded in it a number of diamonds as shown. These are so arranged 


f°rF l K° J . eCt 8 lg lt - v fr 5* In botl ‘ lts lnner and outer edges. Annular spaces are tl lt 
left between core and core-barrel, and betw een the latter and the walls of the Ik V 
Ihese spaces permit the ingress and egress of the water used in removing the deb. 
Irom the hole and, at the same time, prevent the core from binding in the barrel 
the latter in the hole. ° 


Fig. 1 


Fig. 3 



CORK B1T7 



CORK LIFTKR. 



BORING RKA 


J “ 8t above tbe core-bit,” the 44 core-lifter,” Fig 2, is screwed 
t e barrel. This is a tube about 8 ins long and of the same outer diam as 
barrel. Inside it is slightly coned, with the base of the cone upward and i 
nished with a loose split-r.ng, II, toothed inside, and similarly coiied VhHe 

oiVterwfvHtfder ? hnl * at ! d -,nai„S ToosS ! 


outer cylinder; but when the drilling is stopped, a^d the drin-Jod b^gin^To'l 
Weled <■'' 


beveled shape is pressed hard against tf,e cfreofrock, which Is pulIeKartcl 
to its foot by the power which lifts the drill-rod. p 

Art. 7. This power is supplied by a rope-drum, fastened to the ton of t 

driir r oTl ‘‘Th^rone f S th « dri i' and worked h - v the 8ar »e engine which rotates i 
’ a P the drum P a8ses »P to a pulley at the top of a tfcrrir 

and thence down to the upper end of the drill-rod. The considerable heiirht of i 
demek enables from *Mo 50 feet of the drilbrod to Is, removS , f n one p fee 
Art. 8. Above the “ core- liter ” is tlie 44 corp-hurvoi »» mu ‘ ' 

iron tube from 8 to 16 ft long. core-Parrel.” This is a wroug 

grooved outside, to permit the ascent of the water and debris from the iVii f 1P i” 
sometimes set with diamonds on it, outer surface, to preVenJ SSS The hi! Uf 
and barrel are of uniform outer diam, a little less than Hin.„ /<• ,, e , ,’^ 

S r bar a rel 0f fr ° m ab ° Ut ^ ins for 2-inch barrel to 5^ insfor 














MACHINE ROCK-DRILLS. 


653 


* f ?.Herfo.- a ,«d will. h»l« whie “an™,h« to pi‘o“!"Lm 

e„ce. ° ’ dIU “ arU ‘ ed Wlth dla,1,onds ’ sorne of which project beyond its circum- 


Art. 9. 

end,” 


■ence. 

rSmS" ThV^»if,.VT 0,, ; 0S , a ‘ » s e < “‘■ ,, of f "> m 200 l “ 4 »» revolutions 
rs either fixed or?>srfnat-n’ wh ’ c '. '! ls rot »ted, consists usually of two cylin- 
ri^ht ^gl^^eadi 5 other ^ Rv e m lted b V tean l or ‘““Pressed air, and working 

Ihtvdrauiic plston^of 

S '" ics a,Ml in a, *y direclion, to great depth"; from jB*ol?00*fe£ 
n^ not uncommon. This, with the fact that it brings u unbroki 

WhiCh , 8h ‘n the P,edse " a *™ ” ld "Stiflcatio”tSf" 
p et ated, remieis it very valuable m test-boring.'prospecting of mines &c 

Ehinwllk 4° f sufficient size to boreholes from 6 to 15 ins diam’, for 
1 ,an w ® 1,s * Th , e roundness of the holes bored enables the use of casing of 

» » p“mSh M £"e“ 5S* ““ ‘' 0le; Eb “ ir strai « htness '• atlvnutageo,., in 

*JbJ*nM?e‘r h “ 7^ r ° Ck a . bit drill through 200 ft or more without resetting. 

lsS-d h „' A, a T k f’ 8 i!? ilar dri,ls wil1 wear 0111 hi 10 ft or less. 

d d 7 n by the , Am " Diamond Rock Boring Co, weighing com- 
te about 1400 fts. and costing about $2800, bored, in 1428 hours of actual boring 
doles of 2 ins diam, and aggregating 9141 lineal ft. Average length of hole 172”5 

A Qfi r/1?* r at r?’ 6 nu ft , per hour; « reatest - 12.8. Average total cost, 
ut 96 as per in ft. The rock was principally limestone, wirli some quartz and 

dstone. Ihe holes were bored at angles varying from 0° to 45° with the vertical 
lS H rt : l, g h average we may say that in ordinary rocks, as granite, lime* 
ne. and bard sandstone, these drills will bore deep holes, 2 to 3 ins diam, at from 
o 2 it per liour. and at a cost of from $1 to $2 per ft. 
et Tbcse drills are made of many widely different sizes, and with 

t Iferent mountings, depending upon the nature of the work to be done 
‘ They are made by the Petlna Diamond Drill Co, Pottsville, Pa; American 
mond Rock-Boring Co, office 15 Cortlandt St. New York; M. C. Bullock Mfg Co, 
cago, Ill, and others. These companies usually contract to do the drilling them’ 
/es. They also sell the machines, generally under restrictions as to the location 
extent of the territory in which they are to be used. The prices depend, 

\ great^ extent, upon the nature of these restrictions The caid prices for some 


Diam 
of hole. 

Greatest 
length of 
hole. 

Weight 

of 

machine. 

Card price, 
1888. 

ins. 

ft. 

lbs. 

$ 

i K 

250 

400 

1500 

2 

500 

1000 

2000 

2 

2000 

3500 

4000 

4 

2000 

6200 

6000 


rt. 14. In percussion drilling- machines, the drill-bar is driven 
ibly against the rock by the pressure of steam or of compressed 
', acting upon a piston, P, Fig 4, moving in a cylinder, C C, Figs 4 and 5; and 
:es about 300 strokes per minute. The rotation of the drill-bar is accomplished 
'matlcally, as explained in Art 27. 

>. The cylinder, C C, is free to slide longitudinally in the fixed 
ell, S S, Fig 5, to which it is attached, and which, in turn, is fixed to the 


rt. 15. 

le or she 

id or other stand (see Arts 18 and 19) upon which the machine is supported, 
rt. 16. The drill-rod, R, corresponding to the churn drill, p 651, is 
*ned, by an appropriate chuck, K. to the end of the piston-rod, 0. The drilling 
ra egun with a short drill-rod, and with the cylinder as far from the hole as the 
th of the shell, 3, will permit. As the bit penetrates the rock, the cylinder is 
forward,* either automatically or by hand (see Art 28), as far as the length of 


!v forward, or downward, we mean toward the hole which is being drilled. B\ buek- 
d, or upward, from the hole. 















654 


MACHINE ROCK-DRILLS. 


the shell permits. The drilling is then stopped, by shutting off the steam,* and tl 
cylin ler is run back, by reversing the motion of the feeding apparatus. The she 
drill-bar is then removed, and, if the drilling is to be continued, a longer one is su 
stituted in its place, and the process repeated. 

Art. 17. Inasmuch as the act of drilling wears the edges of the bit, thus rede 
ing its diam somewhat, tlie hole will of course be tapering-, or i 
slightly less diam at bottom thau at top. The second Lit. must therefore be < 
slightly less diam than the first; say from fa to y inch less; the third must he hi 
than the second, and so on. On the other hand, in long holes, the drill-bar w 
seldom be in a perfectly straight line, so that the bit, instead of striking always 
the same spot, will describe a circle, and thus enlarge the bole. 

Art. IS. The shell, S, in which the cylinder slides, is provided with an arrant 

i'll it JTLiV 1 rluninml a no in L'i.w ^.. .. l... 


n ii«tiiuw cut, or against me noor ana ceiling or a tunnel-Heading, Ac, in which c« 
one of its ends is provided with a screw which is run out so as to cause the two en 
of the col to press firmly against the opposite rock walls; or rather against wood 
blocks which are always placed between each end of the col and the rock. In a 
case, the supports of the drill are so jointed that it can bore in any direction. 

Art. 19. Frequently the drill is clamped to a short arm, which, 
turn, is clamped to the column, and projects at right angles from it. The arm m 
be slid lengthwise of the column, and may be revolved around it, and thus may 
placed in any desired position, and there clamped. This gives the drill a great 
range ot motion, and enables it to bore holes over a greater space than would othi 
wise be possible without moving the column. 

_Art. 20. In tunnels, one or more drills may be mounted upon a (Irill-ca 
riag-e, travelling upon a railroad track running longitudinally of the tunn 
Upon this track the carriage is moved up to the work, or run back from it when 
blast is to be fired. The gauge of the track may be made wide enough to admit 
a second track, of narrower gauge, running underneath the drill-carriage. Up 
said narrower track the cars are run which carry away the debris. Drill-carriag 
are less commonly used in this country than in Europe. 

'file pressure used in the cylinders of percussion drills 
usually from about bO to 70 lbs per sq inch. In ail hour, one will dri 
a hole h orn 1 to 2 ins diam, and from 3 to 10 ft deep, depending on the character 
the rock and the size of the machine at from 10 to 25 cts per lin ft with labor 
$1 per day. A hit requires sharpening at about every 2 to 4 ft depth 
hole. One blacksmith and helper can sharpen drills for 5 or 6 machines 
Art. 22. The bits are of many different shapes, varying wi 
the natiue of the work to he done. For uniform hard rock, the bit lias two cuttii 
edges forming ;i cross with equal arms at right angles to each other. For seat 
rock, the arms of the cross are equal, but form two acute and two obtuse angles w 

shape^fYheletter Z X ' r ° r ®° ft rock > the cutting-edge sometimes has t 

Art. 23. Each drill requires one man to operate it. Two or three m 
are required for moving the heavier sizes from place to place. One man can at™ 
to a small air-compressor and its boiler 

I”. ssvssr" ““ d - Fls4 *• “ ' ongitudinai 8ectiul ‘ th '““ 

Art. 25. The cylinder, C, is provided at each end with a rubber cusliio' 
N, foi deadening the blows of the piston, which, in all percussion drills is liable 
times, to strike either cylinder-head. The side of each cushion nearest the piston 
pi effected^ a thni mmi plate. The cushions have to be renewed from time to tin 
A Ilie valve, is shaped somewhat like a spool The bolt 

passes loosely through its center and guides it. Steam is admitted'from the hoi 

v a |\! e ex e cent C i ,eS It il drt™ T” “i" °* n® SpaC . e between th « two end flanges of t 
' e , except t It duves the valve alternately from one end of the valve-chest 

the other, and back, according as one end or the other is relieved from oimosi 

sages' 1 1) I?’ and FF' 1 " 0 ] lll T , !i unication .' vith exhaust. E, by way of {lie p 
sa^es, l> u and IF. I) and b comnniuicafe with the ends of the steam-cln 

through passages not shown; while F and F'communicate through sim lai n 
wf S ; WUk the exhaust, E The piston has an annular channel, L L>nciicling 
hatLvci the position ot the ]>iston,one of the passages. D or D' is always by men 
of tins c hannel, in com munication with its corresponding passage, F or F'Jeadi 

“* ““ u .1 S ,1,, eU119 , , Bom „ 


■■■ 













MACHINE ROCK-DRILLS. 


655 


tli> the exhaust. Thus, one or the other end of the valve-chest is always in coni* 
hoi unication with the open air; and to that end the valve is driven L»y the pres of 
suite steam surrounding it, admitting steam to the cyl, 0, from the other end. 

Art. 2/. I lie rotation of tlie piston, and, with it, that of the drill¬ 
in' 11 '’ is effected thus: The spirally-grooved, cylindrical steel bar, A, called a rifle- 
roar, passes through and works in, the rifle-nut, II, which is firmly fixed in 
■ je end of the piston and has spiral grooves corresponding with those on the rifle- 
|«ir. Said bar is fixed, at its upper end, to the ratchet-wheel, J, the pawls of which 
i e so arranged that, on the down stroke of the piston, the rifle-nut, H, acting upon 
i 16 gloves on the rifle-bar, causes it, and, with it, the ratchet-wheel, to revolve 

about their common axis. The weight and mo¬ 
mentum of the piston, &c, are such that it thus 
readily turns the ratchet-wheel without itself 
turning. Thus the bit is prevented from rotating 
while delivering its blow. Hut, on the up stroke, 
the tendency of the rifle-nut is to turn the rifle- 
bar and ratchet-wheel in the opposite direction; 
and as this is prevented by the pawls, the rifle- 
bur remains stationary, while the piston, piston- 
rod, and drill are made to revolve about their 
common axis. 

Art. 28. The feed-screw, M, is col¬ 
lared, at its upper end, to the fixed frame, Q. It 
is thus prevented from moving longitudinally 
when revolved by means of the crank fixed to its 
top. Its lower end works in a nut, T, fixed to the 
cylinder, w'hich last is thus moved longitudinally 
backward or forward as the crank is turned. 



Fig. 5. 



rge drills are frequently furnished with an automatic feeding arrange- 
ient in addition to the hand-crank. In this arrangement, when the cylinder 
juires feeding forward, and when, consequently, the piston is running nearly to 







































































656 


MACHINE ROCK-DRILLS 


the forward limit of its stroke, the piston presses against a cam projecting into tht 
cyl near the forward end, and presenting an inclined plane to it. The motion of 
tiiis cam, by means of an exterior axle, running alongside of the cyl and furnisher 
at its top with a dog, turns a ratchet-wheel fixed to the feed-screw. When desired 
the automatic feed may be thrown out of gear, and the feed moved by hand. 

Art. 29. The tripod leg's consist of wronght-iron tubes, W W. These arc 
screwed at their upper ends into sockets, X X. At their lower ends, they receive 
the pointed and tapering steel bars, Y Y, about 2 or 3 ft long. The legs may Lx 
lengthened or shortened by turning the set-screws, Z Z, thus regulating the distance 
to which the bars, Y Y, can enter the legs. The clamps, 6 6, have L-shaped hooks 
of }/£ inch to 1 inch round Iron forged to them. On these hooks the weights. 
dd. are hung, which hold the machine down against the upward reaction of itt 
blows. 

Art. 30. The following table gives the principal dimensions of these . 
drills, with the diauis and lengths of holes to which each is adapted. 

For prices, apply to the Co at the above address. We give the card prices foi 
1888. These may be taken as giving, approximately, the present range of prices of 
percussion drills of any first-class make. Size H is used for submarine work, heavy t 
tunneling, and deep rock cutting. G and F for tunneling, street grading, quarrying I 
and sewer work. E, D. and C for general mining purposes. B is adapted only foi I 
very light work. In asking for estimates on drills and compressors, give the fulles I 
possible description (accompanied by a sketch) of the work to he done, stating it: I 
present, and proposed extent. State whether the work is on the surface or under " 
ground. State how far the steam or compressed air will he carried. Give depth ol « 
holes to he drilled, nat-nre of rock, Ac. Percussion drills are sold without restrictioi 
as to the purpose or extent to which they are to he used. 

FIST OF INGEItSOLL “ECLIPSE” PERCUSSION ROCK. Jf 

DRILLING MACHINES. 




Letter designating the size of the machine. 

A 

R 

c 

I> 

E 

F 

G 

II 

Inner diam of cylinder.ins. 

\% 

2K 

2% 

3 

3Ji 

3K 


5 

Length of full stroke. “ 

3 

4 

5 

6 

6 

6 K 

Y 

7 f 

“ tted. 

12 

20 

24 

24 

24 

26 

34 

34 1 

“ machine*. “ 

36 

34 

36 

40 

42 

53 

60 

60 i 

Wtof machine, unmouuted....lbs. 

80 

155 

1!)5 

230 

250 

345 

605 

670 

“ tripod, without the wts. “ i 


125 

125 

125 

125 

150 

275 

275 

“ three wts for tripod legs. “ > 

l \ 

250 

250 

250 

250 

350 

400 

400 

“ column, arm and clamp. “ ) 

} 

200 

280 

280 

2 S 0 

420 

420 

420 

Plant of hole drilled.ins. 

K to y. 

% to \\i 

1 to 2 

1 to 2 

1 to 2 

IK to 2 }$ 

2 to 4 

3 to ' 

Maximum depth of vertt hole.ft. 

t 

4 

8 

10 

12 

16 

30 

40 1 

Prices. 


$ 

s 

$ 

$ 

$ 

$ 

* 

Machine, unmount'd,witho'tdrills. ) 


230 

255 

280 

300 

330 

375 

495 

Set of drills for above depth of hole. > 

l \ 

8 

18 

24 

31 

49 

139 

400 

Tripod, with weights. ) 

l 

45 

45 

45 

45 

45 

55 

53 



Column 4}4 in* diam. 

Column 6 ins dinrn. 

Column, 8 ft long, with arm & clamp.. 

l 

80 

80 

80 

j 80" 

"^110 

| 110 

| 110 


* From top of hand’e of feed-crank to lower end of piston nt the end of the down stroke, 
t For greatest advisable length of Uor holes, deduct oue-fourth from these depths. 

J Machiue A is mouuted on a small frame. Price, so mounted, $150. Hole 18 ins deep. Drills $< 

Art. 31. Tlie drills of different makers differ chiefly In th 

methods by which the valve is operated. In some this is done, as in the Trt^et-.sol 
“ Eclipse ” drill, Art 26, by the pres of steam. In others, the valve is moved b 
a lever or tappet, which projects into the cylinder so as to come into con toe 
with, and be moved by, the piston at each stroke. As these strokes are made wit 
great force some 300 or more times per minute, such valve-gear is necessarily subjet 
to great wear. 

Art. 32. In the “ Little Criant Prill,” made by the Rand Drill Co 

office 23 Park Place, New York, the valve, V, Fig 6, is slid backward and tor wan 
m the same direction in which the piston is moving, by the tappet T which i 
pivoted at p. The inclined lower corners of this tappet ride up as they come altei 
nately, in contact with the shoulders, ss, of the piston. 

Art. 33. Tlie s:ime Co have recently, 1881, brought out two nev 
drills, the “ Economizer ” and the ‘-Slugger”: in each of which tit 
valve, as m the Ingersoll “Eclipse” drill, is moved by steam, but upon a quite di 





































MACHINE ROCK-DRILLS. 


657 





i *rent principle. In these two drills, there is no steam cushion for the piston 
) strike ugaiust on the down stroke, the force of which is thus more completely 
1 spended upon the rock. The cushion behind or above the piston, on the return 
, roke, is formed by exhaust steam. Both of these drills cut off steam before 
io completion of either stroke, thus using the steam expansively. On the down 
# roke, the “Economizer” cuts off earlier than the “Slugger.” Hence its name, 
ejt both machines the point of cut- 
11. f is fixed when the machine is 
ej ade. 

s Art. 34. In the improved 
.'liirleigh drill, the valve, V, 
i g 7, is moved by two tappets, 

1”, which are alternately struck 
ip the ends of the piston, P. liur- 
Idgh Rock-Drill Co, mfrs, 
rl tchbnrg, Mass ; office 115 Liberty 
! , New York. 

Art. 35. Tn the “Dynamic” 

: ck-clri 11, invented by Prof J>e 
i olson Wood, and developed 

: d manufactured by the Gray- Fi°\.6 

on A Denton Mfg Co, 15 

irtlandt St, New York, tile valve is attached to a valve-piston, V, Fig 8, which 
moved backward and forward by steam, which is admitted so as to act alternately 
i »on its two ends. The admission of 
is steam is controlled by a small 
xiliary valve, a. A hub on the 
<ck of the auxiliary valve fits in the 
]iral groove shown on the plug, n. 

. iis plug is constantly pressed down- 
ird (as the Fig stands) by steam 
.easing upon its upper shoulder, but 
] is lifted at each forward stroke by 
e conical surface of the piston, P, 

“essing against its foot. It thus 
fjves constantly up and down, carry- 
g the valve, a, with it. By turning Fig.7 

e plug, n, by means of the adjusting- 

on, s, the hub of the valve is made to occupy a higher or lower point in the spiral 
oove, and thus the stroke of the piston may be varied, or may be confined to any 
rt of the cylinder. 

In this drill, unlike the Ingersoll, Art 27, the piston rotates while making 


Fig. 8 


i|8 downward stroke. The piston-rod, o, is made lighter than in 

her drills. This gives a greater surface under the piston for the pressure of 
e steam ou the up stroke, and, consequently, greater lifting power. This is use- 
when the drill sticks in the hole. 

The tripod legs are of bar iron. Their length is adjustable. 



































































































658 


MACHINE ROCK-DRILLS. 


Art. 36. The hand rock-drilling* machine of the Pierce We 
Excavator Co, New York, and Long Island City, N Y, is a percussion drill. It 
worked by a crank which turns a disc about 2 ft in diam. The disc has a semi-circul 
8lot, in which works the arm which raises the drill-rod. This arm, in rising,compress 
a coil-spring, which, on the down stroke, drives the drill against the rock. An ir 

r 


ball, weighing 30 lbs or more, is furnished with each machine. This ball may 
screwed to the top of the drill-rod, for giving greater force to the blows of the dri 
The ball may be used without the spring, by disengaging the latter. j 

The drill makes about 40 strokes of 10 or 12 ins per minute; and bores holes fr< I 
% to 2% ins diam. It can be arranged to drill to depths of 30 ft and over. T 
sharpening the bits, it has an emery wheel attached, which is turned by the crai 
The latter, at such times, is thrown out of gear with the disc. 

The drill is mounted on a rectangular two-legged frame, about 5 ft hi 
by 2 ft wide, made of irou tubes. To the top of this frame a third leg is attach) 
by adjusting which the angle of the drill-rod with the vert may be changed. Li 
other permission drills worked by hand-power, this one ceases to work to advanta 
when said angle exceeds about 45°. These drills cost, ready for work, in 18 
$225 each. They weigh from 200 to 400 lbs. They are moved from place to pit 
like a wheelbarrow, the disc serving as a wheel. 

Art. 37. Channeling consists iu making long, deep, and narrow cuts 
the rock. Iu this way large blocks can be gotten out without blasting and the c< 
sequent danger of fracture. This is ordinarily done by boring a row of holes abc 
an inch apart in the clear, and then breaking down the intermediate spaces 
means of a blunt tool, called a broach. This is called broach cliaiinelin 
For this purpose a steam drilling machine is mounted upon a hor bar resting up 
two pairs of legs. The hor bar is placed over the intended row of holes, and t 
drill is slid along upon it from one hole to the next. In using the broach , ihe rot 
ing apparatus is thrown out of gear, so that the edge of the broach maintains 
position iu line with the row of holes. 

Art. 3S. The Saunders patent cli aim cl in s* machine, of t 

Ingersoll Co, consists of a rock-drilling machine, having, in place of tlie usual drj 
ing-bit, a gang of tools consisting of a number of chisels, clamped together side 
side, and thus forming a cutting tool about 7 ins long by % inch wide. This t< 
lias as many cutting-edges (each as long as the tool is wide) as there are chisi 
The machine is supported upon a carriage, moving on a track parallel with t 
channel to be cut. The tool is of course not rotated; but the rifle-bar. A, Fig 4 
employed to move the carriage along the track about an inch after each blow. '1 
carriage remains stationary while a blow is being struck. Under favorable circu 
stances this machine lias cut trom 80 to 100 sq ft of channel per day of i 
hours. Its weight, including carriage, is about 5000 lbs. Cost, 1888, about $23 

A valve is provided, by which, if desired, the steam may he shut olF In 
the piston on the down stroke,so that said stroke may be made with ouly the wei 
of the piston, rod, and drill. 

Art. 39. The Ingersoll Co have a special appliance, designed by Mr W 
Saunders, C E, for drilling and blasting' rocks under water, ei 

when they are covered by a considerable depth of mud. 

Art. 40. Air compressors for rock-drills, as made and used in this coi 
try, are mostly hor, direct-acting engines. That is, the axes of the steam- and i; 
cylinders are hor; and the piston-rod passes directly from the steam-cylinder i 
the air-cylinder. A fly-wheel is attached, by a crank and connecting-rod, to 
piston-rod. Sometimes the steam-engine is separate from the compressor, and 
power is conveyed to the latter by belts or gearing; or water-power may be used 
the same way. The air is forced into a receiver, which is generally a plate c 
cylinder, 3 or 4 ft in diam, and 5 to 12 ft long. 

It the air- or pumping-cylinder of the compressor is so arranged as to take in 


D 


Li 


Ki 


- 



L it has two, and thus practically consists of two single compressors, i 

“duplex.” 

The valves may be either “ 


, poppet ’ valves, held in place by springs, t 

operated by the pressure of the air itself; or slide valves, operated'by eccenti 
and rods, as in steam-engines. 

The compression of the air develops lieat. This is removed either by cans 
cold water to circulate through the air-piston, and through jackets surrounding 
atr-cyUnder; or l>y injecting it into the air-cylinder in the form of spray. Or h 
methods may be used together. 

r, 4 , 1 \„ €w, ! , * >r ?, ssors are furnished by the Ingersoll Rock-D 

Co, 10 Park Place, New York; Rand Drill Co, 23 Park Place, New York; Rurle 









! Rock-Drill Co., 115 Liberty St., New York; Graydon & Denton Manufacturing Co., 
J y () rk r a„iother8 NeW Y ° rk ’ Clayt ° D Alr Cjm P lt ‘^or Work,, office, 43 Dey St., New 

The following partial list of Clayton compressors, compiled from data 
^iven by the makers, shows the dimensions and performance of each 
v We give also a list of their receivers. For prices, apply to the Co as above! 

id 

IdLAYTOJf DOUBLE-ACTING AIR-COMPRESSORS, Partial List. 

Fi 


M 

hi[ 


n't* 

lifi 




MACHINE ROCK-DRILLS. 


659 


Duplex. Direct-acting* Compressors. 

iljiDiam of steam-cylinders.ins. 

“ air “ .ins. 

Length of stroke.ins. 

til 

Vi Number of revolutions per minute. 

1 

Hub ft of free air compressed per minute.Actual. 

^Approximate wt of compressor.lbs. 

Approx number of rock-drills- with 3-inch cyls sup¬ 
plied with air at 60 to 80 lbs per sq inch. 


>,■141 
ota 
:lli 

fl f I 
I ft 
le 

iSj 

M 

mil 

4 

"Hub ft of free air compressed per minute.Actual. 

Approx wt of compressor. 

ft Approx number of rock-drills with 3-inch cyls sup¬ 
plied with air at 60 to 80 lbs per sq inch. 

tot 


Single Direct-acting* Compressors. 

Diam of steam-cylinder.ins. 

“ air “ .ins. 

Length of stroke.ins. 

Number of revolutions per minute. 


Number, designating the size 
of the machine. 


1 

*'A 

4 

7 

8 

10 

14 

18 

8 

10 

14 

18 

12 

13 

15 

24 

f 120 

100 

100 

80 

< to 

to 

to 

to 

(HO 

130 

120 

90 

136 

210 

438 

900 

3000 

7000 

15000 

25000 

2 

4 

8 

18 

8 

10 

14 

18 

8 

10 

14 

18 

12 

13 

15 

24 

(120 

100 

100 

80 

to 

to 

to 

to 

1 140 

130 

120 

90 

68 

105 

219 

450 

1650 

3850 

8250 

13750 

1 

2 

4 

9 


* The price of a compressor alone, to be worked by a separate steam-engine or water-power, is of 
ff.’jourse less than that of the above compressor and engine combined. 


Air-Receivers; vertical and horizontal. 


Diameter 

inches. 

Length, 

Feet. 

Approximate 
weight, tbs. 

Diameter, 

Inches. 

Length, 

Feet. 

Approximate 
weight, B,s. 

33 

5 

700 

40 

8 

1675 

30 

7 

890 

40 

10 

1900 

36 

8 

1560 

40 

11 

2000 

40 

6 

1600 

40 

12 

2100 


The Air-Receivers have brass-face pressure gauge, glass water-gauge, safety-valve, 
blow-off valve, try-cocks, flanges and connections to automatic feed on compressor. 
















































660 


EXPLOSIVES. 


GUNPOWDER. 

The explosive force of powder is about 40000 fbs, or 18 tons, per square 
ineh. Its weight averages about the same as that of water, or 62)4 ft>s per 
cubic foot-hence, 1 R> = about 28 cubic inches. In ordinary quarrying, a cubic 
yard of solid rock in place, for about 1.9 cubic yards piled up after being quar¬ 
ried,) requires from % to % !b. In very refractory rock, lying badly for quarrv- 
ing, a solid yard may require from 1 to 2 ft>s. In some of the most successful > 
great blasts for stone for the Holyhead Breakwater, Wales (where several 
thousands of lbs of' powder were usually exploded by electricity at a single 
blast.,) from 2 to 4 cubic yards solid were ioosened per lb’; but in many instances 
not more than 1 to 1)4 yards. Tunnels and shafts require 2 to 6 lbs per solid 
yard; usually 3 to 5 Ids. Soft, partially decomposed rock frequently requires 
more than harder ones. Usually sold in kegs of 25 lbs.* 

» 

Weight of powder In one foot depth of hole. 


Diameter of hole 

1 in 

1)4 ins 

1)4 ins 

2 ins 

2)4 ins 

i 

3 ins 

■Weight, of powder 
avoirdupois 

01b 5oz 

0ft) 8oz 

01b 11 oz 

lib 4oz 

21b 

i* 

2ft) 13oz 

Diameter of hoie 

3% in 

4 ins 

4)4 ins 

5 ins 

5)4 ins 

6 ins , 

Weight of Powder 
avoirdupois 

3lb 14oz 

51b Ooz 

61b 6oz 

71b 14oz 

9ft) 8oz 

jo 

lift) 5oz 


.$2.50 per keg of 25 lbs. 

. 2.00 “ “ “ 11 

Ini 

Itti 


•Price, 1888. in Atlantic cities, 
“A” powder (Saltpetre) 
“B” powder (Soda). 






























MODERN EXPLOSIVES. 


661 


MODERN EXPLOSIVES. 


Art. 1. Most of the explosives, which, of late years, have been taking 
he place of gunpowder (p 660), eonsist of a powdered substance, partly saturated 
vitli nitroglycerine,a fluid produced by mixing glycerine with nitric and sulphuric 
cids. 

Art. 2. Pure nitro-gdyeerine, at 60° Fah, has a sp grav of 1.6. It is odor- 
ess, nearly or quite colorless, and has a sweetish, burning taste. It is poisonous, 
ven in very small quantities. Handling it is apt to cause headaches. It is insoluble 
n water. At about 306° Fah it takes lire, and, if unconfined, burns harmlessly, 
mless it is in such quantity that a part of it, before coming in contact witli air, be- 
omes heated to the exploding point, which is about 380° Fah. 

N-G, and the powders containing it, are always exploded by means of 
harp percussion. See Arts 36, Ac. After N-G is made, great care is required to 
rash it completely from the surplus acids remaining in it from the 
rocess of manufacture. Their presence, either in the liquid N-G, or in the powders 
ontainiug it, renders the N-G liable to spontaneous decomposition, which, by rais- 
njg the temperature, increases the danger of explosion. 

"Art. 2. N-CJ freezes at about 45° Fah. It is then very difficult of ex- 
llosioil, and must he thawed gradually, as by leaving it fora sufficient length 
f time in a comfortably warm room, or by placing the vessel containing it in a sec- 
nd vessel containing hot water, not over 100° Fah; but never by exposing it to 
atense beat, as in placing it before a fire, or setting it on a stove or boiler. Extra 
trong caps are made for exploding N-G and its powders when frozen. 

Art. 4. N-G, owing to its incompressibility, is liable to explosion 
hrougli accidental percussion. This, and its liability to leak- 
render it, inconvenient to transport and handle. Hence it is rarely used in 
he liquid state in ordinary quarrying and other blasting. In the oil regions of 
’enita.it is largely used in oil wells, in order to increase the flow. For this 
urpose it is confined in cylindrical tin casings, from 1 to 5 inches diani, called ter¬ 
edo-shells. These are suspended from, and lowered into the well by means of, a 
ord or wire wound on a reel; and are destroyed when the charge is exploded, 
'hey are about 1 inch less in diam than the well, and contain usually from one to 
wenty quarts = 3 lbs, 5% *>/, to 66 lbs, 6% oz of N-G. They are pointed at their 
>wer ends, in order to facilitate their passage through the oil or water which may 
e in the well. When a greater charge than about 66% Ibsis required, two or more 
f these shells are placed in the well, one on top of another, the conical point on 
he lower end of each one fitting into the top of the one next below. In this case, 
he N-G is fired hv means of a cap or series of caps placed in the top of the charge 
Before it is lowered. When the charge is in place, the caps are exploded by elec- 
ricity led to them by conducting wires, as in Art 37, or (as in the method more 
ommonly practised) by letting a weight fall on them. 

When a well has been repeatedly torpedoed, and a cavity has thus been formed in 
t so large that the space surrounding a torpedo would interfere too greatly with 
lie effect of the explosion of the N-G on the walls of the well, the latter is placed 
irectiy in the well, by lowering a tin cylinder, filled with it, and provided with an 
utomatic arrangement which allows the N-G to escape when at the bottom of the 
.ell. The N-G is then fired by a torpedo suspended on a line, and having caps 



t is also used for increasing the flow of springs of water. It of course cannot be used 
n iier or iiutvard holes, such as often occur in tunneling, &c. 

Art 5. N-G explodes so suddenly that very little tamping: is re- 
iHired. Moist sand or earth, or even water, is sufficient. This, with the tact 
hat N-G is unaffected by immersion in water, and is heavier than water, render it 



asings, however, necessarily leave some spaces 
nd these diminish considerably the effect of the letter. , 

Art 6 The great explosive force of N-CS is due partly to the very 
■arge volume of gas into which a small quantity of it is couverted by explosion, and 







662 


MODERN EXPLOSIVES. 


partly to the suddenness with which this conversion takes place, the gases lieiiv 
liberated almost instantaneously,* while with gunpowder their liberation require! 
a longer time. The suddenness of the explosion increases its effect, not only bj 
applying all of its force practically at one instant, but also by greatly heating th 
gases produced, and thus still further increasing their volume. 

Art. 7. The liquid condition of N-G is useful in causing it to fill the drill 
hole completely, so that there are no vacant spaces in it to waste the fore 
of the explosion. On the other hand, the liquid form is a disadvantage, because 
when thus used without a containing vessel in seamy rock, portions of the N-G leal 
away and remain unexploded and unsuspected, and may cause accidental explosioi 
at a future time. 

Art. 8. N-O is stored in tin cans or earthenware jars. Ill 

properly washed from acid it does not injure tin. For transportation, these cans o 
jars are packed in boxes with sawdust, or in padded boxes, and loaded in wagons 
The It It companies do not receive it. 

Art. 9. When N-G and its compounds are completely exploded, the g’ase? 
{fiven out are not troublesome, but those resulting from incomplete explosion 
such as generally takes place, or from combustion, are very offensive. 

Art. 10. For convenience, we apply the name 4 ' dynamite” to any explo 
sive which contains nitro-glycerine mixed with a granular absorbent; »‘tru« 
dynamite” to those in which the absorbent of the N-G is *• Kieselguhr,”t o 
some other inert powder which takes no part in the explosion; and fc ‘fal*« 
dynamite ” to those in which the absorbent itself contains explosive substance, 
other than N-G. 

Art. 11. The absorbent, by its granular and compressible condition 
acts as a cushion to the Jf-G, and protects it from percussion, and fron 
the consequent danger of accidental explosion. 

N-G undergoes no change in composition by being absorbed; and it then freezes 
burns, explodes, Ac, under the same conditions as to pressure, temperature, Ac, as 
when in the liquid form. The cushioning effect of the absorbent merely renders il 
more difficult to bring about sufficient percussive pressure to cause explosion. The 
absorption of the N-G in dyn enables the latter to be used in hor holes, or in liolco 
drilled upward. 

Art. 12. N-G and dyn explode much more readily when rigidly 
confined, as by a metallic vessel, or by the walls of a hole drilled in rock, than 
when confined by a yielding substance, as wood. Therefore the fact that dyn, not 
being liquid, can be packed in wooden boxes, renders it safer than N-G which has to 
be kept in stone or metal vessels. 

Art. 13. True dynamites must contain at least about 50 per 
cent of N-G. Otherwise the latter will be too completely cushioned by the absorbent 
and the powder will be too difficult to explode. False dynamites, on the contrary 
may contain as small a percentage of N-G as may be desired; some containing as 
little as 15 percent. The added explosive substances in the false dynamites genemlb 
contain large quantities of oxygen, which are liberated upon explosion, and aid ii j 
effecting the complete combustion of any noxious gases arising from the N-G. 

Art. 14. Dynamites which contain large percentages of 
explode (like the liquid N-G. Art 6) with great suddenness, tending to shatter 
the rock in their vicinity into small fragments. They are most useful in very hard 
rock. In such rock, No 1 dynamite, or that containing 75 per cent of N-G is 
roughly estimated to have about 6 times the force of ail eoual wt 
of gunpowder. 

For soft or decomposed rocks, sand, and earth, the lower grades 

of dynamite, or those containing a smaller percentage of N-G, are more suitable 
1 hey explode with less suddenness, and their tendency is rather to upheave large 
masses of rock, Ac, than to splinter small masses of it.' They thus more nearly re¬ 
semble gunpowder in their action. 

Judgment must be exercised as to the grade and quantity of explosive 
to be used in any given case. Where it is not objectionable to break the rock into 
small pieces, or where it is desired to do so for convenience of removal the limber 
shattering grades are useful. Where it is desired to get the rock out in large masses’ 
as in quarrying, the lower grades are preferable. ’ 

lor very difficult work in hard rock, and for submarine blasting, the highest 
grades, containing 70 to 75 per cent of N-G, are used. A small charge of these does 
the same execution as a larger charge of lower grade, and of course does not require 


* Such sudden liberation of gas is called “ detonation.” 

is i,n ea, '. thy ' sillclo , us , limestone, composed of the fossil remains of small shells 
Tad ia New Jersey! reoe l ,tacle for nitro-glycerine. Kieselguhr is found in Hanover, Germany 












MODERN EXPLOSIVES. 


663 


s:he drilling of so large a hole. In submarine work their sharp explosion is not 
'? leadened by the water. 

ie For general mil road work, ordinary tunneling, mining of ores, Ac, the aver- 
i«e grade. containing 4u per cent of N-G, is used ; lor quarrying, <55 per 
kent; for blasting stumps, trees, piles, Ac, 30 per cent; for sand and 
earth, 15 per cent. 

e,| Art. 15. Dynamite, like N-G, can be readily exploded under 

i water, provided it is so immersed as not to be scattered; but long exposure 
mlo water is injurious to it. In the higher grades, the water, by its greater 
iftinity for the absorbent, drives out the N-G. In the lower grades it is apt to wash 

i iway the salts used as additional explosives. 

)r Art. 16. In dyns containing a large percentage of N-G, the latter is liable 
j to exude in liquid form, or to “leak,” especially in warm weather, and then to 
ixplode through accidental percussion. The same danger exists, even though the 
* percentage of N-G be small, if the absorbent has but small absorbing power, and is, 
abonsequently, easily saturated. 

Art. 17. True dyil resembles moist brown sugar. Its properties are 
>generally those of the N-G contained in it. Thus, it takes fire at about <550° F, and 
e purns freely. It freezes at 45° F, and is then difficult to explode. It is not exploded 
irlpy friction, or by ordinary percussion, but requires, for general purposes, a strong 
e ;ap, or exploder, containing fulminating powder, see Arts 36, 38, Ac. It may, how¬ 
’s -ver, be exploded by a priming of gunpowder, tightly tamped, and fired by an ordi¬ 
nary safety-fuse. _ 

nj Art. 18. The charge should fill the cross section of the 
n hole as completely as possible. If water is not standing in the hole, the cartridge 
should be cut open before insertion, so that the powder may escape from it and fill the 
«. hole; or the pow’der may he simply emptied from the cartridge into the hole, 
n Art. 19. for blasting ice in place, holes are cut in it, and a number of dyn 

ii cartridges (one of which must contain an exploding cap) are tied together and low- 
„ . re d from 1 to 5 ft into the water. They are fired as soon as possible atter immer- 
i lion, to avoid the danger of freezing. Electrical exploders (Arts 37, Ac,) are best 


or sub-aqueous w'ork, 

Art. SO. Dyn is useful for breaking np pieces ©f metal, such as old 
cannon, condemned machinery, “salamanders” (masses of hardened slag) m blast 
uniaces, Ac. In cannon, the dyn is of course exploded in the bore. In other pieces, 
unall holes are generally drilled to receive it; but plates,even of considerable thick¬ 
ness, may be broken by merely exploding dyn upon their surface. 

Art. 21. for blasting trees ©r stumps, one or more cartridges are 
ired in*a hole bored in the trunk or roots, or under the latter. This shatters both 
trunk and roots. A t ree may be felletl neatly by boring a number of 
unall radial holes into it, at equal short dists in a hor line around its circumf, and, 
by means of an electric battery (Arts 37, Ac), exploding simultaneously a small 
charge of dyu in each. Or a single long cartridge may he tied around the trunk of 
i small tree, and fired. 

.>.» Piles may be blasted in the same way as trees; or a hole may 
oe bored for the cartridge in the axis of the pile; or the cartridge may be simply 
tied to the side of the pile at any desired ht. . * 

Art. 23. The higher grades of dyn, like N-G, require but little tamp- 
in**’. Use a wooden tarn ping-bar, never a vietallic one, for any explosive. It 
i charge of dyn “ bangs lire,” it is dangerous to attempt to remove it. Remove 
the tamping, all but a few ins in depth, on top of which insert another cartridge, 
containing an exploder, and try again. See electrical exploders, Arts o7, Ac. Dyn, 
like N-G, if frozen, must be thawed gradually , by leaving it in a warm room, tar 
from the fire; or by placing it in a metallic vessel, which is then placed in another 
vessel containing hot water. The water should not ho hotter than can beanie by 
the hand. Otherwise the N-G is liable to separate from the absorbent. Ihe jn-U in 
Jvn may freeze without cementing together the particles of the absorbent; m 
which case the pow’der of course is still soft to the touch. An overcharge of 
N-G or of dyn, is liable to be burned, and thus wasted, giving oft oftensivo gases. 

Art. 24. Dyn is sold in cylindrical, paper-covered cart¬ 
ridges. from % to 2 ins in diam, and 6 to 8 ins long, or longer. I hey aie iur- 
nished to order of any required size, and are packed in boxes containing 2d lbs or .0 
lbs each. The layers of cartridges are separated l»y sawdust. 

Art. 25. Some of the It It companies decline to carry dyn or N-G m any 
diape. Others carry dyn under certain restrictions, based upon State law’s; pio- 
£ling that it must be dry (i e, that no N-G shall be exuding from it); that boxes 
,nd cars containing it shall he plainly marked with some cautionary words, as ex- 
dosive? “dangerous,” Ac; that the cartridges shall he so packed m the boxes, and 
Ihe boxes so loaded in the cars, that both shall lie upon their sides, and the boxes 






664 


MODERN EXPLOSIVES. 


be iu no danger of falling to the floor; that caps, Ac, shall not be loaded in the saint 
car with dyn, Ac, Ac. 

Arl. 26. A great many varieties of<Iyn are made. They diffei 
(generally hut slightly) in the composition of the absorbent, and iu the method ot , 
manufacture. Each maker usually makes a number of grades, containing diflereni , 
percentages of N-G, Ac, and gives to his powders some fanciful name. 

Art. 27. The following table of explosives made by the Repauni], 
Chemical Co, at Thompson’s Point, N J, office 'Wilmington, Del. and known as ” At- f 
las ’' ponders, gives the percentage of N-G in each, and the approx card t 
price for 1888. It gives a general idea of the range of American dyns. 


Brand. 

Percentage 
of X-U., 

Card price, 
1888 . 

cts per tb. 

Brand. 

Percentage 
of N-U. 

-— 

Card price, 

1888 . 

cts per lb. 

A 

75 

35 

D + 

33 

20 

R + 

60 

31 


R 

50 

27 

E + 

27 

18 

c+ 

45 

25 

E 

20 

16 

c 

40 

23 




The absorbents contain: in “A” brand, 18 per cent wood pulp and 1 
per cent carbonate of magnesia; in “C” brand (the average grade), 46 per cent 
nitrate of soda (soda saltpetre), 11 per cent wood pulp, and 3 per cent carbonate of 
magnesia; in “E brand, 62 per cent nitrate of soda, 16 per cent wood pulp, Ac, anc 
2 per cent carbonate of magnesia. 

Art. 28. ‘‘Miner’s Friend” powder, made by the Hecla Powder Co 
office 239. Broadway, New York, contains nitrate of soda, wood pulp, resin, and car¬ 
bonate of magnesia. It freezes at 42°, and is then, like other dyn, difficult to ex¬ 
plode. When used under water, the cartridges should not be broken, because tin 
Powder is injured by direct contact with water. Their “ Hecla ” powder is n 
lower grade. It is in granulated form, like ordinary blasting powder, but is said ti 
be much stronger. It is intended as a substitute for it. 

Art. 29. “ Giant ” powder is made by Atlantic Dynamite Co, office 24f 
Broadway New York. No 1 is dyn proper, containing 75 per cent N-G, and 25 pei 
cent Kieselguhr obtained near their works in New Jersey. Their lowest grade 
branded M, contains 20 per cent N-G. The name “giant powder ” was originalh 
applied to dynamite in general. J 

t, Art \ 0t *J er brands are “ Hercules” powder, Hercules Powder Co 4< 
Prospect St, Cleveland, O ; and « Juclson K R P powder.” Atlantic Dvn Co 
^ or k, a substitute for ordinary blasting powder. It is put up in water proo; 
papei bags, of ’ ^J -34 and 25 lbs each, and these are packed in wooden boxes hold 
lng ;»0 lbs each. The same Co furnish also “Judson F F F dynamite ” t 
higher grade, in cartridges of the usual shape, packed in 50-lb boxes. 
v 1 r m 31 \t 44 Kaekarock ” cartridges, furnished by Rendrock Powder Co, 21 
Paik 1 lace, New York, are said to contain no N-G, and to be entirely inexplosiv. 
until immersed, for a few seconds, in an inexplosive liquid furnished by the stum 
Co. they are then allowed to stand for 15 mins, after which they may be used a 
anytime. I hey are fired in the same way as dyn, and can be used underwater 
dyn” * C Uim that the y “approximate N-G in strength, and are stronger thai 

f T ,,e fo * lo wins: explosives are made and used in 
Fui ope, but have not yet been regularly imported into the U S 

C ompressed gun-cotton, made‘at Stowmarkot, Eng, is cotton dipped ii 
* m ‘ x T? mt t u- t Dd ■ nl £ hnri ? acids ’ *«* reduced to a fint’pulp. and made ini, 

thl Ll-i nf lnS , thlck V a . nd 11IS dlam ’ or larger. It is generally used wet, foi 

t e sake of greater safety. It then requires extra strong caps or primers. Roughb 
speaking it is about as strong as dyn No 1, but is less shattering in its effect. Rein- 
lighter than dyn, it requires larger holes; and, owing to its rigidity is less easih 
lmtTfT’/T does "Ot fit the hole so completely. When dry, it is very inflammable 
and is saffiTo handle bUniS harmlessl > ‘ 11 contain s »o liquid, to freeze or to exude 

VlQ A '* t * 'V}r Tonite consists of finely divided gun-cotton mixed with nitrate ot 
baiyta. It is made by the Cotton Powder Co, Limited, at Faversham Eng. It it 
compressed into candle-shaped cartridges having, at one end, a recess for the recep 
ton of an exploder containing fulminate of mercury. The cartridges weigh abou 
the same as dyn. They are generally made waterproof. ° 8 






















MODERN EXPLOSIVES 


665 


Art. 34. Forcite, Eitliofracteur, and Dualin are foreign makes of 
intro-glycerine explosives. In Dualin the absorbent is sawdust. It lias greater 
bulk than dyn for a given wt, and requires larger holes. 

Art. 3 d. Explosive gelatine is made by the Nobels Explosives Co, Lim 
(office Glasgow, Scotland), at their several works in England. It is a transparent, 
pale yellow, elastic substance, and is composed of 90 per cent N-G and 10 per cent 
, gun-cotton. It is less sensitive than dyn to percussion, friction, or pressure, and is 
not affected by water. Its specific gravity is 1.6. It burns in the open air. For 
complete detonation a special primer is required. The addition of a small propor¬ 
tion of camphor renders it still less sensitive, and increases its explosive force. The 
camphor evaporates to some extent. 

In some experiments on the power of different explosives to increase the contents 
of a small cavity in a leaden block, explosive gelatine caused an increase 50 per cent 
greater than that caused by dyn No 1. In hard rock the diff would probably have 
been greater. The increase was 10 per cent less than that caused by N-G. 

Art. 36. The cap or exploder, used with ordinary safety fuse for ex¬ 
ploding N-G and dyn, is a hollow copper cylinder, about ^ inch diam, and an inch 
or two in length. It contains from 15 to 20 per cent, or more, of fulminate of mer¬ 
cury, mixed with other ingredients into a cement, which fills the closed end of the 
cap. The cap is called “ single-force,” “ triple-force,” &c, according to the quantity 
of explosive it contains. 

The end of the fuse, cut off square, is inserted into the open end of this cap, far 
enough to touch the fulminating mixture in it. In doing this, care must be taken 
not to roughly scratch the latter. The neck of the cap is then pinched, near its 
open end, so as to hold the fuse securely. The cap, with the fuse thus attached, is 
then inserted into the charge of N-G or dyn, care being taken not to let the fuse 
come into contact w r ith the explosive, which would then be burned and wasted. If 
a dyn cartridge is used, the fuse, with cap, is first inserted into it. The neck of the 
cartridge is then tied around the fuse with a string, and the cartridge is then ready 
to be placed in the hole and fired. 

Art. 37. Tlie Siemens magneto-electric blasting appa¬ 
ratus. now in general use, consists of a wooden box about as large as a transit- 
box. Outside it has two metallic binding-posts with screws, for attaching the two 
wires leading to the exploder. From the top of the box projects a handle at the 
end of a vert bar. This bar, which is about as long as the box is high, is made so 
as to slide up and down in it, and is toothed, and gears w ith a small pinion inside 
the box. When a blast is to be fired, the bar is drawn up, by means of the handle, 
as far as it will come. It is then pressed quickly down to the bottom of the box. 
In its descent it puts into operation, by means of the pinion, a magneto-electric 
machine inside the box. This generates a current of electricity, w hich increases in 
force with the downward motion of the bar, but which is confined to a short circuit 
of wire within the, box, until the foot of tlie bar strikes a spring near the bottom of 
e box, breaking the short circuit and forcing the electricity to travel through the 
n two longer “ leading wires,” which lead it from the two binding-posts on the outside 
f, of the box to the cap or exploder placed in the charge. 

Art. 38. Tlie cap used with this machine is similar to that used with safety 
, .fuse (Art 36), except that its mouth is closed with a cork of sulphur cement, through 
n which pass the two wires leading from the electric machine. The ends of these 
wires project into the fulminating mixture in the cap. They are ^ inch apart, but 
are connected by a platinum wire, which is so fine as to be heated to redness by the 
(Current from the battery. Its heat ignites the fulminate and thus explodes the cap. 
These exploders, called platinum caps, or (improperly) platinum fuses, cost, 
1888, about 5 to 11 cts each, depending upon the length (from 4 to 16 ftj of the two 
cotton-covered wires attached to them. With gutta-percha-covered wires, 20 to 40 
cts. The outer end of each of these short wires is connected with the electrical ma¬ 
chine by a cotton-covered 44 leading wire ” costing 1 cent per ft. 

Art. 39. Where a number of holes are to be tired simul¬ 
taneously (thus increasing their effect), each hole has a platinum cap inserted 
into its charge, and one of the short wires attached to each cap is joined to one of 
those of the next cap, so that at each end of the series of caps there is one free end 
of a short wire. Each of these two ends is fastened to the end of one of the leading 

• wires, placing the whole series ‘‘in one circuit.” Where the holes are too far apart 

* for the caps to be thus joined by the short wires attached to them, the ends of the 
^ latter are connected by cotton-covered 44 connecting wires,” costing about 
I 30 cts pev pound. 






666 


MODERN EXPLOSIVES, 


Art. 40. The magneto-electrical machine weighs ahont 16 lbs, and 

costs, 1888, size No o, $25. It can fire about 12 caps at once. A larger size, No 4, 
costs $50; and a still larger one, said to be capable of firing over 50 holes at once. 
$100. . 

Frictional electric blasting- machines, costing about $75 each, are 
now (1884) nearly obsolete. 

Caps for ordinary fuse and for electrical firing, fnseR, wires, electrical machines 
Ax, are made by Laflin & Rand Powder Co, 20 Murray St. New York, and are sold by 
most of the makers of, and dealers in, explosives, rock-drilling machines, &c. 

Art. 41. Simultaneous firing of a number of holes can be conveniently accom¬ 
plished only by electricity. Electric blasting apparatus is specially useful for 
blasting under water, where ordinary fuses are apt, especially at great depths, to 
become saturated and useless. 

If an electrical machine fails to fire a charge, it is known that the charge cannot 
explode until the attempt is repeated. Therefore no time need be lost, aud no risks 
ruu, on account of “hanging fire.” 






H, r> Oil F .«■ 8 . ' 'i t 


■*a l.tftt «#-( 





COST OF STONEWORK 


667 


ml 

le, 

ire 


■■ 

bj 

TO- 

for 
i, to 

!0t 

iu 


Cost of quarrying? stone. After fhe preliminary expenses of purchasing 
lie site of a good quarry; cleaning off the surface earth and disintegrated top rock • 
nd providing the necessary tools, trucks, cranes, &c; the total neat expenses for 
ettvng out the rough stone for masonry, per cub yard, ready for delivery, may be 
onghly approximated thus: Stones of such sizes as two men can readily lift, meas- 
led in piles , will cost about as much as from to the daily wages of a quarry 
iborer. _ Large stones, ranging from 34 to 1 cub yd each, got out by blasting, from 
to 2 daily wages per cub yd. Large stones, ranging from 1 to 134 cub yds each, in 
hich most of the work must be done by wedges, in order that the individual stones 
hall come out in tolerably regular shape, and conform to stipulated dimensions; 
•om 2 to 4 daily wages per cub yard. The smaller prices are low for sandstone, 
diile the higher ones are high for granite. Under ordinary circumstances, about 
Vz cul) yds of good sandstone can be quarried at the same cost as 1 of granite; or, 
1 other words, calling the cost of granite 1, that of sandstone will be%: so that 
he means of the foregoing limits may be regarded as rather full prices for sandstone; 
itlier scant ones for granite; and about fair for limestone or marble. 


Cost of dressing? stone. In the first place, a liberal allowance should be 
lade for waste. Kven when the stone wedges out handsomely on all sides from 
;ie quarry, in large blocks of nearly the required shape and size, from % to ^ of 
l ie rough block will generally not more than cover waste when well dressed. In 
,ioderate-sized blocks, (say averaging about % a cub yard each.,) and got out by 
lasting, from % to 3^ will not be too much for stone of medium oharacter as to 
jraight splitting. About the last allowance should also be made for well-scabbled 
abble. The smaller the stones, the greater must be the allowance for waste in 
iressing. In large operations, it becomes expedient to have the stones dressed, as 
jir as possible, at the quarry; in order to diminish the cost of transportation, which, 
i hen the distance is great, constitutes an important item — especially when by land, 
nd on common roads. 


A stonecutter will first take out of wind; and then fairly patent-hammer dress, about 8 
10 sq ft of plain face in hard granite, in a day of 8 working hours; or twice as much of such iufe- 
)r dressing as is usually bestowed on the beds and joints ; and generally on the faces also of bridge 
asonry, &c, when a very fine finish is not required. In good sandstone, or marble, he can do about 
more than in granite. Of finest hammer finish, granite , 4 to 5 sq ft. 

Cost of masonry. Every item composing the total cost is liable to much 
iriation; therefore, we can merely give an example to show the general principle 
pon which an approximate estimate may be made; assuming the wages of a 
iborer to be $2.00 per day of 8 working hours; and $3.50 for a mason. The 
lonopoly of quarries affects prices very much.* 

Cost of ashlar facing- masonry. Average size of the stones, say 5 ft 
ng, 2 ft wide, and 1.4 thick; or two such stones to a cub yd. Then, supposing the 
one to be granite or gneiss, the cost per cub yd of masonry at such wages 

,11 be, Getting out the stone from the quarry by blasting, allowing y± for waste iu 


dressing: \ % cub yds, at $3.00 per yard. $4 00 

Dressing 14 sq ft of face at 35 cts. 4.90 

“ 52 “ beds and joints, at 18 cts. 9.36 


Neat cost of the dressed stone at the quarry. 18.26 

Hauling, say 1 mile; loading and unloading. 1.20 

Mortar, say.40 

Laying, including scaffold, hoisting machinery, superintendence, &c. 2.00 


Neat cost. 21.86 

Profit to contractor, say 15 per ct. 3.28 


Total cost. 25.14 

Dressing will cost more if the faces are to bo rouuded, or moulded. If the stones are smaller than 
e have assumed, there will be more sq ft per cub yd to be dressed, &c. 

If in the foregoing case, the stones be perfectly well dressed on all sides, including the back, the 
st per cub yd would be increased about $10; aud if some of the sides be curved, as in arch stones, 
.y $12 or $14; aud if the blocks be carefully wedged out to given dimensions, $16 or $18; thus 
aking the neat cost of the dressed stone at the quarry say $28, $31, or $35 per cub yd. 


* The blocks of granite for Bunker Hill monument averaging 2 cub yds each, were 
tarried by wedging, aud delivered at the site of the monument, at a neat actual cost of $5.40 
■r cub vd • by the Monument Association ; from a quarry opened by themselves for the purpose. The 
ssociation received no profit; their services being voluntary. The average contract offers for the 
me were $24.30! The actual cost of getting out the rough blocks at the quarry was $2.70. Load- 
g upon (rucks at quarry, about 15 cts. Transportation 8 miles by railway and common road, $2.55. 
otal, $5.40. In 1825 to 1845 ; common unskilled labor averaging $1 per day. 

Ill 1888, Granite blocks about a cub yd each, with dressed beds and joints, 

ut with only a 2 incli draft around the showiug-face, (which is left rough,J.are del’d on the wharf at 
hilada, from Port Deposit, Md, by McOleuahau k Bros, at $16 each. 






















668 


COST OF STONEWORK. 


The item of laying will be much increased if the stone has to be raised to great heights; or if it ] 
to ke much handled; as when carried in scows, to be deposited in water-piers, &c. Almost evt 
large work presents certain modifying peculiarities, which must be left to the judgment of theeni 
neer and contractor. The percentage of contractors' profit will usually be less on large works tl) 
on small ones. 


Cost of ashlar far ins: masonry. If the stone be samlston i 

with good natural beds, the gettiug out mar be put at $3.00 per cubic vard. Pace dressing at 26 < i 
per sq ft: say $3.64 per cubic yd. Beds and joints 13 cts per sq It; say $6.T6 per cub yd. The nt 
cost, laid, $17.00. i 

And the total cost of large well scabbletf ranjrf ■ 
sandstone masonry in mortar, may be taken at about $10 per cub yf r 


Cost of large soabbled granite rnbble. such as is generally used 

backing for the foregoing ashlar; stones averaging about % cub yd eacli: 

Cost per 

Labor at $1 per day. cub vd of 

. masonry. 

Getting out the stone from the quarry by blasting, allowing for waste in 

scabbling; I^-cub yds at $3.00. $3.43 

H?.uling l mile, loading and unloading . 1.20 

Mortar; (2 cub ft, or 1.6 struck bushels quicklime, either in lump or ground ; 

and 10 cub ft, or 8 struck bushels of sand, or gravel; and mixing). 1.50 

Scabbling; laying, including scaffold, hoisting machinery, &c. 2.50 


Neat cost..... 8.63 

Profit to contractor, say 15 perct. 1.30 


Total cost. 9.93 

Common rubble of small stones, the average size being such as t\ 
men can handle, costs, to get it out of the quarry, about 80 cts per yard of pi] 
or to allow for waste, say $1.00. Hauling 1 mile, $1.00. It can be roughly scabble 
and laid, for $1.20 more; mortar as foregoing, $1.50. Total neat cost, $1.70; or, wi 
15 per ct profit, $5.40, at the above wages for labor. 



layi 

Neat scabbled irregular range-work costs from $2 to $3 more per yd tnan ruonie: according i 
ter of the stone &c. The layingof thin walls costs more than that of thick ones, such as abutments , 

The cost of plain 8 inch thick ashlar facings for dwellings Ac 

Philada, in 188 s , is about as follows per square foot showing, put up, including everything. Sat 
stone, $1.50 to $2.25. Pennsylvania marble, $2.50. New England marble, $2.75 to $3.25. Granii 
$2.25 to $2.75. If 6 ins thick, deduct one-eighth part. First Class artificial StOll 
could be made and put up at one-third the price. See p 681. North It i vcr blllC StOll 
liagS, 3 ins thick, for footwalks, put down, including gravel Ac, 70 cts per sq foot. Bclg'ia I In 

street pavement, with gravel, complete, $8.50 per sq yard in Eastern citie p 


When dressed ashlar facing is backed by rubble, the expense per cub yard of t 
intire mass will of course vary according to the proportions of the two. Thus 

ishlar Sit #12 ner vd is haolred hv on nmnl thiMrnoce mlikln * i, „ ’ 


ent 

ashlar at $12 per yd, is backed by an equal thiekness'of rubble at $5, the mean co i 
will be ($12 + $5) -t- 2 = $8.50; or if the rubble is t wice as thick as the ashlar the 
($12 + $5 + $5) -rii = $7.83, &e. Much compound walls are weak at 
apt to separate in time, as also walls of cut stone backed by concrete, or by bricl 
from unequal settlement of the two parts. 

Ai times the contractor must be allowed extra in opening new quarries; in formi 
short roads to bis tvork ; in digging foundations ; or for pumping or otherwise draining them, v ii 
springs are unexpectedly met with ; for the centers for arches, &c; unless these items are expres: 
included in the contract per cub yd. 


For quantify of masonry in walls of wells, see p 158. 

Approximate cost of buildings per cubic foot, at Philada pne 

in 1873 ; including every cub ft of space from roof to cellar floor. Plain brick dwellings, such as m< 
of those lu 1 htlada, 12 to to cts. Better class, highly finished throughout, i5 to 18 cts. First cla 11 
with cut stone fronts, 20 to 30 cts. Plain brick churches, public schools, court-houses, theaters <S 
12 to 16 cts. Ornate Gothic churches with much cut stone facing, 30 to 45 cts, exclusive of spin 
Large plain brick or rubble H R shops, depots, station-houses, &c, 9 to 12 cts; or with oruanieni 
finish and best materials, lo to 20 cts. First class city stores, marble fronts, high stories fire-nro. 
(so called,) throughout^ roof; best materials and workmanship, 18 to 25 cts. 

Small buildings cost more per cub ft than large ones of the sac 1 

hmsh. Also isolated or corner buildings, cost more than those which have two party-walls. 

In Philada dwellings of brick, the carpentry and lumber usually co 
each about one-fourth of the entire building. Memorial Hall of the Cente 
mal Buildings cost 68 cts per cub ft, exclusive of iron dome. 


* In Philadelphia, in 1888, cellar and other walls of rough rubble, $3 to $4 per perch of 22 cub ft 
.ea/L Outside waljs a ith a facing of broken range rock-work of saud S to..e, (as common in Gm 
chinches,) to $7 per 22 cub tt, including everything. 




















MORTAR, BRICKS, ETC. 


MORTAR, BRICKS, &c. 


609 


ClltH 


used 


usually exceed that 


11 4rt. 1. Mortar. The proportion of 1 measure of quicklime, either in ir- 
l *(ular lumps, or ground,* and 5 measures of sand, is about the average used for 
muon mortar, by good builders in our principal Atlantic cities; and if both 
mgfkte rials are good, and well mixed (or tempered) with clean water, the mortar is 
j 'tainly as good as can be desired for such ordinary purposes as require no addi- 
n of hydraulic cement. The bulk of the mixed mortar will us 
the dry loose sand alone about Y part. 

Quantity required. 20 cub ft, or 16 struck bushels of sand, and 4 cub ft. or 

struck bushels of quicklime, the measures slightly shaken in both cases, will make abt 22 cub ft of 
rtar; sufficient to iay 1000 bricks of the ordinary average size of 814 by 4 by 2 ins, with the coarse 
rtarjoints usual in interior house-walls, varying say from fg to inch. With such joints, 100Q 
:h bricks make 2 cubic yards of massive work. Nearly one-third of the mass is mortar. For 
side or showing joints, where a whiter and neater looking mortar is required, house-builders in- 
ase the proportion of lime to 1 in 4, or 1 in 3. For mortar of fine screened gravel, for cellar-walls 
stone rubble, or coarse brickwork. 1 measure of lime to 6 or 8 of gravel, is usual; and the mortar 
;ood. In average rough massive rubble, as in the foregoing brickwork, about one-third the mass is 
rtar: consequently a cubic yard will require about as much as 600 such bricks; or 10 cubic feet. (8 
uck bushels) of sand; and 2 cub ft, or 1.6 bushels of quicklime. Superior, well-scabbled rubble, 

efully laid, will contain but about A of its bulk of mortar; or 5^ cub ft sand, and 1.1 cub ft lime, 
cub yard. 

’or public engineering works, especially in massive ones, or where exposed to dampness, an addi- 

n should be made in either of the foregoing mortars, of a quantity of good hyd 
ment, equal to about Y of the lime; or still better. Yt of the lime should be 
iited. and an equal measure of cement be substituted for it. If exposed to water while 
Ite new, use little or no lime outside. 

f With bricks of by 4 by 2 ins, the following are the quantities of mor- 
r ami of bricks tor a cubic yard of massive work. 


,’ili 


11 


i! 


11 

oie 

ltl» 

lion 

:i» 

citi 

rfl 

A 


Thickness 
of Joints. 

1 
8 
1 

4 
3 
8 
1 

2 

5 
8 


Proportion of Mortar 
in the whole mass. 


No. of Bricks 
per cub yard. 


No. of Bricks 
per cub foot. 



. about i . 

.638. 


it 


. 574 . 


it 

n 3 

. 522 . 

. 19.33 





u 

u 1 

. 475 . 

. 17.60 

it 

.. 4 

.. 433 . 




In estimating for bricks in massive work, allow 2 or 3 per et for waste : 

j i,i common buildings, 5 per ct. or more. Much of the waste is incurred in cutting bricks to fit 
gles, &o. In Philadelphia a barrel of lump lime is allowed for 1000 bricks; or for 2 perches (25 cub 
» ac h) of rough cellar-wall rubble. Somewhat less mortar per 1000 is contained in thin walls, than 
i massive engineering structures; because the former have proportionally more outside face, which 
JU S not require to be covered with mortar; but thin walls involve more waste while building; so that 
th require about the same quantity of materials to be provided. Careful experiments show that 
iriar becomes harder, and more adhesive to brick or stone, if the proportion of lime is increased. 
St -nee. on our public works the proportion of oue measure of quicklime to 3 of sand, is usually spec¬ 
ie! ill, hut probably never used. 

Ume is usually sold in lump,by the barrel,* of about 230 lbs net, 

260 lbs gross. A heaped bushel of lump lime averages about 75 lbs. Ground quicklime, 
#l l,se, averages about 70 lbs per struck bushel: and 3 bushels loose just fill a common flour barrel, but 
!i " ,m 3.5 to 3.75 bushels, or 245 to 280 lbs can readily be compacted into a barrel. 

iiieueral remarks on mortar anti lime. On too great a pro- 

i tion of our public works, the common lime mortar may be seen to be rotten and useless, where it 
s been exposed to moisture; which will be carried by the capillary action of earth to several feet 
ove the natural surface: or as far below the artificial surface of embankments deposited behind 
‘Iff hutments, retaining-walls, &c. The same will frequently be seen in the soffits of arches under em- 
ukmeuts. Common lime mortar, thus exjiosed to constant moisture, will never harden properly. 
,- e ,j when very old and hard, it absorbs water freely. Cement also docs so, but hardens. 

lirickdust. or burnt clay, improves common mortar; and makes it hydraulic, 
localities where sand cannot be obtained, burnt clay, ground, may be substituted ; and will gen- 
■ 3 ') ally give a better mortar. 




B* 

las v 
Ji 


■its 


jt 


Protection of quicklime from moisture, even that of the air, is 
isolutely essential, otherwise it undergoes the process of air-slacking, or 


coi * price of quicklime In lump in Philadelphia, about 25 cts per bushel. Bar 
(band 81.40 per cubic yard. J. Hex Allen, 1319 Washington Ave. 


























670 


MORTAR, BRICKS, ETC. 


spontaneous slacking, by which it becomes reduced to powder as when slacked by water as ust ‘ 
but without heating, anil with but little swelling. As this air slacking requires from a few month: 
a ^• e j lr /. or mnre ' depending on quality and exposure, it gives the lime time to absorb sufficient cai bo 

acid from the air to injure or destroy its efficacy. But quick linit* will ke« 
S’ootl for a loiijf time if first ground, and then well packed in air-tig f 

barrels. The grinding also breaks down refractory particles found in all limes, and which injure ® 
mortar by not slacking until it has been made and used. For the same reason it is better that li 
should not be made iuto mortar as soon as it is slacked, but be allowed to remain slacked for a dav 
two (or even several) protected from rain, sun, and dust. 

rime Nlackcd in great bulk may char or even set fire to wood. 

^' 1 ,ne J )as ^ e and mortar will keep for years, and improve, if w 1 

buried In the earth. Also for months if merely covered in heaps under shelter, with a thick layer 
sand. The paste shrinks and cracks in drying ; but the sand in mortar prevents this. 

As approximate averages varying much according to the character a l ' 

degree of burning of the limestone: and to the fineness or coarseness of the snud, one measure 
good quicklime, either in lump, or ground; if wet with about M a measure of water, will within 1 
than an hour, slack to about 2 measures of dry powder. Aud if to this powder there be added abi 
.4 more measures ot water, and :i measures of dry sand, and the whole thoroughly mixed, the res 
w .fLr a measures of mortar. Or the same slacked dry powder, with about 1 measure 1 

• 3 JnT Ur K S ° f , S , and ’ WlM " lilkeabolu5 ^ measures of mortar. In both cases the bulk 
the mortal will be about % part greater than that of the dry sand alone. If f; of a measure of wa 
be used for slacking, the result, instead of a dry powder, will be about lV^ measures of stiff paste; 
•ThouJ ,h*\°L e n ‘ eHSUr ® ° f wat , er {oT . packing, the result will be about 1 * measures of thin paste, 
abnnJ , 00n,,8 * M,ee f ? r w,tb the sand - Very Pure. fat limes, slack quickly, aud ma 

Slow sl-ick'in 10 J ' ueasu !' es powder; while poor, meagre ones, require more time, aud swell le 
Mow slacking, and small swelling, m case the lime has been properly burnt, are not in general b 

maker abettor 1 ° n 5 h<? cont, ' a , r y’ u * uaIly indicate that it is to seme extent hydraulic. In this case 
wfthe wmt Zn far 1 especially for works exposed to moisture, or to the weather. Very pure lin 

never bemused 'withoutcemeiTtr P0SUre3 ’ ° r “ bad "“"^-limes ; and in important wbrL, shot 

hnSt?? 1 , a PPf. ar8 to be about the same as that from the purest limestone 

Cha v‘ k 18 8tl mor ? ln I erl0 , r - and win not bear more than about I 1 * measures of san 
ne '' e ' becomes very hard. Madrepores (commonly called coral) appear to furnish a lii 
intermediate between those of chalk and limestone. They require to be but moderately burnt. 

per*c!b*ft UVera ® e wei *» ht of common hardened mortar is about 103 to 115 1 
t, l *, 0 VV? ™, e r ely c . ommon mortar made so thin as to flow almost like creai 

It is intended to fill interstices left in the mortar-joints of rough masonry ; but uuless it contain- l 
arge amount of cement, it is probably entirely worthless; since the great' quantity of water injur I 

low the iime es Besides d i ? or ?.? ver ' its ingredients separate from each other; the sand settli ng t * 
i Besides this, it will never harden thoroughly in the interior of thick masses of ml 

. y ' mdeed, the same may probably be said of any common lime mortar. In such positions it h 
been found to be perfectly soft, after the lapse of many years. positions, it n 

1 mu sai , 1( ^ an( * ^J e wa ter for lime mortar, should he free from cluv an 
salt. Ihe clay may be removed by thorough washing; but it is extremely di 

r?m .-n 0 t g l l nd ,u f th ® ,! t lt from seashore Si ‘n d . even by repeated washings. Knough will general 
r‘" ,n tok ®«P ,he work damp, and to produce efflorescences of nitre on the surface • whether wi 
Mowlr «hnniH n |f«. t |^ 0 r i ar J Slack ' n S b y salt water gives less paste than fresh. 

/ lr be nl,xed upon the surface of clayey ground ; but a rough board, brick or str 

platform should be interposed. Pit sand sifted from decomposed gneiss, and other allied rocks is , 1 
m0Uar: US shar " fugles making with the lime a more coherent mass than t^roun^ 
fn hrick’Lm L e - r ° t r h sea f an ' 1 - Morta r should he applied wetter in hot than in cold weather • espeeia 
Injured! ° therWlse the water ,s t0 ° muoh absorbed by the masonry, and the mortar is there 


/he tenacity, nr cohesive strength, that is, the resistance to a mi 

bftom It 

per sq inch; or about one third of a ton per sq ft. in mortar 6 months old. ’ 10 be but 5 B 

!naT ranSV ?r!* e s ,fengrth of good common mortar 6 months old. A bar 

inch square and 12 ins clear span, breaks with a ceuter load of 4 to 8 lbs. 

file lime in mortar decays wood rapidly, especially in Hoe, 

damp situations. Still the soaking of timber for a week or two in a sokitiou of quicklime in wat 
moUtnre^h™ h aS % P re ; ,ervatlve - Iron, so completely embedded in mortar as to exclude air an 
So, probably, wifhotbe, m«au! ^ '' bUt if the m0rtar admits moisturu tb « Bon decay 

The adhesion to common hricks, or to rough nibble it an 

age wilt average about % of the cohesive strength at the same age • or sav 12 to 24 b „*« f l f" 

wetting ■s^especfal'h' 1 n^c^Tarv^f ve^v^hot^eath^^t^pl^ve^^tli'^h |! °^ C ^ * " M '' 

injr the mortar by the rapid absorption and evaporation of itswate^The ‘od'ho.ln/T" kU 
smooth hard pressed bricks, o, to smoothly dressed or sawed sfone ifSfns^abKUs*® ^ 






MORTAR, BRICKS, ETC. 


671 






mi ^^ Bricks, size, wt, «fcc.* A common size in our eastern cities is 

1 ! 'j V X 2 >ns ; which is equal to 66 cub ins ; or 26.2 bricks to a cub ft; or 707 bricks to a cub 
#l d. 8 or the number required with mortar, see table, p 669. 

f u ordering a large number a minimum limit of dimension should be specified in order to prevent 
ua. A brick % inch less each way than the above, contains but 52.5 cub ins; thus requiring full 
■ >er ?. ent bricks to do the same work ,* in addition to 25 per ct more cost for laving, which is 
' erally paid for by the 1000. J 


li 

a 

in 

e 

k 

Hi 


rile weight of a good common brick of 8.25 X 4 X 2 ins, will aver- 

about 4.o n»s; or 118 lbs per cub ft — 3186 lbs or 1.42 tons per cub yard; or £.01 tons per lOOO. 

g'oou pressed brick of the same size will average about 5 lbs, = 131 lbs 
r cub It = 3537 lbs or 1.58 tons per cub yd ; or 2.23 tons per 1000. 
Immersed in water, either of them will in a few minutes absorb from 


o lb of water ; the last being about ^ of the weight of a hand moulded one; or A of its own 
k. Since the weight of hardened mortar averages but little less than that of good common brick, 
may for ordinary calculations assume the weight of such brickwork at 1.4 tons per cub yard ; 1.3 
s per perch of 25 cub ft; or 116 lbs per cub ft; or for machine-moulded, at 1.56 tons per cub vd - 
i tons per perch; or 129 lbs per cub ft. 

llowing for the usual waste in cutting bricks to fit corners, jambs, &c, the average number of 

X 4 X 2, required per sq foot of wall is as follows: 


‘| Thleknessof Wall. No. of Bricks. 

8J4ins,orl brick... 14 

j 12 % “ or 1>4 “ . 21 

,i 17 “ or 2 “ . 28 

J 25 ^" or 3 •* . 42 


[.aying per day, a bricklayer, with a laborer to keep him supplied with materials, will in 
1 moo house walls, lay on an average about 1500 bricks per day of 10 working hours. In the neater 
hr faces of back buildings, from 1000 to 1200 ; in good ordinary street fronts, 800 to 1000; or of the 
f finest lower story faces used in street fronts, from 150 to 300, depending on the number of angles 
In plain massive engineering work, he should average about 2000 per day, or 4 cub yds; and 
, 'arge arches, about 1500, or 3 cub yds.f 

[ ince bricks shrink about yj part of each dimension in drying and burning, the moulds should be 
ut -jL part larger every way than the burnt brick is intended to be. 
ood well-burnt bricks will ring when two are struck together. 

t the brick-vards about Philadelphia, a brick-moulder’s work is 2333 bricks per day; or 14000 per 
k. He is assisted by two boys, one of whom supplies the prepared clay, moulding sand, and 
er; while the other carries away the bricks as they are moulded. A fourth person arranges them 
-ows for drying. About % of a cord, or 96 cub ft or wood, is allowed per 1000 for burning 0 Where 
1 is used, the kilns are fired up with anthracite, and the finishing is done with bituminous. Oue 
of coal, in all, makes 4500 bricks. 


avifl^ Wlttl brick. Tn our cities this is done over a 6-inch layer of gravel, which 
aid be free from clay, and well consolidated. With bricks of 8%X4X2 ins. with joints of from 
4 inch wide, a sq yard requires, flatwise, as is usual in streets, 38 bricks; edgewise, 73; endwise, 
An average workman, with a laborer to supply the bricks and gravel, will in 10 hours pave about 
) bricks; or 53 sq yds fiat, 27 edgewise, 13 endwise. When done, sand is brushed into the joints. 

krt. 3. The crnstliaig- strenigrth of brieBtS of course varies greatly. A 
,'ter soft one will crush under from 450 to 600 Tbs per sq inch ; or about 30 to 40 tons per sq it: « bile 
st-rate machine-pressed one will require about 200 to 400 tons per square foot. This last is 
ut the crushing limit of the best sandstone; % as much as the best marbles or limestones ; and J j 
nuch as the best granites, or roofing slates. But masses of brickwork crush uuder much smaller 
is than single bricks. In some English experiments, small cubical masses only 9 inches on each 
e, laid in cement, crushed under 27 to 40 tons per sq ft. Others, with piers 9 ins square, aud 2 ft 
is high, in cement, only two days after being built, required 44 to 62 tons per sq ft to crush them. 
>ther, of pressed brick, in best Portland cement, is said to have withstood 202 tons per sq ft; and 
h common lime mortar only as much. See page 436. 

; must, however, be remembered, that cracking and splitting usually commence under about one- 
f the crushing loads. To be safe, the load should not exceed % or jyj- of the crushing one; and 
vith stone. Moreover, these experiments were made upon low masses ; but the strength decreases 
h the proportion of the height to the thickness. 

he pressure at the base of a brick shot-tower in Baltimore, 216 feet high, is estimated at 6H tons 


iTlie Peerless Brick Co, office No. 1003 Walnut St, Philada, make superb smooth 
t shining) bricks of various shapes and colors (as white, black, gray, buff, brown, red, &c) for 
aniental architectural purposes. Their standard size is 8% X 4!^ X 2% =82 cub ins, or % larger 
n the above 8)4 X 4 X 2 ins. The color extends throughout the body of the brick. With a few 
these judiciously distributed among common bricks, beautiful architectural effects maybe pro- 
:fcd, both indoors and out, at far less cost than in stone. For prices and illustrated catalogue, 
Ircss as above. 

'he same Co will, if a sufficient order is given, furnish voussolr bricks for specified radii, but of 
quality and finish of ordinary good hard brick, at from 25 to 50 per ct advance on prevailing 
irket rates of common plain ones. 

Prices in Philada in 1888. Bricks alone; Salmon, or soft, $7 per 1000. Hard brick, $10. 
;k stretchers (generally used for the facings of baok buildings, &c), $14 Paving brick, $14. 
■ssed, (for lower stories of first-class fronts,) $26. 

Bricklaying; including mortar; averaging an entire dwelling, $7 per 1000. Best pressed bricks 
Srst-class fronts, $15. “ Tuck ” fronts, laid with steel wire, special rates. 

















MORTAR, BRICKS, ETC. 




Tensile strength of brick, 40 to 400 lbs per sq inch ; or 2.6 to 26 tons per sq 

The English ro«l »1 brickwork is 306 cub feet, or 11^ cub yards; ai 
requires about 4500 bricks of the English standard size; with about 75 cub ft of mortar. The Engli 
hundred oj lime, is a cub yd. 

Frozen niortsir. There is risk in using common mortar in cold weather. If the 
should continue long enough to allow the frozeu mortar to set well, the work may remain safe , but 
a warm day should occur between the freezing and the setting of the mortar, the suu shining on o 
side of the wall may melt the mortar on that side, while that on the other side may remain froz 
hard. In that case, the wall will be apt to fall; or if it does not, it will at least always be weak; 
mortar that has partially set while frozen, if then melted, will never regain its strength. By t 
writer’s own trials hydraulic cements seemed not to be injured by freezing. 


per sq ft; and in a brick chimney at Glasgow, Scotland, 468 feet high, at 9 tons. Professor Rank 
calculates that in heavy gales this is increased to 15 tons, on the leew ard side. The walls of both 
of course much thicker at bottom than at top. With walls 1U0 feet high, of uniform thickness, 
pressure at base would be 5.4 tons per sq ft. 

With our present imperfect kuowledco on this subject, it cannot be considered safe to expose ev 
first-class pressed brickwork, in cement, to more than 12 or 15 tons per sq ft; or good haud-mould( 
to more than two-thirds as much. 


Experiments for rendering briek masonry impervious t 

Water. Abstract of a paper read before the Americau Society of Civil Engineers, May 4, 18“ 
by William L. Dearborn, Civil Engineer, member of the Society. 

The face walls of the Back Bays of the Gate-houses of the new Croton reservoir, located nor 
of Eighty-sixth Street, in Central Park, were built of the best quality of hard-burnt brick ; laid 
mortar composed of hydraulic cement of New York, and saud mixed in the proportion of one measu 
of cement to two of sand. The space between the walls is 4 ft; and was filled with concrete. The f; 
walls were laid up with great care, and every precaution was taken to have the joints well filled a 
insure good work. They are 12 ins thick, and 40 ft high ; and the Bays when full generally have 36 
of water in them. 

When the reservoir was first filled, and the water was let into the Gate-houses, it was found to filt 
through these walls to a considerable amount. As soon as this was discovered, the water was drat 
out of the Bays, with the intention of attempting to remedy or prevent this infiltration. After cat. 
fully considering several modes of accomplishing the object desired. I came to the conclusion to t 
“ Sylvester's Process for Repelliug Moisture from External Walls." 

The process consists in using two washes or solutions for coveriug the surface of brick walls; o 
composed of Castile soap and water; and one of alum and water. The proportions are : three-qu: 
ters of a pound of soap to one gallon of water; and half a pound of alum to four gallons of wate 
both substances to be perfectly dissolved in the water before being used. 

The walls should be perfectly clean and dry ; and the temperature of the air should not be beh 
50 degrees Fahrenheit, when the compositions are applied. 

The first, or soap wash, should be laid ou when at boiling heat, with a flat brush, taking care i 
to form a froth on the brickwork. This w r ash should remain tweutv-four hours ; so as to oecome d 
and hard before the second or alum wash is applied; which should be done in the same manner 
the first. The temperature of this wash when applied may be60° or 70°; and it should also rema 
twenty-four hours before a second coat of the soap wash is put on ; and these coats are to be repeat i 
alternately until the walls are made .mpervious to water. 

The alum and soap thus combined form au insoluhle compound, filling the pores of the masonr 
and entirely preventing the water from penetrating the walls. 

Before applying these compositions to the walls of the Bays, some experiments were made to tt 
the absorption of water by bricks under pressure after being covered with these washes, in order 
determine how many coats the wall would require to render them impervious to water. 

To do this, a strong wooden box was made, put together with screws, large enough to hold 2 brici ;j 
and on the top was inserted an iuch pipe forty feet long. 

In this box were placed two bricks after being made perfectly dry, aud then covered with a coat- 
each of the washes, as before directed, and weighed. They were then subjected to the pressure o 
column of water 40 feet high : and, after remaining a sufficient length of time, they were takeu 
and weighed again, to ascertain the amount of water they had absorbed. 

The bricks were then dried, and agaiu coated with the washes and weighed, and subjected to pri 
ure as before : and this operation was repeated until the bricks were found not to absorb any watt 
Four coatings rendered the bricks impenetrable under the pressure of 40 ft head. 

The mean weight of the bricks (dry) before being coated, was 3% lbs; the mean absorption w 
one-half pound of water. An hydrometer was used in testing the solutions. 

As this experiment was made in the fail and winter. (1863.) after the temporary roofs were put 
the Gate-house, artificial heat had to be resorted to, to dry the walls and keep the air at a pro| 


to 


k 


temperature. The cost was 10.06 cts per sq ft. As soon as the last coat, had become hard the wat 
was let iuto the Bars, and the walls were found to be perfectly impervious to water, aud they st 
remain so in 1870, after about 6H years. 

Brick arch (footway op High Bui non). The brick arch of the footway of High Rrldge is t 
arc of a circle 29 ft 6 in radius ; and is 12 in thick ; the width on top is 17 ft; and the length cover 
was 1381 ft. 

The first two oourses of the brick of the arch are composed of the best hard burnt brick, laid ed;) 
wise in mortar composed of one part, by measure, of hydraulic cement or New York, aud two pai 
of sand. The top of these bricks, and the inside of the granite coping against which the two t 
Courses of brick rest was, when they were perfectly dry, covered with a coat of asphalt one-half 
inch thick, laid on when the asphalt was heated to a temperature of from 360° to 518° Fahrenheit 

On top of this was laid a course of brick flatwise, dipped in asphalt, and laid when the asphalt w 
hot; and the joints were run full of hot asphalt. 

On top of this a course of pressed brick was laid flatwise in hydraulic oement mortar, forming t 
paving and floor of the bridge. This asphalt was the Trinidad varietv : and was mixed with 10 ; 
cent, by measure, of coal tar; and 25 per cent of sand. A few experiments for testing the streng 
or this asphalt, when used to'oement bricks together, were made, and two of them are given belov 

Six bricks, pressed together flatwise, with aspha't. joints, were, after lying six months, brokt 
The distance between the supports was 12 ins ; breaking weight, 900 tbs ; area of single joint. 2b>4 
ins. The asphalt, adhered so strougly to the brick as to tear away the surface iu many places. 










CEMENT, CONCRETE, ETC. 


673 


wo bricks pressed together end to end. cemented with asphalt, were, after lying 6 months, broken, 
he distance between the supports was 10 ins; area of joint, 8J4 sq ins; breaking weight, 150 tbs. 
he area of the bridge covered with asphalted brick, was 23065 sq ft. There was used 91200 lbs of 
halt, 33 barrels of coal tar, 10 cub yds of sand, 93800 bricks. 

he time occupied was 109 days of masons, aud 118 days of laborers. Two masons and two labor- 
will melt aud spread, of the first coat, 1650 sq ft per day. The total cost of this coat was 5.25 
ts per .sq ft, exclusive of duty on asphalt. There were three grooves, 2 ins wide by 4 ins deep, 
de eutirely across the brick arch, and immediately under the first coat of asphalt, dividing tue 
n into four equal parts. These grooves were filled with elastic paint cement, 
his arrangement was intended to guard against the evil effects of the contraction of the arch in 
ter; as it was expected to yield slightly at these points, and at no other point; and then tue 
itic cement would prevent any leakage there. 

he entire experiment has proved a very successful one, and the arch has remained perfectly tight, 
it proposing the above plau for working the asphalt with the brickwork, the object was to avoid 
euding on a large continued surface of asphalt, as is usual in covering arches, which very fre- 
ntly cracks from the greater contraction of the asphalt than that of the masonry with which it is 
jontact; the extent of the asphalt on this work being only about one-quarter of an inch to each 
;k. This is deemed to be an essential element in the success of the impervious coveriug.” 

cheap and effective process for preventing the percolation of water through the arches of aque 
its, and even of bridges, is a great desideratum. Many expensive trials with resinous compounds 
e proved failures. Hydraulic cement appears to merely diminish the evil. Much of the trouble 
robably due to cracks produced by changes of temperature. 

The white efflorescence so common on walls, especially on those of brick, 
lue to the presence of soluble salts in the bricks and mortar. These are dissolved, 
d carried to the face of the wall, by rain and other moisture. Sulphate of magne- 
(Epsom Salt) appears to be the most frequent cause of the disfiguration. In many 
oces mortar lime is made from dolomite, or magnesian limestone, which often con¬ 
ns 30 per cent or more of magnesia; which also occurs frequently in brick clay, 
al generally contains sulphur, most frequently in combination with iron, forming 
* well-known “ iron pyrites ”. The combustion of the coal, as in burning the llme- 
ne or clay, in manufactures, in cooking etc, converts the sulphur into sulphurous 
<1 gas, which, when in contact with magnesia and air, as in the lime or brick kiln, 
in the finished wall or chimney, becomes sulphuric acid and unites with the mag- 
da, forming the soluble sulphate. We are not aware of any remedy that will pre¬ 
fit its appearance under such circumstances; but the formation of the sulphate may 
prevented by the use of limestone and brick-clay free from magnesia. See p. 678. 


Irt. 4. Hydraulic cements.* Certain limestones, when burnt, will not slack 
b water; but when the burnt stone is finelv ground, and made into a paste, jt possesses the pro- 
ty of hardening under water: and is therefore called hydraulic cement. So long as the propor- 
i of those ingredients which impart hydraulicity, is so small that the burut stone will slack ; but 
1 make a paste or a mortar, which will harden under water, it is called hydraulic lime . This does 
harden so promptly, or to so great a degree, as the cements. Hydraulic limes slack more slowly, 
! swell less, iu proportion to their hydraulicity ; some requiring many hours. Artificial hydraulic 
les and cemeuts, of excellent quality, may be made by mixing lime and clay thoroughly together; 
u moulding the mixture into blocks like bricks; which are first dried, then burut, aud finely 
uud. The celebrated artificial English Portland cement, is made by grinding together iu water 
lk and clay. The fine particles are floated away to other vessels, and allowed to settle as a paste ; 

,ich is then collected, moulded, dried, burnt, and ground. Natural Port- 
mi is that made from limestone, or other material of very rare occurrence, 

tch combines naturally that proportion of lime and clay which gives the above artificial 1 ortlauds 
ir pre-eminence. This alone constitutes its difference from our common natural hyd cemeuts. 

fVeiglits of cements, p. 382. Saylor’s Portland, about 120 lbs. per struck bushel. 

he writer found by 10 years' trial that if, after setting, dampness is absolutely excluded. Cements 
‘.serve Iron, lead, zinc, copper, and brass; aud that Plaster of Paris preserves all except 
i, which it rusts somewhat unless galvanized. Lime-mortar probably preserves all of them, 
ept free from damp. 



Prices ofliyd cements in Philada, 1888, by the large importing firm of 
uel II French’& Co, corner of York Avenue and Callowhill St. huglish 
irtland cement, $2.60 to $3.00 per barrel of about 400 lbs gross; according to 
LlTtv and Quantity German Portland. $2.40 to $2.75 per barrel of about 
mJeroi q Savior’s American Portland, $2.35 to $2.05 per barrel of 
i lbs o -oss lto'sendale, per barrel of about 300 Its net, $1.15 to $1.40. Ot her 
S cements, per barrel of about 300 lbs net, $1.00 to $EiO. Ground oal- 
ned plaster of Paris : selected, barrel of 300 lbs net, $2.00 to $2.2o. 
immercial, barrel of about 200 lbs net, $1.2 ) to $l.xa 

kmerlean Improved Cements Co., Egypt,]■!**» £ 16 S ' 3d St *’ 

ila., make "Giant” Portland. Price per barrel of 400 ft* net, $-.40 
“ Improved Union.” “ ' 300 1. 

“Union” “ “ 300 “ L1 ° 

3ee “Mr. Eliot ('. Clarke,” p 678. 










674 


CEMENT, CONCRETE, ETC. 


appreciabh - deteriorate Tor six months; but after 14 or 16 months, Gillmore says it is unfit for use 
important works. Hut in lumps, kept dry, it will remaiu good for 2or 3years; aud may be groi 
as required for use. 

Good Portland cement is stated by good authority rather to improve by free exposure to the 
under cover; but whether this is correct or not, we cannot say. 

Restoration l».y rehurning may be effected. Tf the injured ground 

meut is spread in a thin layer, on a red-hot iron plate, for about 15 minutes, its good qualities wil 
in a great measure restored. The time should be ascertained by trial. If it has been actually x 
and lumpy, or cemented into a mass, it should first be broken into small pieces, aud then grouud. i 
these pieces may be first kiln-burnt at a bright red-heat for about 1J>$ hours; aud then ground. I 

Art. 5. For roughcasting. or stuccoing the outside of walls, very II 

hydraulic cements are fit. Mr. Dowuing, iu his work on “ Country Houses." excepts that from Ber I 
Connecticut, as the only one within his exteuded knowledge, that is suitable. Portland cemen i 
•aid to be good for that purpose. A wall with a northern exposure iu Philada was coated with i i 
1860; and appears to be iu perfect condition in 1880. 

Quantity required. A barrel of cement, 800 lbs; and 2 barrels of sand, 
bushels, or 7J4 cub ft;) mixed with about % a barrel of water, will make about 8 cub ft of rnori 
sufficient for 

192 sq ft of mortar-joint H inch thick = 21^ sq yards. 

288 “ “ *« “ ^ “ •* = 32 “ “ 

384 “ “ “ “ - 42J$ “ “ 

768 “ “ « “ % “ “ = 85X “ “ 

Or, to lay 1 cubic yard, or 522 bricks of 8bf by 4, by 2 ins, with joints % inch thick; or a cubic y 
of roughly scrabbled rubble stonework. The quantity of sand may be increased, however, to 3 c 
measures for ordinary work. 

Pointing' mortar. Gen Gillmore recommends “1 part by weight of gc 

cemeut powder, to 3 or 3^ parts of sand. To be mixed under shelter, and in quantities of only : 

3 pints at a time, using very little water, so that the mortar, when ready for use, shall appear rat 
incoherent, aud quite deficient iu plasticity. The joints being previously scraped out to a dept! 
at least ^ an iuch, the mortar is put in by the trowel; a straight edge being held just below the joi 
if straight, as an auxiliary. The mortar is then to be well calked into the joint by a calki 
iron and hammer; then more mortar is put in, aud calked, until the joint is full. It is then rub 
and polished uuder as great pressure as the mason can exert. If the joints are very fine they sho 
be enlarged by a stonecutter, to about y 3 ^ inch, to receive the pointing. The wall should be well 
before the pointing is put iu, aud kept in such condition as neither to give water to, nor take it from 
mortar. In hot weather, the pointing should be kept sheltered for some days from the sun, so as 
to dry too quickly.” Why not finish joints at once, without subsequent pointing ? Author. 

Art. 6.* Folor is no indication of strength in livd cements. The fin 
fliey are ground the belter. At least 90 per ct should pass through 

sieve of 50 meshes per lineal inch, of Wire No 35 Amer wire gauge (.0056 inch thick); or 2500 ui'es 

per sq inch. Weight is a good indication when equally well ground. At* 

cake of good cement paste placed in water as soon as it admits of so doing safelv, aud left in it 0 ! 

week, should show no cracks. New cement is not as good as when a fj 
weeks old. The term Setting does not imply that the cement has harder, 
to any great extent, but merely that it has ceased to be pasty and has become brittle, quick sett i 
cements may do this sufficiently to allow small experimental samples to be lifted and handled c: 
fully within five to thirty minutes; while others may require from one to eight or more lid 

Sioiv setting does not indicate inferiority, for many of the v' 

best are the slowest setting. A layer of very quick setting cement may partially set. especial! 
warm weather, before the masonry is properly lowered and adjusted upon"it, and any dlsturbai 
after setting: has commenced is prejudicial. Such are to be regarded with suspicion, and e] 
mined to longer tests than slow ones. Still, quick setting ones are best in certain cases, as w 
exposed to running water, &c. They may be rendered slower by adding a bulk of lime paste equal ! 
or 15 per ct of the cemeut paste, without weakening them seriously. As a general rule term I 

set anti harden better in wa ter than in air, especially in wa < 

weather. If, however, the temp for the first few days does not exceed 55° to 65° Kah, there seem.' 1 
be no appreciable difference in this respect; but iu warm air cement dries instead of setting, and t 
loses most of its strength. In hot weather every precaution should be used against t 

The time reqd to attain the greatest hardness is many years, \ 

after about a year the increase is usually very small and slow, especially with ueat cement. M< 
over, any subsequent increase is a matter of little importance, because generally bv that. time, 
often much sooner, the work is completed and exposed to its maximum strains. Sand retai 
setting, and weakens the cement paste. But although with sand the strength of the mortar t 
never attain to that of the neat paste, yet it increases with age in a greater proportion; so tin 
ueat paste which at the end of a year would be but twice as strong as in 7 days, may with sand v 
a mortar which at the end of a year will be 3. 4. or 5 times as strong as it was itt 7 days. Good P 
lands neat usually have at the end of a year from 1.5 to 2 times their strength at the end of 7 da 
and the Americau natural cements, Rosendale, l.omsville, Cumberland, the. from 2.5 to 3.5 tin 
but inasmuch as Portlands average (roughly speaking) about 5 or 6 times the strength of the ott 
in 7 days, they still average about 2.5 to 3 times as strong in a year or longer. Cements of the st 

class differ much in their rapidity of hardening. One may at the end 

a month gain nearly one-half, and another not more than one-sixth of its increase at the end ( 
year, ai which time both may have about the same strength. Hence, tests for 1 week or 1 mouth 
by no means conclusive as to their final comparative merits. 

There seems to be a period occurring front a few weeks to several months after having been laid 
which cement and its mortars for a short time not only cease from hardening, but actually li 
Strength. They then recover, and the hardening goes on as before. It has been suggested t 
this opittiou has originated iu some oversight of the experimenters, but the writer believes it to 
founded on fact. In his expts with various hvd cements or the consistence of mortar, even with 
sand, the writter detected no change of bulk In setting. 


* See “ Mr. Eliot C. Clarice ”, p G78. 








vaiviiiax, UOiNCKETE, ETC. 


675 


rMi^Ara'soc 1 ?’ V" E :£ 8ee ,,is v , erv illstrul:t i'’e paper in 

>6 tbs per sq inch, while that kept iu water of 700 9Q u if at 40 ’ a tensile strength of but 
loulded in air at 60°, after 6 dirs in wnr PP nr k ??’ ^ ^ as rauc ^- Other bars 

i water at 70° required 254 B)s ^or about 2 25 WVt £ ll, ^ s ten sile per sq inch, while those 

tting, as the temp exceeds about 65° to 70° But if mixed onlv iii^msl? \7^ r S in 9 instead of 

mmmmmmm 

gh temps. But the reverse occurs wfth suTh U Snafuralce 

some being largely increased by such drying. Experiments iu Europe with Portlands keDt3*™n&s 
water, seemed to show the weakest period for such to be at 2 days’ exposure when ** 

t half as great as when first taken from the water But on the tth wflen the stret >gth vu 

first; and the strength then increased with time as if there had been noln^iTuptiot^ 008 *^ tha “ 

The efTeets of cold, although it retards the setting, do not annear to he 

lous otherwise. If the cemeut mortar even freezes almost as quickly as the masonnMs laid wirh 
it does not seem to depreciate appreciably. The writer has found this to be the case also with iw 

, i r in ar ’ e R r n u e “, h0U ? aft f fr , eeziu « the tem f became so high as to .soften the froxen mortar 
uu. But although the mortar of either lime or cement mav not be therphv iniurpd *h n * », 

Uy in thin brick walls, may be ruined and overthrown Thus if ««nn|^ ured ’ the wot*, espe- 
i ^thickness of such a wafl be frozen, fleet? H ^L^toX^Mrut 

V ha l face ’" h,le tbe mortar behind it remains hard, it is plain that the wall will be liable tn settle 
; the heated face, and at least bend outward if it does not fall. The writer ha’ obstrved that coat 
,s of cement appUed to the backs of arches on the approach of winter, and left unprotected were 
tuely broken up and worthless on resuming work the next spring. Tlic 
Pitting' of sand and cement in freezing weather seems to be a had nr»e- 

b coo laCe ^ ia i C j ld Wat ! r ’ Bu . 1 for use out of water Mr Maclav says they maf be 

ated to 50 or 60 . Cold water tor mixing’ is probably no farther iniu- 

us than that it retards the setting. All cements when mixed with sand to a proper eonsistence^or 

1,^ ,al1 *2 P ,e ® es lf placed in water before setting has commenced, 

tlands do so even without sand; but U. S. natural cements of good quality do not. 

I*-/-*-* Stre_ngth of cements. The strength as before stated is much 

j b> , th * tem P ?. f th . e al ^ aQd " ater . as als ° by the degree of force with which the cement is 
ssed into the moulds; by the extent of setting before being put into the water, and of drving when 
eu out, and still more by the consideration of whether or not it sets while under the influence of 
ssnre which increases the strength materially. Ou this account cements in actual masonry may 
ler ordinary circumstances give better results than in door expts. These causes, together with the 

f ee of thoroughness ol the mixing or gauging, the proportion of water 

d aud other considerations may easily affect the results 100 per ct, or even much more. Hence 
discrepancies in the reports of different experimenters. ’ 

®« m * Portlands require more water than the common U. S. 

lents, and shrink less in mixing. See next Art. Also, mortars require more than concrete, espe- 

liy when the last is to be well rammed, in which case it should be merely 

st. so as barely to cohere when pressed into a ball by hand. If more water is present, theconsoli- 

iou by ramming is proportionally imperfect. To assure himself that 
e quality of cement furnished is equal to that contracted for, the engineer 

, uld reserve the right to bore with a long auger into any part of each barrel, and to reject everv 
i rel of which the sample drawn out does uot satisfy the stipulations. On works using large quan- 
:s, there should be one person specially detailed to this dutv. One advantage of very strong 
ents is their economy, even at a higher cost, in allowing the use of a larger proportion of the 
iper ingredients, sand, gravel, and broken stone. Almost any common U. S. cement, if of good 
hty, will with 1.5 or 2 measures of sand give a mortar strong enough far most engineering pur¬ 
ls ; but a good Portland will give one equally strong with 3 or 4 meas of sand; and will, therefore 
qually cheap at twice the price; beside requiring the handling, storing, and testing of onlv half 
number of barrels. ° J 

fter what has been said it is plain that great latitude must be allowed in attempting to prepare a 
e of approximate average strengths. The writer can pretend to nothing more than the following 
ch is deduced from reliable reports, aided by a few experiments of his own ou transverse strengths’ 
mmary of which last forms the last column. ° 

one measure of cement slightly shaken be mixed to a paste with about .35 meas of water if a 
mon U. S. cement, or about .40 meas if Portland, in the shade, and in a temp of from 60° to 90°, 
paste will occupy about .7 meas if common, and about .86 if Portland, when well pressed into 
ien moulds by the fingers (protected from corrosion bv gloves of rubber or buckskin). If then 
ved front 30 minutes to some hours (according to its setting properties) to set; then removed from 
noulds, and at the end or 24 hours total, placed in water of the above limits of temp for 7 davs, 
broken at once when taken from the water, the samples will generally exhibit about the following 
lgths. Those for compression are supposed to be cubes ; and those for transverse strength iu the 
[5 were beams 1 inch square, and 12 ins clear span, loaded at the center. 


* See “ Mr. Eliot C. Clarke ”, p 678. 

47 







CEMENT, CONCRETE, ETC, 


t)7(5 


Table A. Average Ultimate Strengths of neat Cements aftei 
6 days in water, aud broken directly from the water. 


Portlands, artificial, either foreign, or the 

Tensile, lbs 

per sq in. 

Com pres, lbs 
per sq in. 

Compres tons 
per sq ft. 

Trnnsv. 1" ) 

1" X 12". B>i 

“ National ” of Kingston, N.Y. 

200 to 350 

1400 to 2400 

90 to 154 

25 to 45 

“ Saylor's natural, Coplav, Penn.. 

170 to 370 

1100 to 1700 

71 to 109 

26 

U. S. common hydraulic cements. 

40 to 70 

250 to 450 

16 to 29 

3 to 7 


All below the lowest of these should be rejected ; the average of the table maw _be considered fair 

and all above the highest superior. After only 24 hours in water th 

strength of the common U. S. cements averages about half that for 6 days, but with considerab! 

variations both ways. In like manner at the end of a year neat Portland 

average from 1.5 to 2 times as strong as in 6 days; and onr common cements from 2.5 to 3.5 timet 

The London board of works require that Portlands after 7 days in wate 

shall have at least 35 lbs transverse, and 350 lbs tensile strength. Some have reqd 500 or more ter 

sile to break them. For Portlands ihe writer found the transverse 

strengths of several well known English brands moulded as before described, to be 26 to 40 lbs after 
days in water; National Portland of Kingston, N. Y., 40 and 46; Saylor’s Portland (only 2 trials) 2 i 

Tbs.‘ Toepffer, Grawitz, A- Co, of Stettin, Germany, warrant all thei 
Portland (known as the ‘‘ Stern ” brand) equal to 56# lbs tensile after 7 days in water. Some of it ha i 
borne 760 Bis. 

Mr. J. Herbert Shedd, as Engineer of the Water Works and Sewers o 

Providence, R. I., rejected all Rosendale which when mixed to a stiff paste, and allowed 30 min in ai 
to set, and then put into water for only 24 hours, broke with less tension thau 70 lbs. At first h 
found some- that broke with 10 to 15 lbs; some that would not set at all in water; and but little thf 
bore 30 lbs. Now samples frequently bear 100 Bis or more; but that usually sold still rarely exceec 

40 to 50, and frequently scarce half as much. The Sewer Department o: 
St. I.oiiis. Missouri, requires all Louisville, Kentucky, cement to hear at leas 

4n lbs tensile after 24 hours iu water. Some of it now shows as high as 100 or more; and 60 or 7 
would have been adopted as the mininum, but for the fear that it would have encouraged the makin 
of too quick setting cement. Most of that sold will probably not exceed 30 lbs. 

Art. 9. Cement mortar is cement mixed with water and sand only 

The writer found that for making cement pastes of about equal consistency and fit for mortar 1 
themselves, the English Portlands, slightly shaken in the measure, required an average of about 
of their own bulk of water; and the U. S. common cements about .35. The Portland pastes whe 
thoroughly mixed and slightly pressed by hand into a box shrank about one-eighth of their bulk i 
dry shaken cement; and the others about one-fourth ; or in other words the common U. S. ccmen 
shrink about twice as much as the Portlands ; and these are about the proper proportions to assun 
in estimating the quantity of cerneut for theoretically filling the voids in sand. 

But, when sand In added, more water is reqd. It is impossible to la 

down rules for all cases, but as a very rough average, mortar will require an addition equal to ahoi, 
.2 of the bulk of dry sand ; varying of course with the weather, &c. Trial on the work in baud 
better than rules. 

Any addition of sand weakens eement. especially as regards te 

sion ; as it does also lime mortar. But economy requires its use. Sand also retards the setting, 
that cement which by itself would set in half an hour, may not do so for some days if mixed with 
large proportion of sand. This weakening effect will of course vary with different cements, and wi 

many circumstances inferable from Art. 7, Ac. As a roue'll average tl 
following is perhaps not far from the truth as regards either tensile or transverse strength when n 
rammed. See p 682. 


Sand. 

0 


i 

i H 

2 

3 

4 j 5 

6 

7 i 8 

Strength. 

1 

% 


A 


.3 

X A 1 \ 

% 

1 i 1 / 


Tensile Strength of Cement Mortars,* 

of medium coarse sea-beach sand, and good Rosendale, and English Portland c 
ments; being averages of about 25000 experiments in the years 1878 to 1882. Tl 
area of breaking section was 2.25 sq ins. The preportions of «md and cement we 
by measure. The mortar was rammed into the moulds, and the specimens we 
immersed in water as soon as they would bear handling, and so remained for \ da 
or 1 week, or for 1, 6, or 12 months. The strengths are in lbs l>er sq inch. 


* From experiments by Mr. Eliot C. Clarke, C. E ; of Boston. See also p 678. 







































CEMENT, CONCRETE, ETC. 


677 


Rosen dale. 


I>. 

1W. 

Neat 

1M. 

651. 

1 Y. 


Cen 

1W. 

(lent 1. 

1M. 

San 

6 51. 

d 1. 

1 Y. 


Ceir 

15V. 

lent 1. 

1M. 

Sand 

6 51. 

1.5. 

1 Y. 

71 

92 

145 

282 

290 


56 

116 

180 

236 

41 

90 

135 

210 

Ce 

ment 

22 

.1. 

49 

Sand 2. 
105 1169 

Cen 

12 

oent 1. 

25 

Sand 3. 

65 | 121 

Cen 

tient 1. 

10 

Sand 5. 

36 | 80 

>2 

Neat 
303 j 412 

468 J |94 


Por 

Cement 1. 

160 j 225 

tland 

San 

347 

1. 

d 1. 

387 


Cen 

lent 1. Saud 

I 

1.5. 

Cement 

|l26 

1. 

163 

Sand 

279 j 

2. 

323 

Cement 1. 
95 | 130 

San 

198 

d 3. 

257 

Oement 1. 
55 | 78 

Sand 5. 

116 j 145 


The crushing' strength. For each proportion of sand we may take the 
iogth preceding it in the table, p 676. Moreover the crushing strength with sand increases with 
much more rapidly than the tensile; and the more so the greater the proportion of sand. 


>s a general rule with cements of good quality we shall have mortars fit for most engineering pur- 
es ir we do not exceed from 1 to 1.5 measures of dry sand to 1 of the common cements; or from 2 
! of sand to 1 of Portland. 

The shearing strength of neat cements averages about one-fourth of the 

sile. 

The adhesion of cements to bricks or rough rubble, at dif- 

:ut qges, and whether neat or with sand, may probably be taken nt an average of about three- 
rths of the cohesive or tensile strength of t he cement or mortar at the Rame age. If the bricks and 
ie are moist and entirely free from dust when laid, the adhesion is increased ; whereas if very dry 
i dusty, especially in hot weather, it may be reduced almost to 0. The adhesion to very bard, 
>ot.h bricks, or to finely dressed or sawed masonry is less. 

The voids in sand of pure quartz like that found on most, of our sea- 

res, when perfectly dry and loose, occupy from .303 ofthe mass in sand weighing 115 tbs per cub ft, 
515 in that weighing 80 lbs. Usually, however, such dry sand weighs say from 105 lbs with voids of 
,: to 95 lbs, with voids .424 ; the mean being 100 lbs, with voids .394.* But the wet sand in mortar 
jpies about from 5 to 7 per cent less space than when dry ; the shrinkage averaging say 6 per ct 
r V part; thus making the voids .304 or the 105 lb sand when wet; and .364 of the 95 lb ; the mean 
which is .334. But to allow for imperfect mixing, &c, it is better to assume the voids at .4 of the 

• sand. Moreover, since the cements, as before stated, shrink more or less when mixed with water, 
til worked up into mortar, it would be as well to assume that to make sufficient paste to thoroughly 

the voids, we should not use a less volume of dry common oement, slightly shaken, than half the 

• Ik of the dry sand; or than .45 of the bulk if Portland. The bulk ofthe 
5 ixed mortar will then be about equal to or a trifle less than that of the dry 

d alone. 

; The best sand is that with grains of very uneven sizes, and sharp. The 

k.-e uneven the sizes the smaller are the voids, and the heavier is the sand. It should be well 
' ihed if it contains clay or mud, for these are very injurious to mortar or concrete. 


If greater accuracy Is desired pour into a graduated cylindrical 

suring-glass 100 measures of dry sand. Pour this out, and fill the glass up to 60 measures with 

Her. Into this sprinkle slowly the same 100 measures of dry sand. These 

now be found to fill the glass only to say 94 measures, having shrunk say 6 per ct; while the 
»r will reach to say 121 measures; of which 121—94 = 27 measure's will be above the sand; leaving 
27 = 33 measures filling the voids in 94 measures of wet sand; showing the voids in the wet sand 

>e If = .351 of the wet mass. If the sand is poured into the water 

ily, ajr is carried in with it, the voids will not be filled, and the result will be quite different. 

,iinee a cubic foot of pure quartz weighs 165 lbs. it follows that 

J 3 weigh a cubic foot of pure dry sand either loose or rammed, then as 165 is to the wt found, so is 
the solid part of the sand. And if this solid part be subtracted from 1, the remainder will be the 
»)|s, as below. 


Vt in lbs per cub ft dry 80 
Proportion of solid .485 
proportion of voids .515 


85 

90 

95 

100 

105 

110 

115 

.515 

.546 

.576 

.606 

.036 

.667 

.697 

.483 

.454 

.424 

.394 

.364 

.333 

.303 





































































678 


CEMENT, CONCRETE, ETC. 


The common (not Portland) cements, when used as mortar for brie« cps ' whic 

ally near sea coasts, and in damp climates, by wl^^^^orescences 3, 

sometimes spread over the entire exposed face of the work, and also mjiirc the or cK. • I 

occurs in stone masonry, but to a much less extent, and is confined to the mortar joints j and l J 
only porous stone. It is usually a hydrous carbonate of soda or of potash often containing ot ■ 
Gen'I Gillmore recommends as a preventive to add to every 300 lbs (Ibarrel) of th ® 

100 lbs of quicklime, and from 8 to Pi lbs of any cheap animal fat. The fat to he wel1 incorpor* 
with the quicklime before slacking it preparatory to adding it to the cement. This i* , h 
tard the setting, and somewhat diminish the strength of the cement It is jf, j 

linseed oil at the rate of 2 gallons to 300 lbs of dry cement, either with or 

exposures prevent efflorescence, but like the fat it greatly retards setting, and weakens, bee also 
673. __ , yv , „ | y 

r , Eliot C. Clarko, to whom we are already indebted for tobies on pp *5« 
and 68*2, lias published, in Trans Am Soc C E April 1885, the results of a series of e: 
pertinents made for the Boston Main Drainage Works. From his paper we condem 
as follows, by permission. 

Variations in sliadr in a given kind of cement may indicate diffs in the charaot 
of the rock or degree of burning. Thus, with Rosendale, a light color generally i 
cated an inferior or under-burned rock. A coarse-ground cement, light in color ai 
weight, would be viewed with suspicion. 

Finer sieves than No 50 (about 50 meshes to the lineal inch) should be 
No 126 <120 meshes) was used in the experiments. 

The highest strength was obtained by the use of just enougl) water to th< 
oughly dampen the cement. An excess of water retards setting. American cemen 
needed more water than Portland ; fine-ground more than coarse; quick-setting mo 
than slow. Neat Itosfendale, a year old, was strongest with 35 per cent water. Ne 
Portland, same age, with 20 per cent. 

The liner the sand, the less is the strength. 

Salt, either in the water used for mixing, or in that in which the cement is la * 
retards setting somewhat, but has no important effect upon the strength. 

Adding clay gives a much more dense, plastic, water-tight paste, useful for pi; r 
ter or for stopping leaks. Half a part of clay did not seem to weaken mortar ma 
rially, except in the case of sample blocks exposed to the weather for 2% years afl 
a week’s hardening in water. 

A year’s saturation in fresh or salt water, and in contact with oak, hai 
pine, white pine, spruce or ash. did not affect the mortars. 

With sand, fine-ground cements make the strongest mortar; but when test 
neat, coarsc-ground cements are strongest. This is especially the case wj 
Portlands. 

Good results were obtained from ntixing cements. A mortar of half a p: 
each of Rosendale and Portland, and two parts sand, was stronger, at 1 wk. 1 me 
mos and 1 year, than the average of two mortars, one of 1 part Rosendale and one 
1 part Portland; each with 2 parts sand. Mixtures of Roman {quick setting) a 
Portland (slow) set about as quickly as Roman alone, and were much stronger. 

Portland resisted abrasion best when mixed with 2 parts sand ; RosendA 
with 1 part. A little more or less sand rapidly reduced the resistance in both cash 

Cements expand in setting? ; but not more than 1 part in 10U0 of any git' 
dimension. 

Art. 10. Cement concrete or Beton, is the foregoing cement mor 

mixed witt) gravel or broken,stoue, brick, oyster shells, &o, or -with all together. In concrete a 
mortar, it is advisable on the score of strength that all the voids he tilled or more thau filled. Tl 
of broken stone of tolerably uniform size and shape are about .5 of the mass j with more irregula 
of size and shape they may decrease to .4. Those of gravel vary like those of sand, and had 111 
better be taken at .5 when estimating the dry oement. We shall tfien pave as follows. 


For 1 cub yd of concrete of stone, gravel, and sand, witbo 

voids. 

1 cub yJ broken atone with .5 of its bulk voids, requiring -Scuh rd travel. 

0 5 cub yd gravel “ ,5 “ " “ 

6.25 ebb yd sdtta .5 " “ “ 

• «»lll Oljsi bOUMMl ** II 


.5 cub yd gravel. 

.25 cub yd wind. 

.125 (or cub yd dry cement. 


Iti» probable that mistakes have occurred from Inadvertently assuming that becstusi 
voids in a broken mass.Qonstitute a certaiu proportion of the bulk of said mass; therefore, the orig 
solid has swelied in onlyihat same proportion. Thus, if a sol id cubic yard of stone be broken into a 
irregular’pieee's, which have among themselves about the same proportions of large audsmall one 
usually occurs in quarrying, or in railroad rock cuttings; and if these be loosely thrown iuto a b 
the .47 of this heap, or rather less than half of it, will be voids. Bufit ijocs not follow, therefore, 
the'nriginnl solid sub yd h£s swelled only .47. or nearly'one-half, or makeh ontv 1 Hi ctlb yds of bri 
stone ; although many young engineers would probably Consider this a very full allowance; and w 
suppose that tfiey were quite just to the company, if they counted for the contractor one solid 
of-6xcavktibn fot every 115 yds Of fragments. Now, it Is plain that if .47 of tli«s broken heap are v 
the remaining .53 must be' stone. But these .53 constituted the original solid cubic yard ; and 
still remain equal to it in actual solidity. Hence we must say as follows: If .53 of the broken i 
occupies pne cuh yd of actual space, how much space will the whole mass occupy ; or, 




Of the 
broken mass. 


Cuh vd 
of space. 


Entire 

broken mass. 


Cub yds 
of space. 


.53 


1 X 1 


,53 


— 1.9 















CEMENT, CONCRETE, ETC. 


679 


Hence, we see that a solid cub yd of stone, when so broken, swells to 1.9, or nearly 2 cub yds ; and 
nee a proper allowance to a contractor, would be 1 cub yd solid, for every 1.9 cub yds of pieces ; or 
e yds of pieces must be divided by 1.9 for the solid yards. 

If we know that a cubic yard of any stone, breaks to, say 1.9 yds, then to find the proportions of 
ids, and solid, in the broken mass, proceed thus : The solid part of the broken mass must occupy 1 
b yd of space; aud the question is what part of 1.9 yds does this 1 yd constitute. The answer i 3 

- — .53; therefore, 53 hundredths of the broken mass is solid; and of course the remaining 47 hun- 

edths are voids. 

If a cubic foot solid weighs N lbs ; but when broken up. or ground, only n lbs per cub ft, then n 
vided by N. will be the proportion of solid in the broken mass. 

If the broken stone is loosely piled up, it will occupy a little less space, say about 1 8 cub yds; in 
lich case the voids will be .44 ; and the solid. .56 of the mass. We will here' venture to express our 
ubts whether hard rock when blasted and made into embankment, settles to less than 1% yds for 
ery solid yd. Mr KUwood Morris gives as the result of certain embankment of hard sandstone, 
ide under his supervision, an increase of bulk of y 5 ,-; or in other words, that 1 cub yd of rock in 
ice, made ly 5 ^-, or 1.417 yds of embankment. This corresponds to very nearly .7 solid ; and .3 
ids; while 1% yds to 1 solid, corresponds to .6 solid; and .4 voids. The rough sides of rock excava- 
ns, make it difficult to measure them with accuracy; and we cannot but suspect that something 
this kind has interfered with the results obtained by Mr Morris. He, however, may be right, and 
wrong. 

3y some careful experiments of our own, an ordinary pure sand from the sea shore, perfectly dry, 
d loose, weighed 97 IBs per cub ft; and its voids were .41, and the solid .59 of the mass. By thorough 
iking, and jarring, it could be settled the .1333 part, (halfway between i, and \i,) and no more, 
then weighed 112 lbs per cub ft; and its voids were then .32; and the solid. .68 of the mass. 
Vnother pure quartz sand, of much finer grain, perfectly dry and loose, weighed but 88 lbs per cub 
the voids were .466; and the solid .534 of the mass. Ry thorough shaking and jarring it could 
reduced; like the former, only the .1333 part; it then weighed 101.6 lbs per cub ft; and its voids 
re .384; and the solid .616. Another, consisting of the finest sifted grains, of the last, weighed 82 
per cub ft; so that its voids, and solid, each were very nearly .5 of the mass. This could be com- 
;ted about % part; and then weighed 98per cnb ft. 

The first, or coarsest of these sands, when quite moist, but not wet, perfectly loose, weighed but 86 
per cub ft; or 11 lbs less than when dry. It could be rammed in thin layers, until it settled one- 
h part; and then weighed 107)4 lbs per cub ft. Voids .348. solid .652. 

'he second sand, similarlv moist and loose, weighed but 69 lbs per cub ft; or 19 lbs less than when 
•. It could be rammed in thin layers, until it settled % part; and then weighed 103^ Bis per cub 
Voids .373, solid .627. 

7one of these sands when dry, and loose, if poured gently into water to a depth of 15 inches, set- 
1 more than about one-fifteenth part; the coarsest one, considerably less. 

lere the .125 cub yd of dry cement constitutes one-eighth of the single mass; or one-tifteenth of 
the dry ingredients as measured separately. 

or 1 cub yd of concrete of broken stone and sand without 

voids. 

I 

■ :ub yd broken stone, with .5 of its bulk voids requiring | .5 cub yd sand. 

mb yard sand, “ .5 “ “ “ “ | -25 cub yd dry cement. 

1 The strength of concrete is affected by the quality of the broken stone, 

well as by that of the cement, the degree of ramming, &c. Cubes of either of the above with Port- 
. d, as weil as one composed of 1 meas of good Portland to 5 of sand only, well made, and rammed, 
m’ld either in air or in water require to crush them at different ages, not less than about as follows. 

Age in months.1 3 6 9 12 

Tons j»er sq ft .15 40 65 85 100 

l Tnder favorable conditions of materials, workmanship and weather, the strengths may be from 50 

■ 100 per ct greater. For transverse strength as beams see p 682. 

If not rammed the strength will average about one-third part less. 
jWitli common U. S. cements, if of good quality from .2 to .3 of the 

jjngtta of Portlaud concrete may be had. 

Slow setting- cements are best for concrete, especially when to he 

I lined. 

It may not be amiss to state here that when masonry is backed by 
tiierete the two are liable in time to crack apart from unequal settlement, 
>ecially if the ramming has not been thorough; also that in variable climates 
it iron cylinders tilled with concrete are frequently split horizon- 

y by unequal expansion and contraction. Iu such structures it is safest to consider the cyls as 
re moulds for the concrete: and to depend upon the last only for sustaining the load. 

The concrete for the New York City docks consists of 1 measure 

either English or Saylor's Portland, 2 of sand, 5 of broken stone (hard trap). That made of Eng- 
( Portland, after drying a few davs, and then being immersed 6 weeks, required about 30 tons per 

ft to crush it. Saylor’s would probably require the same. At the Missis- 
pni Jetties, (see “South Pass Jetties” by Max E. Schmidt, C. E., Trans Am 

*C E, Aug 1879) Saylor's Portland 1; sand 2.76; gravel 1.46; brokeu stone 5. 

[n the foundations of the Washington Monument at Washington, D.C., 
30) English Portland (J. B. White & Bros) 1 ; sand 2 ; gravel 3 ; broken stone 4; and according to 

jovt. Report, has a crushing strength of 155 tons persq ft when 7.5 months old. 
t Croton Dmn, N. Y., (1870) Rosendale 1; sand 2; broken stone 4.5. Some 

;he same work, and deposited under water, hud 6 meas of stone; and at the end of a year had be- 
ae so hard that it was found necessary to drill and blast a portion that had to be removed. 








680 


CEMENT, CONCRETE, ETC. 


Lime with cement weakens all of them, but General Q. A. Gillmore, our 

best authority, repeatedly states that even in important concrete work in either the air or water (pro¬ 
vided the water does not come iuto coutact with it until setting takes place), from .25to even .0 of the 
neat cement paste of the U. S. common cements may be replaced by lime paste without serious dnni 
nutiou of either strength or hydraulicity ; and with decided economy. It retards the setting which 
is often of great advantage, especially with quick setting cements which at times cannot on that ac 
count be advantageously used without some lime. 

Moulded blocks of Portland concrete of even 50 tons tvt can generall} r be 

handled and removed to their places in from 1 to 2 weeks. ,,,, 

It;t 111111 i ti j;’ of concrete, when properly done, consolidates the mass about j sl 

5 or 6 per ct, rendering it less porous, and very materially stronger. The rammers are like those L 
used ill street paving, of wood, about 4 ft long, 6 to 8 ins diam at font with a lifting haudle, and shod 
with iron; weight about 35 lbs. They are let fall 6 or 8 ins. The men using them, if standing « 
on the concrete, should wear india-rubber boots to preserve their feet from corrosion by the cement n 

Ramming: cannot be done tinder water, except partially, when the 11 

concrete is enclosed in bags. A rake may, however, be used gently tor levelling concrete under water y 

Blake’s Stone Crusher (Co, New Haven, Connecticut), is useful lor w 

breaking the stone more cheaply than by hand on a large work. The two sizes best adapted to this 



engine man, 1 or 2 men to break the larger stones to a size that will enter the machine, 1 driver U 
horse-cart, 1 man to feed the stone into the machine, 2 to keep him supplied with stone, 1 at th(, tt 
screen, 2 wheeling away the broken stone to the stone-heap, 1 or 2 to receive it at the heap. . Say l(|j SI 

or 12 men in all. The size of the broken stone for concrete is gen 

erally specified not to exceed about 2 ins on any edge; but if it is well freed from dust by screening o 
washing, all sizes from .5 to 4 ins on any edge may be used, care being taken that the other ingredi e 
euts completely- fill the voids. 

Concrete is g-ood for bringing up an uneven foundation t< 
a level before starting the masonry. By this means the number of horizon ti - 
joints in the masonry is equalized, and unequal settlement is thereby prevented. 

Concrete may readily be deposited under water in the usm 

way of lowering it, -soon after it is mixed, in a V shaped box of wood or plate-iron, with a lid tli 
may be closed while the box descends. The lid however is often omitted. This box is so arrange 


that on reaching bottom a pin may be drawn out by a string reaching to the surface, thus permitiir ta 


til) 


one of the sloping sides to swing open below, and allow the concrete to fall out. 1 he box is the 
raised to be refilled. In large works the box may contain a cubic yard or more, and should be st 
pended from a travelling crane, by which it can readily be brought over any required spot in t 
work. The concrete may if necessary lie gently levelled by a rake soon after it leaves the box. 
consistency and strength will of course be impaired by falling through the water from the box ; a 
moreover it cannot be rammed under water without still greater injury. Still, if good it will in di 
time become sufficiently strong for all engineering purposes. Concrete has been safely deposited ; 
the above manner in depths of 50 ft. 

The Tremie, sometimes used for depositing concrete under water, is a bo 

of wood or of plate iron, round or square, aud open at top and bottom; and of a length suited to tl 
depth of water It may be about 18 ins diam. its top, which is always kept above water, is hoppe 
shaped, for receiving the concrete more readily. It is moved laterally and vertically by a travellir 
crane or other device suited to the case. Its lower end rests on the river bottom, or on the depositor 
concrete. In commencing operations, its lower end resting on the river bottom, it is first entire w 
filled with concrete, which (to prevent its being washed to pieces by falling through the water in t 
tremie) is lowered iu a cylindrical tub, with a bottom somewhat like the box before described, whi. 
can be opened when it arrives at its proper place. After being filled it is kept so by throwing fret 
concrete iuto the hopper to supply the place of that which gradually falls out below, as the tremie 
lifted a little to allow it to do so. The wi of the filled tremie compacts the concrete as it is deposite 
A tremie had better widen out downwards, to allow the concrete to fall out more readily. See “ Gi! 
more on Cements." 

The area upon which it is deposited must previously be surround 

by some kind of enclosure, to prevent the concrete from spreading beyond its proper limits; and 
serve as a mould to give it its intended shape. This enclosure must be so strong that its sides m 
not be bulged outwards by the weight of the concrete. It will usually be a close crib of timber 
plate-iron without a bottom; and will remain after the work is done. If of timber it may require 
outer row of cells, to be filled with stone or gravel for sinking it into place. Care must be tukeu"* 
prevent the escape of the concrete through open spaces under the sides of the crib or enclosure, 
this eud the crib may be scribed to suit the inequalities of the bottom when the latter cannot read 
be levelled off. Or iuside sheet piles will be better in some cases; or an outer or inner broad flap 
tarpaulin may be fastened all around the lower edge of the crib, and be weighted witli stone or gra 
to keep it iu place on the bottom. Broken stone or gravel or even earth (the last two where then 


ifirsi 


H. 

line 


i«kli 


#pit 

| 

in 


no current) heaped up outside of a weak crib will prevent the bulging outwards of its sides by 
pressure of the concrete. After the concrete has been carried up to within some feet of low wal 9 
and levelled off, the masonry may he started upon it by means of a caisson (page 636); or bv met 
diving dresses. Or if the concrete reaches very nearly to low water, a first deep course of stone n ! ,il 
be laid and the work thus brought at once above low water without any such aids. 

The concrete should extend out, from 2'to 5 feet (according 
the case) beyond the base of the masonry. All soft mild should he reniOV< 4 », 
before depositing concrete. Rag'S partly filled with Concrete, and merely thr< 
into the water may be useful in certain cases. If the texture of the bags is slightly open, a pori W 
of the cement will ooze out, aud bind the whole into a tolerably oompact mass. Such bags, by the 
of divers, may be employed for stopping leaks, underpinning, and various other purposes, that n 
suggest themselves. Such bags may be rammed to some extent. 

Tarpaulin may be spread over deep seams in rock to prev< 
the loss of concrete; and in some cases, to prevent it from being washed away by springs. 


Jf« 


"le 


1*1 











CEMENT, CONCRETE, ETC. 


681 


oucrece; especfa i™!?derwater 0,n for J udgment in the various applications of 

•ea of 400 ft by 100 ft; forming. ^ U we^e a P arM&'ff 4 t0 5 t l hick,iess of ^ feet, over an 
ader water; an immense mm.M i,. *1* ul”? le artlficlal _ 8to . ne °f that size. It was deposited 



'ape. The last deposits of concrete were then faced wtfh ° r enc,osure8of timber, conforming wlc 
,e " U -V b ““* °f cement concrete deposited betweenblanks ^ A moTrn °„ f b “ ildi °g s re also fre- 

the building goes on. Flues may be made in these l„ii!! tf a mou ! d • and w h«eh are moved upward 

tn afterwards be lifted oit and'beTsS’S“i! a 


ome of th 0 i,- ^ out ; aQd be used for the next course above. The 
d. is of concrete “'rhi* ■?’ 142ft d « m ’ and now neari y 2000 years 

Alma, at Paris, have Arches of llKiudHI LTs^an^ofcon^e”* Ka P 0,e0n > and Pont 


-* V* VUUVICICt 

HhoS p?«Sf*^ 80 ribM5*,e , 5tete B COn , Cr « te ’ Gen Gillmore gives the 

reening, into different sizes, “ the concrete § was nrennfed'h aud ^ bble3 being first separated by 
rough boards, in a layer from 8 to 12 ius thick ^rhe^man^ spr . eadlu g out the gravel on a platform 

es on top. The mortal was Sen Spreadoverthe^fravel ^u^Ho m,T the bo . t i? m> a ' ld the 
re then mixed bv 4 men : 2 with shovels n.,d ■> Zuh __ uniformly as possible. The materials 


n with a shoveller,’ and were leZireTto rZ Sir , aea „ w . lth hoes worked eacb iu eonjunc 

d spread, or rather scattered on^he platform bv a jerkiui“LoUon"*Vlf e ’? ortar ' as lt was turned 
; oud time, in the same manner, but in the^posite dWtlon • i„d , he ^ was turned over a 

te L h a e 

'■ticular attention of the o-erseer." 'it is hard work. b ° rer ’ yet he seIdom acquires it without the 


i '.lass ntta&sw 

;r,:i h the , horiz T 7 his e-vliuder is made to revolve 15 or 20Times per min by deans of f 
pie leather strap or band around its outside; and to which motion is given by a locomotive which 
m mo r, r 6 * orked a heavy mill for mixing the mortar. This simple machfne^X turns Tut 
ft HflinVexpense!^ ° f C ° UCrete itt 10 h ° UrS 5 aad wflB “ worked in coLect^wU^a IrLr Lill. 

cemen^andTfses to*the* t surfacr at Th^ SpeC,a "^ h D the sea - a p “lpy gelatinous fluid exudes from 
' ! h surface - Tbl8 causes the water to assume a milky hue; faeuce the term 

hi." ’ which French engineers apply to this substance. As it sets very imperfectly and 
nod > e I ?atJ anetl tp af cen ?. e , n f s scarcely at all, its interposition between the I a vers'of concrete^even 
,chof,r antltleS ; WH1 hav f, a te,,duncy t0 lesseu. more or less sensibl>, the con tin uky and 
ngth of the mass. It.is usually removed from the enclosed space by pumps. Its proportion is 

n i y toTu°c S h 6d d by redu ? ln K/he area of concrete exposed to the water, by using large iLxes say 
n 1 to l]4 cub yds capacity, for immersing the concrete.'* J 6 * 

Weight of good concrete 130 to 160 lbs per cub ft, dry, 

}©st off concrete $5 to $9 per cub yard if roughly deposited ; and $9 to $15 
irst made into blocks; depending on size, cement, locality, wages, &e. 

* 9 oitf,,et s beton. The artificial stone which bears this engineer’s 
ne has for several years been used in France with perfect success, not only for 
ellings, depots, large city sewers, <fcc, but for the piers and arches of bridges 
tit-houses, &c. Bridge arches of 116 ft span, and of low rise, have been built of 
It is composed of 5 measures of sand, 1 of sifted dry-slaked lime, and from 
4 measure of ground Portland cement. Or of sand 6, cement 1, lime 

bese are first well mixed togeiher dry. and then placed in a mixing-mill; at the same time 
nkhng them with .3 to A measure of water, so as to moisten them slightly, without wetting 
rh ^' ar « tb 5". thoroughly incorporated by mixing, until they form a stiff pasty mass, 
illy coherent. I his is then placed m a mould, in successive thin layers, each of which is well 
pacted by blows of a 1G ft rammer. The top of each layer may be scored or cross-cut to make 
next one unite better with it. Owing to the small proportion of water, it sets soon' and may 
Tally be taken from the mould in from a few hours to a few days, depending on the size of the 
k , and left to harden. River sand is the best, inasmuch as it. requires less lime and cement 
‘ Pit sand, to make equally good stone.(?> The cement should be a rather slow-setting one; and 
it and the lime should be screened, to exclude lumps. About bushels, or 1% cub ft of 
Jry materials, make 1 cubic foot of finished stoue, weighing about 140 fiis; resisting 100 to 
tons per square foot at 3 months old. 250 to 400 in 2 years. Arches of it are made no thicker 
i brick one*. An arch, pier, wall, foundation, Ac, may he built of it, as one stone instead of 
parate blocks. In sewers the eenters may be struck within 10 to 15 hours after the arch ia 
hed; and the water may be admitted within a week or less. The distinctive feature* of Coignet’s 
a are: the very small proportion of water; the thorough incorporation of the ingredients - and 
sousolidatton of the separate layers by ramming. It is difficult for a person who has never *een 
process, to credit the rapidity, facility, and economy with which blocks of good stone can be 
e by it. Its cost, as compared with perfectly plain dressed granite, does not exceed one-half- 
e for ornamental work it compares even far more favorably. Hence the Coignet beton or artifl- 
stone, is nothing more than good, well-prepared mortar, mixed with very little water: and well 
med into moulds, in successive layers. A mixture of 1 measure of hydraulic cement, and 3 
<ures of sand, similarly treated, has been successfully used in the U. S., for some years in 
lings of all kinds. Ornamental work can be furnished at the price of stone; and will answer 
Hy well. For full Information, see Gillmore’s “ Coignet Beton.” 






682 


CEMENT, CONCRETE, ETC. 


Transverse Strength of Concrete Beams.* 

Averaee results of 24 beams, 10 ins square, made of good Roseudale and English 
Portland cements, pit sand and screened pebbles, few exceeding 1 inch diam. The 
l»eams were buried for 6 months, in a pit 4 ft deep, in gravelly soil, exposed to the 
rain, snow, Ac. A first set of beams all broke on being taken from the moulds aft< 

7 or 8 days, although carefully handled. To avoid this, the bottom of the pit itsel 
was rammed to a smooth, hard surface: immediately upon which a new set w 
made by ramming the concrete into 2 inch planed plank moulds without botton 
The moulds were removed after 24 hours, ami when all were done the earth w 
filled in over the undisturbed beams. Very little of the soil adhered to them. Thei 
wt in all cases when tested was about 150 lbs per cub ft, or 520 lbs wt of 5 ft cleat 
span of beam; one half of which, or 260 lbs, must be deducted from the cen breakp 
loads of the 5 I t spans below ; and 124 lbs from the 2 ft 4.5 ins ones. The coeflicieul 
or Constant e is the cen breakg load in lbs for a beam 1 inch square, and 1 fl 
clear span, like those in table p 493 ; and like them is found 1 y the formula at toj 
ef p. 492. Its use is shown by the formula at foot of p 492. 


Proportions of inate- 
rials by measure. 

Center Breaking 
load in lbs, including half 
wt of beam. 

i 

Constant e 

■ * 

Cement. 

Sand. 

Pebbles. 

Span 2 ft 4.5 ins. 

Span 5 ft. 

* 

■ a 

Rosendale 1 

2 

5 

1782 

690 

S.7 

“ 1 

3 

7 

all broke in 

mndling 

1995 

.JO 01 

Portland 1 

3 

7 

3926 

9.8 

“ 1 

4 

9 

3648 


8.1 it 

u 1 

6 

11 

2822 

1190 

6.2 

— 


* This nsefhl table and that on p 677 were kindly furnished us by Bitot < 
Clarke, Esq., then Principal Ass’t in charge of the Improved Sewerage Works < 


Boston, Mass.; for which the experiments were made. 

Clarke ”, p 678. 


See also “ Mr. Eliot 

























RETAINING-WALLS 


683 


KET AININ 6 - W ALLS, 


Art. 1. 



^ s Peak only of walls sustaining earth; for those sustaining re a ter, 

e pp _29 to 232, and 236. A retaining-wall is one for sustaining the pres 
earth, saud, or other filling or backing, deposited behind it after it is built- in 
stinotion to a face-wall, which is a similar structure for preventing the fall of 
1 th which is m its undisturbed natural position, but in which avert or inclined 
;i pe has been excavated. The earth is then in so consolidated a condition as to exert 
k* itahiing one* 6141 pre8 ’ “‘d* therefore the wall may generally be thinner than a 

Dus, however, will depend upon the nature and 
idtiou of the strata in which the face is cut. If 
* strata are of rock, with interposed beds of clay, 

-tin or sand ; and if they dip or incline toward the 
11, it may require to be of far greater thickness 
fin any ordinary retaining-wall; because when the 
i n seams of earth become softened by infiltrating 
In, they act as lubrics, like soap, or tallow, to fa- 
1 tate the sliding of the rock strata; and thus bring 
tf enormous pres against the wall. Or the rock may 
■ set in motion by the action of frost upon the clay 
ims; or, as sometimes occurs, bv the tremor pro- 
led by passing trains. Even if there be no rock, 
ll if the strata of soil dip toward the wall, there 
1 always be danger of a similar result; and addi- 
nal precautions must be adopted, especially when 
If strata reach to much greater height than the 

‘11- A vertical wall Las both c o 

Ids vert. 

Experience, rather than jheo- 

/, must be our guide in the building of 
th kinds of wall. We recommend that 
3 hor thickness a b, Fig 1, at the base of a 
rt or nearly vert retaining-wall c d b a, 

lich sustains a backing of either sand, gravel, or earth, level with its top c d 
in the tig, should not be less than the following, in railroad practice, when the’ 
nidations are not more than about three feet deep. 

When the hack in; 

mped from, carts, cars, &c. 

ill of cut- stone, or of first-class large ranged r ubble, 

in mortar . a.b .35 of its entire vert height d b. 

“ good common scabbled mortar-rubble, or brick. .4 “ “ « 

“ well-scabbled dry rubble .5 “ “ “ “ 

Aith good masonry, however, we may take the height d s instead of d l, and then 
i above proportions of d s will give a sufficient thickness at the ground-line o s 
i Table, p 690. 

When the backing is somewhat consolidated in hor layers, 

-h of these thicknesses may be reduced, but no rule can be given for this. 

['he offset o e, in front of the wall, is not included in these thicknesses. 

Chen, however, the backing is a pure clean sand, or gravel, we should use only the full dimen- 
is; inasmuch as the tremor, caused by passing trains, would neutralize any supposed advantage 
n ramming materials so devoid of cohesion. Such sand may he rammed with much advantage 
the purpose of compacting it in foundations; but a diff principle is involved in that ease. When 
i done even with cohesive earths, with a view of saving masonry in retaining-walls, it is probable 
i t the expense will generally be found quite equal to that of the masonry saved. See Rem 4, p 691. 
he base aft in Fig 1, is -A_ of the height hd. In the foregoing thicknesses at base, the back d 6 
he wall is supposed to be vert; and the face ea either vert, or battered (sloped or inclined back¬ 
'd) to an extent not exceeding about 13 >^ inches to a foot; which limit it is rarely advisable to ex- 
j i in practice, owing to the bad effect of rain, &e. upon the mortar when the batter is great. The 
;e of a vert wall need not in fact be as thick as one with a battered face; but when the batter does 
exceed 1.5 iilches to a foot, the diff is very small. See Table, Art 7 

Iem. 1. A mixture of sand, or earth, with a larjre proportion 

to usd bowlders, paving pebbles, &c, will weigh considerably more than the materials ordinarily 
d for backing; and will exert a greater pres against the wall; the thickness of which should be 
I reased, say about one-eighth to one-sixth part, when such backing has to be used. 

If.m. 2. The wall will be stronger if all the courses) of masonry he laid 
Ith an inclination inward, as at oeb; especially if of dry masonry, 
if time cannot be allowed fas it always should be, when practicable) for the mor- 
to set properly, before the backing is deposited behind it. The object of inclin* 


is deposited loosely, as usual, as when 




















684 


RETAINING-WALLS 


ing the courses, is to place the joints more nearly at right angles to the directio 
/P, Figs G. 7, and 8, of the pres against the back of the wall; and thus diminis 
the tendency of the stones to slide on one another, and cause the wall to bulge. 

When the courses are hor, there is nothing to pr< 
vent this sliding, except the friction of the stones, one upon the other, when of dr 
masonry; or friction and the mortar, when the last is used, lint if, as is frequentl 
the case, (especially in thick and hastily built walls,) this has not had time to liarde 
properly, it will oppose but little resistance to sliding. Hut when the courses ar 
inclined, they cannot slide, without at the same time being lifted up the incline 
planes formed by themselves. In retaining-walls, as in the abuts of importan 
arches, the engineer should place as little dependence as possible upon mortar; bu 
should rely more upon the position of the joints, for stability. 

An objection to this inclining of the joints in dry (without mortar) walls, is that rain-water, fallin 
eu the battered face, is thereby carried inward to the earth backing: which thus becomes soft, an ‘ 
settles. This may be in a great measure obviated by laying the outer or face-courses hor; or b j 
using mortar for a depth of only about a foot from the face. The top of the wall should be proteote I 
by a coping c d, Kig 1, which had better project a few ins in front. After the masonry lias beet ! 
built up to the surface of the ground, the foundation pit should be tilled up; and it is well to con 
solidate the tilling by ramming, especially in front of the wall. 


The back d b of the wall Kliould be left rough. In brickwork i 

would be well to let every third or fourth course project au inch or two. This increases the frictioi 
of the earth agaiust the back, and thus causes the resultaut of the forces acting behind the wall t 
become more nearly vert; and to fall farther within the base, giving increased stability. It also con 
duces to strength not to make each course of uniform height throughout the thickness of the wall 
but to have some of the stones (especially near the back; sufficiently high to reach up through two o 
three courses. By this meaus the whole masonry becomes more effectually interlocked or honde< 
together as one mass: and therefore less liable to bulge. Very thick walls may consist of a faciu, 
of masonry, and a backing of concrete. 


Rem. 3. It is the pres itself of the earth against the back, that creates the friction, which in tnri 
Inodities the action of the pres; as the wtor pres of a body upou an iucliued plaue produces frictioi 
between the body and the plane, sufficient, perhaps, to prevent the body from sliding down it. A re 
taining-wall is overthrown by being made to revolve around its outer toe or edge e, Kig 1, as a ful 
irum, or turning-point; but in order thus to revolve, its back must first plainly rise; and in doiu; 

so must rub against the backing, and thus encouuter aud overcome this friction. Th 
friction exists the same, whether the wall stands firm or not; as in the case of th> 
body on au inclined plane ; the only diff is that in one case it prevents motion ; am 
in the other only retards it. 

Whore deep freezing: occurs the back of the wall shouk 

be sloped forwards for 3 or 4 ft below its top as at c o, which should be quite smootl 
so as to lessen the hold of the frost and prevent displacement. 



Rf.m. 4. When the wall is too thin, it will generally fail 
by bulging- outward, at about % of its height above the 
ground, as at a, in Fig 2. A slight bulging in a new wall 
does not necessarily prove it to be actually unsafe. It it 
generally due to the newness of the mortar, and to the ] 
greater (ires exerted by the fresh backing; and will ofte< 
cease to increase after a few months. It need not excit 
apprehension it it does not exceed inch for each foot ii 
thickness at a. See Remark 3, Art 7, p G91. 

Art. 2. The young engineer need not in practice concern himself particulnrlv about the precis: 
sp grav op his backing, or about the angle op slope at which it will stand ; for the material whicl 
he deposits behind his wall one day. may be dry and incoherent, so as to slope at \ % to 1; the nex 
day rain may convert it into liquid mud. seeking its own level, like water; the next it may be ice 
capable of sustaining a considerable load, as a vert pillar. 

Moreover, he cannot foretell what may be the nature of his backing; for. as a general rule, thi 
must consist of whatever the adjacent excavation may produce from time to time ; sand to-day. rocl 
to-morrow, &c. Retaining-walls are therefore usually built before the engineer knows the characte 
of their backing: so that in practice, these theoretical considerations have comparatively but littl 
weight. Theory, uncontrolled by observation and common sense, will lead to great errors in ever 
department of engineering ; but, on the other hand, no amount of experience alone will compeusat 
for au ignorance of theory. The two must go hand-in-hand. 



Again, tiie settlement of the booking- under Its own wt. aide 

by the tremors produced by heavy trains at high speed; its expansion by frost, o 
by the infiltration of rain; the hydrostatic pressure arising from the admission o 
the latter through cracks produced in the backing during long droughts ; as well a 
its lubricating action upon it, (diminishing its friction, and giving it a tendency t 
elide,) Ac, exert at times quite as powerful an overturning tendency as the legitimat 
theoretical pres does. The action of these agencies is gradual. Careful observatioi 
of retaining-walls year after year, will often show that their battered faces are be 
coming vertical. Then they will begin to incline outward ; and eventually the wa) 
will fail. Theory omits loads that may come on backing increasing its pres. 













RETAINING-WALLS. 


685 


to Assuming the theoretical views advanced by Professor Moseley to be correct as 
is theories, the thicknesses which we have recommended in Art 1, for mortar walls 
correspond to from 7 to 14 times; and for dry walls about 10 to 20 times, the pres 
assigned by him; and we do not consider ours greater than experience has shown 
If necessary. See lable 3. Retaining-walls designed by good engineers, but in 

tl t f° c J ose accordance with theory, (which assumes that a resistance equal to twice 
le tj'e theoretical pres is sufficient,) have failed; and the inference is fair that many of 
j, those which stand have too small a coefficient of safety. 


ft 

tf 


t 


The fact is, (or at least so it appears to us,) there must be defects in the theoretical assumptions of 
some of the most prominent writers who give practical rules on this subject. Thus Poncelet who 
certainly is at their head, states that his tables, tor practical use, give thicknesses of base for sus* 
taiuing 1 times the theoretical pres; and this he considers amply safe. Yet. for a vert wall of ci t 


.ase for sustaining dry sand level with the top, as in Fig 1, is .35 of the vert height; 

" 11 ^ ei l f° UI1 ^ that when not subject to tremor , a wooden model of a vert 

t? nut. /H tii!S nnr f'lin ft unH with •» Iwion ox i • . u l . i . . .... 


granite, his base 
and for brick. 

wail weighing but 28 lbs per cub ft, and with a base of .35 or its height, balanced perfectly dry sand 
sloping at to 1, and weighing 89 lbs per cub ft. r 1 J 


Now, THE RESISTANCE OP SIMILAR WALLS, OF THE SAME DIMENSIONS, 
varies as their specific gravities ; and, since granite weighs about Ki5 
lbs per cub foot, or 6 times as much as our model, it follows, we conceive, 
that a wall of that material, with a base of .35 of its height, must have 
; a resistance of 6 times any true theoretical pres, instead of onlv 1.8 
* times; and that his brick wall must have about 5 times the mere bal- 
i ancing resistance. Our experiments were made in an upper room of a 
h strongly built dwelling ; and we found that the tremor produced by pass- 
■I iug vehicles in the street, by the shutting of doors, and walking about 
r the room, sufficed to gradually produce leaning in walls of considerably 
r more than twice the mere balancing stability w hile quiet; and it appears 
it to us that the injurious effects of a heavy train would be comparatively 
quite as great upon an actual retaining-wall, supporting so incohesive 
a material as dry sand. 

Since, therefore, Poncelet’s wall is in this instance sufficiently stable 
for practice, it seems to us that his theory, which neglects the effect of 
tremors, &c, must be defective. He also gives 1 of the height as a suf¬ 
ficiently safe thickness for a vert granite wall supporting stiff earth; but 
we suspect that very few engineers would be willing to trust to that pro¬ 
portion, when, as usual, the earth is dumped in from carts, or cars; espe¬ 
cially during a rainy period. If deposited, and consolidated in layers, 
theory could scarcely assign any thickness for the wall; for the hacking thus becomes, as it were, a 
mass of unburnt brick, exerting no hor thrust; and requiring nothing but protection from atmospheric 
influence, to insure its stability without any retaining- wall. It is with great diffidence, and distrust 
in our opinions, that we venture to express doubts respecting the assumptions of so profound an in- 
i vestigator and writer as Poncelet; and we do so only with the hope that the views of more compe¬ 
tent persons than ourselves, may be thereby elicited. Our own have no better foundation than ex¬ 
periments with wooden and brick models, by ourselves ; combined with observation of actual walls. 

Art. 3. After a wall a b c o, Fig 3, with a vert back, has been proportioned by 
our rule in Art 1, it may be converted into one with an offsetted 
back. Main o. This will present greater resistance to overturning; and yet con¬ 
tain no more material. Thus, through the center t of the back, draw any line i n ; 
from n draw n s, vert; divide i s into any even number of equal parts; (in the fig 
there are 4;) and divide .< n, into one, more equal parts; (in the fig there are 5.) From 
the points of division draw hor, and vert lines, for forming the offsets, as in the fig. 

In the offsetted wall, the cen of grav is thrown farther back from the toe o, than 

I iri the other, thus giving it increased leverage and resistance: but within ordinary 
practical limits, the diff is very small; and since the triangle of supported earth is 
greater than when the back is vert, its pres is also greater; so that probably no ap¬ 
preciable advantage attends that consideration. The increase of thickness 
near the base, diminishes, however, the 
leverage v a, Fig 8, of the pres / P, of the 
earth against the back. The center of pressure of 
this pres is in both cases at X / A the vert height, meas¬ 
ured from the bottom; and it is therefore plain that 
the farther back from the front it is applied, the shorter 
must v a become. Moreover, in the offsetted back, the 
direction of the pres becomes more nearly vert than 
when the back is upright. It is to these causes, rather 
than to the throwing back of the cent of grav, that 
the offsetted wall owes its increase of stability over 
one with a vert back. 




Art. 4. When, as in Fig 4, the hacking; is higher than the 

wall, anti slopes away from its inner edge d, at the natural slope d s, of \]/ 2 to 1, we 
are confident that the following thicknesses at base will at least he found sufficient 





















686 


RETAINING-WALLS, 


for vert walls with sand. They are deduced from the experiments just alluded to, 
and are but rude approximations, with no scientific basis. We should not have in¬ 
serted them, but for the fact that we know of no others for this case. See p 689. r 
The first column contains the vert height s v, of the earth, as compared with the 
vert height of the wall; which latter is assumed to be 1; so that the table begins 
with backing of the same height as the wall, sis in Fig 1. These vert walls may be 
changed to others, with battered faces, by Art 8; or without any such proceeding, 
their faces may be battered to any extent not exceeding \]/ 2 inches to a foot, or 1 in 
k, without sensibly affecting their stability, without increasing the base. 


T AISLE 1. (Original.) 


an* 

a v § 

V u 

0> v t£ 

^ > 
oSi 
•c s- 
.Sf-o "3 

*11 

Wall 

of 

Cut Stone, 
in 

Mortar. 

Good 

Mortar 

Rubble, 

or 

Brick. 

Wall 

of 

good dry 
Rubble. 

r “2 

0> v to 
•5-S <0 
“ - ► 

V.n ® 

*7’* cfi 
•C — 

Jg £ is 

Wall 

of 

Cut Stone 
iu 

Mortar. 

Good 

Mortar 

Rubble, 

or 

Brick. 

Wall 

of 

good dry 
Rubble! 

■3 g-Ja 

« a- 

Thickness at Base, iu parts of 

■3 M 

O S ^ 

Thickuess at Base, iu parts of 

> o <~ 

H o o 


the height. 


H o O 


the height. 


i. 

.35 

.40 

.50 

2. 

.58 

.63 

.73 

i.i 

.42 

.47 

.57 

2.5 

.60 

.65 

.75 

1.2 

.46 

.51 

.61 

3. 

.62 

.67 

.77 

1.3 

.49 

.54 

.64 

4. 

.63 

.68 

.78 

1.4 

.51 

.56 

.66 

6. 

.64 

.69 

.79 

1.5 

.52 

.57 

.67 

9. 

.65 

.70 

.80 

1.6 

.54 

.59 

.69 

14. 

.66 

.71 

.81 

1.7 

.55 

.60 

.70 

25. 




1.8 

.56 

.61 

.71 

or more 

.68 

.73 

.83 


Art. 5. But when the slope n r. Fig 5, of V ^ to 1, starts from the outer edge n 
of the wall, greater thickness is required. Poncelet gives the following for this 
case, for dry sand. 

TABLE 2. 



Wheu the earth reaches above the top of the wall, as in Figs 4 and 5. the wall is J 

and the earth that is above the top, is oalled the surcharge. When the surcharge is earefullv deposited 
above the wall so as to slope back at a steeper angle than 1^ to 1. as sav at 1 to 1, theory does not 
requite the wall to be as thick. Notwithstanding Poncelet’s high position, the writer cannot imagine 
that the base of a brick wall need be so great as lj^ times its height for any height of sand whatever. 

Art. 6. On the ttioory of retaining'-walls. Let b c a m, Fig 6, be 

such a wall, upholding backing or filling c s m g ; the upper surf c » of which ie 
hor, and level with the top b c of the wall; and let m s represent the nat slope of the 
earth which composes the backing; mg being hor. 

Abundant experience on public works shows that this slope, whether for sand, gravel, or earth, 
when dry, nifty be practically taken at 1^ to 1 : that is. 1^ hor. to 1 of vert measurement; which 
corresponds to an angle 8 m g of 83° 41' with the hor; which is also about the angle at which brick* 
and roughly dressed masonry begin to slide on each other. This angle, however, varies consider 










































































RETAINING-WALLS. 


687 


^telv'd'iufnf s^uid^oV^plir^h°n^ii ) ^f*' he i < ^ e ^ ree i° f drynes3 ' or dampness, of the material; so that mode- 
t i« Liu d 0 earth %\i.l stand at a slope of 1 to 1, or at an angle of 459, Whatever it mav l.n 

or walls iUs^afestlrtLann SLOPE of ^ he material under consideration. In theoretical calculations 
or walls, n is safest to assume (as we have done throughout) that the hacking is perfectly dry, since 



Fio\7 


•i nV, rt 7 * S tben greatest; unless it he supposed to be so wet as to possess some degree of fluidity. The 
) dofng by'th^waU^ ab ° Ve the nat slope ms ’ tends t0 sIide down said slo P e . but ia prevented from 
It is assumed In all cases that the wall is secured from sliding along its base,Art 9. p 692 that it is 
!Ln„^r? h l ° and that it will ill only by overturning ^ rotating 

mind its toe, a, as a, fulcrum. The thickness necessary to insure safety against the last will also be 
h* fZ pre !!: nt bu ‘S ,n «- Now referring only to Fig 6 with a vert back, if the angle o m s, con- 
nied between the natural slope to *, and a vert Hue to o, drawn from the inner bottom edge m of 
1 ie wall, be divided by a line to t, into two equal angles, o m t, t m s, then the angle o m t is called 
ik angle, and m t the slopk, op maximum prkssukk. The triaugular prism of earth, of which 
mt is a section, or an end view, is called the pkism op max pkks; because, ifcousidered as a wedge 
:ting against the back of the wall, it would produce a greater pres upon it than would the entire 
tangle crnsof earth, considered as a single wedge. For although the last is the heaviest, yet it is 
lore supported by the earth below it. Calculation shows that if we consider the earth o m s to be 
us div into wedges by any line m t, the wedge that will press most against the wall is that formed 
heu to t divides the angle o m «, or the arc o i, into two equal parts. But see Art 11. 

Since to p is hor, and m o vert, the two form an angle of 90°: consequently the angle of max ores 
I plainly found by taking the angle smg of nat slope from 90°, and div the rem by 2, Thus a nat 

I ipe of to 1, or 33° 41Vtaken from 9<P, leaves 56° 18'; and 560 I8< _ 3509 ' the COr- 
cs pond ins- aiiffle o m t of max pres. 

For ease of calculation, only one foot of the length of the wall, and of its backing is usually con- 
: dered. The number of cub ft of wall, or of backing, is then equal to that of the square feet in 
heir respective profiles, or cross-sections. 

| Now, according- to Moseley, if we assume the particles of earth composing the 
Ucking to be perfectly dry, and devoid of cohesion, (or tendency to stick to each 
;her,) which is very nearly the case in pure sand; and if we suppose the wall to be 
uddenly removed, then the triangle of earth emt , comprised between the slope mt 
max pres, and the vert back c m ot the wall, Fig 6, would slide down, under the in- 
Iuence of a force which may be represented by y P, acting in a direction y P, at right 
igles to the face c m of the triangle of earth; (or in other words, at right angles 
> the back of the vert wall,) its center of force being at P, distant way between 
and c, measured from the bottom ; and its amount equal to either of the following: 

Perp pres _ wt °f ilie triangle o f earth c m t X o t 
y P vert depth o m ’ 


No 1. 


or 


No 2. 


Perp pres 

y t 


Wt of a single cub .. - 

ft of the backing ^ si i 0 ^ 


2 . 


See 
■Art. 11, 
p 692. 


In view of the great uncertainty involved in the matter of the actual pressure of 
rth against retaining-walls in : practice (see Art 2. ), and in order to furnish 

simple rule which, although entirely unsupported by theory, is still (in the writer’s 
inion) sufficiently approximate for ordinary practical purposes, we shall assume 
at No 1 of the two loregoing formulas applies heat enough to walls Willi in- 
ined backs c wt, also,as. Figs 7 and 8, (precisely as they are lettered,) at least 
iiilil the back of Ike wall inclines forward as much as tt ins 
1 loot vert, or at an angle cwo of *26° 34'. What follows on 
i:tainin$£-walls will involve this incorrect assumption, an<l 
l list be regarded merely as giving: safe approximation. 

Some apppap to assume this perp pres to be the only one acting against.the bark 
tup Witlt; and hence arrive at erroneous practical conclusions. For.when, in' 
ler to preve.ttt this fvvee from causing the triangle:of earth to slide, we place « 
aining-wall in front of it, then, instead of motion, the force will produce pres of 
■ earth against the wall,, causing friction between the pressed surfaces of the 



















688 


RETAINING-WALLS. 


earth and wall. That is, if a wall were to begin to overturn around the toe a as 
fulcrum, its back c m must of course rise, aud in so doiug must rub against the 
earth filling in contact with it; and this rubbing would evidently act to impede the 
overturning. So long as the wall does not move, the same friction assists in pre¬ 
venting overturning. To ascertain the amount and effect of this friction, let ?/P, Fig 
8 represent by scale, the force perp to' the back c m; and supposed to have been pre¬ 
viously calculated by the foregoing formula No 1. Make the angle y P f equal 

the angle of wall frictiou,* draw y f at right angles 
y P, or parallel to me; make P a; equal to y f and com¬ 
plete the parallelogram P yfx. Then will x P represent 
by the same scale, the amount of the friction 
against the bach of the wall. Since the fric¬ 
tion acts in the direction of the back c m, (see end of Art j 
62, of Force, etc, p 354), it may be considered asj, 
acting at any point P, iu that line. j u 

Hence we have acting at P, two forces; 
namely, the perp force y P, and the friction x P; conse¬ 
quently, by comp aud res of force, the diag/" P of tht 
parallelogram P yfx , if measured by the same scale, wil 
give us the amount of their resultant; which Is th€ 
approx single theoretical force, both in 
amount and iu direction, which the wall 
has to resist, including the wall friction. 

But this force,/P, is also always equal to the 5 



force y P, mult by the nat sec of the angle y P f oi |r 
the wall friction; (or divided by its nat cosine) and 
course may be ascertained thus; 


Approx theoreti¬ 
cal pres f P 


wt of triangle v . v nat sec of angle y P f 
emt aoa 0 / wall friction 


c m t 


vert depth o m 


cos y P f X o m 


ie 


Or finally, if it is assumed, as we do throughout, that the earth is perfectly dry (in ' 
asinuch as its pressure is then the greatest) and that the angles of nat slope, an<W 
of wall friction are then each 33° 4V or 1.6 to 1, then in Figs 6 , 7 and 8 , if the angl 
c m 0 between the back c m and the vert o m does not exceed about 26° 34' we ms 
assume 


Appro^tlieoreticai _ wt wf t r | a „ s i e c in t X .643 


which includes the action of the friction of the earth against the back of the wall 


Rem. 1. When the back of the wall is ofTsetted or stepped, s 

in Fig 3, instead of being simply battered, as in Figs 7 and 8 , the direction of th. 
pres of the earth will be the same as if the back had the batter t», on the priucip 
given iu Art 34, Fig 17, of Force in Rigid Bodies, p 326. 


Rem. 2. Now to find both the overturning- tendency of th 
earth, and the resistance of the wall against being overturned around its toe <1 s' 
a fulcrum, first find the cen of grav g of the wall (p 348), and through it draw 
vert line g h. Prolong/P towards rand draw nr perp to it. By any scale malJ W| 
» n = vvt of wall, and s i = calculated pres fP. Complete the parallelogram st'n 
and draw its diagonal s n, which will be the resultant of the pres/P and of the vf 
of the wall; and should for safety be such that aj be not less than about one-fift 
of a m, even with best masonry and unyielding soil. .Otherwise the great pressure 
near the toe a may either fracture the wall or compress the soil near that poi 1,1 
so that the wall will lean forward. In walls built by our rule, Art 1, or by tab! 11 
p 690, a j will be more than one-fifth of a m. The pres / P if mult by its Jevera 
a v will give the moment of the pres about a; and the wt of the wall mult by 1 
leverage e a will give that of the wall. The wall is safe from overturning in pr 
portion as its moment exceeds that of the pres. It is assumed to be safe agair 
sliding, breaking, or settling iuto the soil. See Art 13, p 231. 


♦This angle of wall friction is that at which a plane of masonry mi 

be Inclined to the horizontal so that dry sand or earth would slide down tt. It Is about the same 
the nat slope, or 33° 41', or 1.6 to 1; and its nat secant is 1.202, and its nat cos .832. 



















RETAIN ING-WALLS. 


G89 


j Rem. 4. If the cnrtU slopes 

1{ A or B, instead of being hor as in Fi 
1{ p tb c m n instead of c in t, in n being 
j, 1 A the point of application will st.il 
■\c) as in 6 , 7, 8 ; but in B it will be j 

• low for Fig 9. 

r Snrcliargcd walls are those ir 
tends above the tops of the walls. 

,. According to theory, when as in 1 
c k of backing, sloping away from 
‘ e max pres against the wall is 
, Gained when the earth reaches to 
J e lev «l of d, where the slope m t d 
max pres intersects the face of the 
' t slope cy; so that if afterward the 
, ’’tb is raised to v, or to any greater 

• ight, no additional pres is thereby 
rown against the back of the wall. 

j. also if the earth slopes from b, or 
>m between c and b, except that 
in the slope in d of max pres must 
j tend up to meet this other slope. 

The approx i mate amonn t 

the oblique pres, when the wall is 
•charged, (as in any of the Figs 4, 

1 ,) may be found on the same priti¬ 
de as when the earth is level with 
top; namely, instead of the trian- 
c to t of earth, Figs 6 , 7, 8 , 9, find 
9 wt of all the earth d n m l, Fig 4, 

<i t r, Fig 5, or c d to, Fig 9 (if the ^ 

j ’charge reaches to d or v, or higher), 
hi ween the slope to d, Fig 9, to t, Figs 4 and 5, of max pres, the back of the wall and 
: l i front slope; omitting any which, like den, Fig 5, rests on the top of the wall 
j d thus adds to its stability) when the slope starts in front of c. Having found 
1 s weight, then for dry Lacking the 

approximately | = 411 of the earth X .643, 

, hiding the action of the friction of the earth against the back of the wall; near 
'Ugh (in the writer’s opinion) for practical purposes in so uncertain a matter; 
; essentially empirical. 

The direction of the pressure thus found will be the same as when the 
fth is level with the top be; namely, as in Figs 6 and 7, first draw a line, as P y, 
p to the back cm, whether vert or inclined. Then draw another line, as P/’ 

, king the angle y P/= the angle of wall friction, which we all along assume to 
33° 41', or 1.5 to 1. Then P/ will give the direction of the pressure. But its 
nt of application will not always be at P (one-third of the height of the wall 
t we to) as heretofore; for in all cases it will be at that point P, or at some 
, ghcr one as h, where the back is cut by a line IPoreA, Fig 9, drawn from the 
, of grav of the sustained earth (omitting any that rests immediately on the top 
■ , and parallel to the slope to d of max pres; and such a line will strike at one- 
| "d the height of the wall only when the sustained earth t c m or dem forms a 
mplete t riangle, one of whose angles is at the inner top edge c of the wall. 

! all other cases said line for a surcharge will strike above P. 


i downward from C, as 
gs 6, 7, 8, use the wt of the 
the slope of max pressure. 
1 be at P (at one-third of 
i litt le higher as explained 




a 




















690 


RETAINING-WALLS. 




Art. 7. On page 683, Fig 1, we recommend that the base o $ at the ground 
line of well built vertical walls should not be less than .35, or .4, or .5 of tin 
height d s above said line, depending on the kind of masonry. Bnt a wall with i 
battered (inclined) frout or face as found by Art 8, (by which the followinj 
table was prepared), will be as strong, and at the same time contain less masour 
than a vert wall, although the battered one will have the thickest base os. 

Table 3, of thicknesses at base o #, Fig 1, and at top c ei, ol 
walls with battered faces, so as to be as strong as vertica 
ones which contain more masonry. 

For the cub yds of masonry above o s per foot run of wall, mult th 
square of the vert height d s by the number in the column of cub yds. The) 
add the foundation masonry below o s. See also Table, p 693. Also study Rems 
and 2, Art 8. J 

(Original.) 


All the walls below have the same strength 
as a vert one whose base os, fig 1= .35 
of its ht d s. 

All the walls below have the 
same strength as a vert one 
whose base o s, fig 1 = .4 
of its ht d s. 

-- 

All the walls below have th 
same strength as a vei 
one whose base os, fig 1: 
.5 of its ht d s. 

Batter, in 
ins to a ft. 

Ci 

Base, in 
pts of 
bt. 

nt stone 
Top, in 
pts of 
ht. 

. 

C yds per 
ft run. 

M 

Base, in 
pts of 
ht. 

ortar ru 
Top, in 
pts of 
ht. 

bble. 

C yds per 
ft run. 

E 

Base, in 
pts of 
ht. 

iry rubl 
Top, in 
pts of 
ht. 

ale. 

C yds pe 
ft run. 

0 

.350 

.350 

.01296 

.400 

.400 

.01482 

.500 

.500 

.01852 

Yz 

.352 

.310 

.01226 

.401 

.359 

.01407 

.501 

.459 

.01778 

1 

.355 

.270 

.01158 

.403 

.320 

.01339 

.503 

.420 

.01709 

^Vz 

.359 

.234 

.01098 

.408 

.283 

.01280 

.506 

.381 

.01643 

2 

.361 

.197 

.01039 

.413 

.246 

.01220 

.510 

.343 

.01580 

l l /z 

.371 

.163 

.00989 

.419 

.210 

.01165 

.516 

.308 

.01526 

3 

.379 

.129 

.00941 

.425 

.175 

.01111 

.522 

.272 

.01470 

%Vz 

.389 

.096 

.00898 

.435 

.143 

.01070 

.528 

.236 

.01415 

4 

.400 

.066 

.00863 

.445 

.110 

.01028 

.537 

.204 

.01372 

5 

.425 

.007 

.00800 

.468 

.051 

.00961 

.555 

.138 

.01283 

Triangle 

.429 

.000 

.00794 

.490 

.000 

.00907 

.612 

.000 

.01133 


circulating : 


•L j i s V. v m;hiu at no less angle Tor a savant than for a 

body else. For practical purposes, we may say that dry sand, gravel and earths, slope at 33° 41 
1^ to 1; as abundaut experience on railroad embkts proves. Poncelet gives tables for walls to su 
port dry earth sloping at 1 to 1. or 4a°; but as we do not believe in the existence of such earth t 
omit such tables. Sand, gravel, and earths tuay be moistened to diff degrees, so as to stand at ai 
angle between hop and vert; and by moistening and ramming, the earths may be con verted into con 
pact masses, exerting little or no pres; and may even so continue after they become drv ; being the" 
in fact, a kind of air-dned brick. It is sometimes difficult to know whether earth nr t. 


f V . Cf’ J ; .‘s f ■ '“*'0 may even -s > continue alter they become drv ; being the 

in fact, a kind of air-dried brick. It s sometimes difficult to know whether earth or sand is perfect 
dry or not, and an exceedingly small degree of moisture will cause them to stand at 1 to 1 in smr 
heap*, such as have probably been observed by the authorities on the subject. The writer found uTi 
fine sand from the seashore, and undercover would u/ m writer iouna tnt 


. -y. ....... . . j ..... ..... .... lue auuiorities on tne sub ect. The writer found tht 

fine sand from the sea shore, and under cover, would stand at 1 U to l _ «nter iouna tin 

at 1 to 1 when the air was damp. Yet no diff whatever ”n its dUrnn dry weath( ' r J, an 

the feeling. Its susceptibility to dampness was of course owing to salt. A few'iiandfuls'o^drv oa^t 
may perhaps be coquetted into standing at 1 to 1 on -i tanlo • h,,t „„ nanaiuis or (lr\ oart 

11 '* S''"’!'."! In Inns' qn*ntltl«, from carl, and aircimm. in Mnpc U 1 £!u\iI7to V* nr 

S'J “ 60 “ el *» 'nlcnlations, .bar. .5,,, 


confined our tables to dry backing As stated in ArV i"^ mve ’ ol ? the score of safety 
eions less than those nil * .i] d m A . r . f cann °t recommend dtme.i 














































RETAINING-WALLS. 


691 


assumed to be applied at % of the height from the bottom; nor indeed, can it be 
calculated at all. 

Rem 2. Wharf walls are an instance where the thickness should be increased 
notwithstanding that the pres of the water in front helps to sustain them. The earth 
behind such walls, is not only liable to be very heavily loaded when vessels are dis- 
ohargmg; but is apt to become saturated with water, especially below low-water 
evel; and thus to exert a very great pres against the walls. Moreover, the water 
$ets under the wall; and by its upward pressure virtually reduces its weight and 
lonsequently its stability. The same cause of course diminishes the friction of the 
r ft U u P° n its base. Such walls are, therefore, very liable to slide, if the foundation 
s smooth, and horizontal; and have done so even when the foundation had a con¬ 
siderable inclination backward, as in Fig 1. See Art 9. 

Rem. 3. A retaining-wall is usually in greater danger Tor a few months after its completion, than 
tter time has been allowed for the mortar to harden perfectly ; and for the backing to settle. When 
here are suspicions of the safety of a new wall, it would be well to place strong temporary shores 
.gainst it, at about. M to of its height above ground. In some cases, permanent buttresses of 
aasonrv may be built for the purpose. They should be well bonded into the wall 
Rem. 4. The pres of the earth backing will be much reduced, if the first few feet of its height be 
oade up in thin hor layers, to be consolidated by being used by the masons instead of scaffolding; aa 
nown at h , Fig 1. Frequently this can be done without inconvenience; and at very trifling cost. 

Art. 8. To change a vert retaining.wall. Into one with a 
►altered face, which shall present an equal resistance 
igainst overturning; although requiring less masonry. 

'his is sometimes termed a transformation of profile. (Original.) 

Let ab oi, Fig 10, be the vert wall. Mult its base 
in',by 1.225; (1.22475 is nearer;) the prod will be the 
mse o e, of a triangular wall b o e, possessing the 
ame stability; and yet not requiring much more 
han half the masonry of the vert one! See Rem 1. 

Cliis being done, suppose a wall to be desired with a 
ace batter, of say 3 ins to a ft; or 1 in 4. From the 
>oint n, where the face of the triangular wall inter¬ 
acts that of the vert one, step off vert any 4 short 
qual spaces; and from the upper one to, step off one 
[pace hor, to v. Through v and n draw the dotted 
ine s t, which evidently will batter 1 in 4. Then is 
sto approximately the reqd wall; but a little 
hicker than necessary. To reduce it, from t draw 
he dotted line t b. Mark the point c, where this 
ine intersects the face a i, of the vert wall; and 
hrough c draw d ?, parallel to s t. Then is b d l o 
he reqd wall. Our fig is drawn in an exaggerated 
banner, so as to avoid confusion in the lines. The 
ase o e of the triangular wall, would not in reality 
e near so great as it is represented. 

It will he observed that as the base increases, the quantity of masonry diminishes. 

Rem. 1. The battered wall will in fact be safer than the vert 
,(>ne. The battered wall has the same moment of stability as the vert one ; and the 
ares of the earth against it also remains unchanged, but the moment or tendency of the pres to upset 
he wall has become less. For let a b m n, Fig 11, represent a vertical wall; aud/o the amount and 
lirection of pres behind it. (For ease of illustration, we have placed o above the true ceu of pres of the 
:arth filling, which would be at one-third of a n above n.) Now, the leverage with which this pre* 
ends to overturn the wall around its toe m, is the dist m s, 
neasured from the toe or fulcrum m, and at right angles to 
he direction /os c of the pres ; and this leverage mult by 
he force / o, gives the overturning tendency or moment of 
aid force. See “Moments and leverage." Again, 

et a ny. represent a triangular wall of the same stability 
ts the other, as found by our rule. Here we still have the 
ame amount fo. and direction fo s c, of pres force against 
he wall; but it now acts to overturn the wall any 
round the toe y : and therefore, with the reduced leverage 
c. Consequent.lv, its overturning tendency is less than 
iefore. Therefore, in ordinary language, we may say that 
he wall is stronger than before, although its moment of 
tability, or standing tendency, has in itself undergone no 
ihange. If the pres / o against the vert back were hor, as 
n the case of water, then its leverage would evidently be 
be same in both walls; and the proportion between the 
iverturning moment of the pres, aud the moments of 
t&biiitv of the two walls, would be constant. P 2'29. 

Rkm. 2. In attempting to reduce the masonry by adopt- 
ng a wall, o b e, Fig 10, of a triangular section : or of one 
early approaching a triangle, special attention should be 
I veil to the qualitv of the masonry near the thin toee; 
rhich will otherwise be apt to crack, or fail under the pres. 

48 


















692 


.RETAINING-WALLS. 


KkM 3. SlORKOVEU, WHEW COMMON MORTAR 19 USED WITHOUT AN ADMIXTURE OF CEMENT, which it DeVCT 
should be, iu retaiuiug- walls, where durability is au object, a great batter is objec¬ 
tionable ; inasmuch as the raiu, combined with frost, &c, soon destroys the mor¬ 
tar. In such cases, therefore, the batter should not exceed 1 or ius to a ft; and 



even then, at least the pointing of the joints, and a few feet in height of both 
the upper and the lower courses of masonry, should be done with cement, or 
cement-mortar. We have observed a most marked diffin the corrosion of the mor¬ 
tar, where, in the same walls, with the same exposure, oue portion has been built 
with a vert face; and another with a batter of but 1J$ inch to a foot. Common 
mortar will never set properly, and continue firm, when it is exposed to mois¬ 
ture from the earth. This is very observable near the tops and bottoms of 
abuts, retaining-walls, &c; the lime-mortar at those parts will generally be 
found to be rendered entirely worthless. A profile somewhat like Fig 12, may 
at times prove serviceable, instead of the triangular. This is the form of the 
Gothic buttress; which probably had its origin in the cause just spoken of. 

Art. 9. A retaining:- wall may slide, without 

in losing: its vertical! ty ; and, indeed, without any danger 
Jp j O 1 /O of being overturned. This is very apt to occur if it is built upon 
*■* a hor wooden platform; or upon a level surf of rock, or clay, 

Without other means than mere friction to prevent sliding. This may be obviated 


by inclining the base, as in Pig 1; by founding the wall at such a depth as to pro¬ 
vide a proper resistance from the soil in front; or in case of a platform, by securing 
one or more lines of strong beams to its upper surf, across the direction in which 
sliding would take place. On wet clay, friction may be as low as from .2 to 
the weight of tne wall; on dry earth, it is about y to %; and on sand or gravel, 

about % to %. The friction of masonry on a wooden 
platform, is about _6_ of the wt, if dry; and % if wet. 

Counterforts, shown in plan at c c c, Fig 13, consist in 
an increase of the t hickness of the wall, at its back, at regular inter¬ 
vals of its length. We conceive them to be but little better than a 
waste of masonry. When a wall of this kind fails, it almost in¬ 
variably separates from its counterforts; to which it is connected 
merely by the adhesion of the mortar; and to a slight extent, by the 


,jn R 




Fujl3 


bonding of the masonry. The table in Art 7 shows that a very small addition to the base of a wall, is 
attended by a great increase of its strength; we therefore think that the masonry of counterforts 
■would be much better, and more cheaply employed in giving the wall an additional thickness, along 
Its entire length; and for the lower third of its height. Counterforts are very generally used in 
retaining-walls by European engineers; but rarely, if ever, by Americans 

Buttresses are like counterforts, except that they are placed in front of a wall instead of be¬ 
hind it; and that their profile is generally triaugular, or nearly so. They greatly increase its strength; 
but. being unsightly, are seldom used, except as a remedy when a wall is seen to be failing. 

Land-ties, or long rods of iron, have been employed as a makeshift for upholding weak re* 
taining-walls. Extending through the wall from its face, the land euds are connected with anchors 
of masonry, cast-iron or wooden posts; the whole being at some dist below the surface. 

Retaining: walls with curved profiles are mentioned here merely to cau¬ 


tion the young engineer against building them. Although sanctioned by the practice of some high 
authorities, they really possess no merit sufficient to compensate for the additional expense and trou¬ 
ble of their construction. 

Art. 10. Among military men, a retaining-wall is called a revetment. When the 
earth is level with the top, a SCJirp revetment; when above it, a COUIltersCRrp 
revetmeut, or a demi-revetment. When the face of the wall is battered, a sloping; and when the back 
is battered, a countersloping revetment. The batter is called the talus. 

Art. 11. The pres against a wall Fig 6, from sand etc level with its top, is not 
diminished by reducing the quantity of sand, until its top width cs becomes less than 
that (c t) pertaining to the angle cm t of maximum pres. The pres then begins to di¬ 
minish, hut in practice the diminution is not appreciable until the width is reduced to about 
one sixth of that (c s) pertaining to the angle cm s of natural slope, or about half of 
c t. The pres then begins to decrease rapidly as the width is further reduced. 

Table 4, of contents in cub yards for each foot in length 
of retaining:-waUs, with a thickness at base equal to .4 of the vert height, 
if the back is vert. If the back is stepped according to the rule in Art 3, p 685, the 
proportionate thickness at base will of course he increased. Face batter, \y inches 
to a foot; or J^gth of the height. Back either vert, or stepped according to the rule 
in Art 3, Fig 3. The strength is very nearly equal to that of a vert wall with ti 
base of .4 its height. See table, p 690. Experience has proved that such walls, 
when composed of well-scabbled mortar rubble, are safe under all ordinary circum¬ 
stances for earth level with the top. Steps or offsets, o c, at foot, Fig 1, are not here 
included. 


ril 



















STONE BRIDGES 


693 


TABLE 4. (Original.) 


Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

1 

.013 

10^ 

1.38 

20 

5.00 

29^ 

10.9 

48 

28.8 

74 

68.5 


.028 

11 

1.51 

hi 

5.25 

30 

11.3 

49 

30.0 

76 

72 2 

2 

.050 

hi 

1.65 

21 

5.51 

31 

12.0 

50 

31.3 

78 

76 1 


.078 

12 

1.80 

hi 

5.78 

32 

12.8 

51 

32 5 

80 

80.0 

3 

.113 

hi 

1.95 

22 

6.05 

33 

13.6 

52 

33.8 

82 

84.1 

hi 

.153 

13 

2.11 

hi 

6.33 

34 

14.5 

53 

35.1 

84 

88.4 

4 

.200 

hi 

2.28 

23 

6 61 

35 

15.3 

54 

36.5 

86 

92.5 

hi 

.253 

14 

2.45 

hi 

6.90 

36 

16.2 

55 

37.8 

88 

96.8 

5 

.313 

hi 

2.63 

24 

7.20 

37 

17.1 

56 

39.2 

90 

101.3 

hi 

.378 

15 

2.81 

hi 

7.50 

38 

18.1 

57 

40.6 

92 

105.8 

6 

.450 

hi 

3.00 

25 

7.81 

39 

19 0 

58 

42.1 

94 

110 5 

% 

.528 

16 

3.20 

hi 

8.13 

40 

20.0 

59 

43.5 

96 

115.2 

7 

.613 


3.40 

26 

8.45 

41 

21.0 

60 

45.0 

98 

120.1 

H 

.703 

17 

3.61 

hi 

8.78 

42 

22.1 

62 

48.1 

100 

125.0 

8 

.800 

hi 

3.83 

27 

9.12 

43 

23.1 

64 

51.2 

102 

130.1 

hi 

.903 

18 

4.05 

hi 

9 45 

44 

24.2 

66 

54.5 

104 

135.2 

a 

1.01 

H 

4.28 

28 

9.80 

45 

25.3 

68 

57.8 

106 

110.5 

hi 

1.13 

19 

4.51 

hi 

10 2 

46 

26.5 

70 

61.3 



0 

1.25 

* 

4.75 

29 

10.5 

47 

27.6 

72 

64 8 




STONE BRIDGES. 


Art. 1. In an arch s t s, Fig 1, the dist eo is called its span ; i a its rise ; t its 
rown ; its lower boundary line, e.ao, its sottit, or intrados ; the upper one, 
!r, its back, or extrados. The terms soffit and back are also applied to the 
itire lower and upper curved surfaces of the whole arch. The ends of an arch, or 
ie showing areas comprised between its intrados and extrados, are its faces; thus 
ie area stsa is a face. The inclined surfaces or joints, re, ro, upon which the feet 
the arch rest, or from which the arch springs, are the skew bar ks. Lines 
vel with e and o, at right angles to the faces of the arch, and forming the lower 
Iges of its feet, (see nn , Fig 2%,) are the springing' lines, or spring's. The 
ocks of which the arch itself is composed, are the arch-stones, or votissoirs. 
ie center one, ta, is the keystone; and the lowest ones, ss, the springers, 
ie term archblock might be substituted for voussoir, and like it would apply to 
•ick or other material, as well as to stone. The parts tr,tr, are the haunches; 
id the spaces t r l, trb , above these, are the spand rels. The material deposited 
these spaces is the spandrel filling; it is sometimes earth, sometimes ma- 
nry; or partly of each, as in Fig 1. 

rn large arches, it often consists of several parallel spandrel-walls, ll, Fig 2^, running lengthwise 
the road wav, or astraddle of the arch. They are covered at top either by small arches from wall to 
. 11 , or bv Hat stones, for supporting the material of the roadway. They are also at times connected 
;ether by vert cross-walls at intervals, for steadying them laterally, as at tt, Fig 2M. The parts 
> e n, gpon, Fig 1, are the abutments of the arch; en, o n, the faces ; gp.gp, the backs; and 
i, p n, the bases of the abuts. The bases are usually widened by fret, steps, or offsets, d d , for dis- 
huting the wt of the bridge over a greater area of foundation ; thus diminishing the danger of sot- 
meat. The distance t a in any arch-stone, is called its depth. 

The only arche3 in common 
e lor bridges, are the circular, 
ften called segmental); and 
e elliptic. 

Art. 2. To find tSie 
gpth of keystone for 
rst-class cut-stone 
relies, whether cir- 
ilar or elliptic.* 

Find the rad co, Fig 1, which 
11 touch the arch at o, a, and 
Add together this rad, and 
If the span o e. Take the sq 
of the sum. Div this sq rt 
4. To the quot add jq of a 
Or by formula, 


t t 



6 Inasmuch as the rules which we give for arches and abuts are entirely original and novel, it may 
; be amiss to state that they are not altogether empirical; but are based upon accurate drawings 































































694 


STONE BRIDGES. 


Depth of key _ VRad + half span 
in feet 4 


.2 foot. 


For soeond-class work, this depth may be increased about ^th part; c 
for brick or fair rubble, about J^rd. See table of Keystones, p b9<. 

I 11 large arches it is advisable to increase the depth of the archstones toward tli 
springs • but when the span is as small as about 6 U to 80 or 100 feet, this is not at a 
necessary if the stone is good; although the arch will be stronger if it ib done I 
practice' this increase, even in the largest spans, does not exceed from to / 2 tti 
depth of the key ; although theory would require much more in arches of great ns 


Rem To flml the rad c o, whether the arch be circular or elliptic. Squa 
half the span e o. Square the whole rise t a. Add these squares together; div tl 
sum by twice the rise i a. Or it may be found near enough lor this purpose by t) 
dividers, from a small arch drawn to a scale. 


Amount of pressure sustained by archstones. In bridges c 
the same width of roadway ; if all the other parts bore to each other the same propo 
tion as the spans, the total pres would increase as the squares of the spans, while tl 
pressure pev sqvlcivg foot would increase as tlie spans. I$ut in practice tne depth of tl 
archstones increases much less rapidly than the span; while the thickness of tl 
roadway material, and the extraneous load per sq ft, remain the same for all span 
Hence the total pressures, at key and at spring, increase less rapidly than the squar 
of the spans ; but wore rapidly than the simple spans; as do also the pressures p 
square foot. Thus in two bridges of the same width, but with spans of 100 and 200 1 
with depths of archstones taken from our table page 697. and uniform from key 
spring; supposed to be filled up solid with masonry of 160 lbs per cub ft. to a level < 
about 15 inches above the crown, (including the stone paving of the roadway); wi 
an extraneous load of 100 lbs per sq ft; the pressures will be approximately as f( 
lows: 



Span 100 ft. 



Span 200 ft. 



AT KEY. 


AT SPRING. 

AT KEY. 

AT SPRING. 


For 1 ft in 


For 1 ft in 


For 1 ft in 


For 1 ft in 



width of 
its entire 

Per sq ft. 

width of 
its entire 

Per sq ft. 

width of 
its entire 

Per sq ft. 

width of 
its entire 

Per sq 


depth. 


depth. 


depth. 


depth. 


Rise. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons 

34 

* 254 

13*4 

58 

18*4 

126 

2934 

179 

42 

1 

TC 

363* 

12k 

57 

19 

112 

2734 

181 

44 

34 

31 34 

11 

5734 

20 

97 

2434 

188 

4734 

34 

25 

9 

6134 

2234 

8034 

21 

207 

5434 

34 

18 

6*4 

6734 

25 

5734 

1534 

230 

6134 


It will be seen that with the same span, the pres at the key becomes less, while tl 
at the spring becomes greater, as the rise increases. Also that when the archstor 
are of uniform depth, the pres at either spring of a semicircular arch is about 4 tin 
as great as at the key ; whereas when the rise is but one-sixth of the span, the pres 
spring averages but about one-third greater than at the key. These proportions va 
somewhat in different spans. 

The greater pres per sq ft at the springs may be reduced by increasing the depth 
the archstones towards the springs. This however is not necessary in moderate spa 
inasmuch as good stone will be safe even under this greater pres. 

By using 1 parallel spandrel n ails, see Fig 2%, p 698, or by partly f 
irig with earth instead of masonry, the pres on the archstones may be diminish 
say, as a rough average, about £ part. 


and calculations made by the writer, of lines of pres, &c, of arches from 1 to 300 ft span, and of ev 
rise, from a semicircle to ^ of the span. From these drawings he endeavored to find proporti 
which, although they might uot endure the test of strict criticism, would still apply to all the cs 
with un accuracy sufficient for ordinury practical purposes. 



























STONE BRIDGES 


695 


Table 1. Of some existing 1 arches, with both their actual and their 

calculated depths (by our rule) of keystone. Where two depths are given in the column of keys the 
smallest is for first class cut-stone, and the largest for good rubble, or brick. Those also which are 
)«uot specified are of first-class cut-stone. C stands for circular, E for elliptic. For 2d class work, add 
about )^th part; and for brick, or fair rubble, about %th. 








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* Granite Width 12 ft. The centers (not struck for 4 months) rested on sand in 1-32 inch sheet- 
iron cylinders 1 ft high, 1 ft diameter. Not injured by a distributed load of 360 

tons nor bv a wt of 5 tons falling 18 ins on key. T otal settlement about inch and a Quarter. Depth 
or archstones 2.67 ft at key ; 3.6 ft at springs. Rise about one-eigbteenth of span. Mortar joiuts full 
quarter inch thick. 



















































696 


STONE BRIDGES. 


The arch on the Bourbonnais Railway, is probably the boldest:* and tub Cabin John arch, by 
Capt, now Gen'l M. C. Meigs, U S Army, the grandest stone one in existence. Pont-y-Prydd, in 
Wales, is a common road bridge, of very rude construction ; with a dangerously steep roadway. It 
was built entirely of rubble, iu mortar, by a common country mason, in 1750; aud is still in perfect 
condition. Only the outer, or showing arch-stones, are 2.5 ft deep: and that depth is made up of two 
stones. The inner arch-stones are hut 1.5 ft deep; and but from ti to 9 inches thick. The stone quar¬ 
ried with tolerably fair natural beds; and received little or no dressing in addition. The bridge is a 
fine example of that iguorance which often passes for boldness. Pont Napoleon carries a railroad 
across the Seine at Paris. The arches are of the uniform depth of 4 ft, from crown to spring. They 
are composed chiefly of small rough quarry chips , or spawls ; well washed, to free them from dirt 
and dust; and then thoroughly bedded iu good cement; and grouted with the same. It is in fact an 
arch of cement-concrete. The Pont i»e Alma, near it, aud built in the same way , has elliptic arches 
of from 126 to 141 ft span ; with rises of -I the span. Key 4.9 ft. These two bridges, considering the 
want of precedent in this kind of construction, on so large a scale, must be regarded as very bold; 
and as reflecting the highest credit for practical science, upon their engineers, Parcel aud Couche. 
Some trouble arose from the unequal contraction of the different thicknesses of cemeut. They show 
what may be readily accomplished in arches of moderate spaus, by means of small stone, and good 
hydraulic cement when large stone fit for arches is not procurable. In Pont Napoleon the depth of 
arch is less than our rule gives for second class cut-stone. 


Rem. Our engineers are usually too sparing of cement. It should 

be freely used, not only in the arches themselves, and in the masonry above them, as a 
protection from rain-soakage; but in abuts, wing-walls, retaining-walls, and all other 
important masonry exposed to dampness. The entire backs of important brick arches 
should be covered with a layer of good cement, about an inch thick. The want of it 
can be seen throughout most of our public works. The common mortar will be 
found to be decayed, and falling down from the soffits of arches; and from the joints 
of masonry generally, within from 3 to 6 ft of the surface of the ground. The mois¬ 
ture rises by capillary attraction, to that dist above the surf of the nat soil; or 
descends to it from the artificial surf of embankments, Ac; therefore, cement-mortar 
should be employed in those portions at least. The mortar in the faces of battered 
walls, even when the batter is but 1 to 1% inches per foot, is far more injured by rain 
and exposure, than in vert ones; and should therefore be of the best quality. See 
Mortar, Ac. 

We have, however, seen a quite free percolation of surface water through brick 
arches of nearly 3 ft in depth, even when cement was freely used. In aqueduct 
bridges, we believe that cement has not been found to prevent leaks, whether the 
arches were of brick, or even of cut-stone. May not this be the effect of cracks 
produced by settlement of the arch; or by contraction and expansion under atmos¬ 
pheric influence? Cement at any rate prevents the joints from crumbling. 

Art. 3. The keystones for large elliptic arches by the best en 
gineers, are generally made about P iu 't deeper than our rule requires; or than is 
considered necessary for circular ones of the same span and rise; in order to keep the 
line of pres well within the joints ; although the elliptic arch,with its spandrel filling, 

Jl. has slightly less wt; and that wt has 

a trifle less leverage than in a circulai 
Fin I- one » a,,( * consequently it exerts les 
P res Poth at the key, and at the skew 
back. See London, Gloucester, an. 
Waterloo bridges, in the preceding 
table. 

■ 

Rbm. Young engineers are apt to affect shallow arch-stones; but It would be far better to adop 
the opposite course; for not only do deep ones make a more stable structure, but a thin arch is a 
unsightly an object as too slender a column. According to our own taste, arch-stones fully % deepe 
than our rule gives for first-class cut stone, are greatly to be preferred when appearance is consulted i 
Especially when an arch is of rough rubble, whioh costs about the same whether it is built up a 
arch, or as spandrel filling, it is mere folly to make the arches shallow. Stability and durabillt; 
should be the objects aimed at; and wheu they can be attained even to excess, without increased oost 
it is best to do so. 



* Built like that at Soupes in the preceding table. 












STONE BRIDGES 


697 


Table 

by Art 2. 
fourth to 
--— 


2. Depths of keystones for arches of first-class cut stone, 
For second class add full one-eighth part; and for superior brick one- 
one-third part, if the span exceeds about 15 or 20 ft. Original. 


tli 


Rise, in parts of the span. 


Span. 

Feet. 

1 

2 

1 

3 

1 

¥ 

1 

T 


Key. Ft. 

Key. Ft. 

Key. Ft. 

Key. Ft. 

2 

.55 

.56 

.58 

.60 

4 

.70 

.72 

.74 

;76 

6 

.81 

.83 

.86 

.89 

8 

.91 

.93 

.96 

1.00 

10 

.99 

1.01 

1.04 

1.07 

15 

1.17 

1.19 

1.22 

1.26 

20 

1.32 

1.35 

1.38 

1.43 

25 

1.45 

1.48 

1.53 

1.58 

30 

1.57 

1.60 

1.65 

1.71 

35 

1.68 

1.70 

1.76 

1.83 

40 

1.78 

1.81 

1.88 

1.95 

50 

1.97 

2 00 

2.08 

2.16 

60 

2.14 

2.18 

2.26 

2.35 

80 

2.44 

2.49 

2.58 

2.68 

100 

2.70 

2.75 

2.86 

2.97 

120 

2.94 

2.99 

3.10 

3.22 

140 

3.16 

3.21 

3.33 

3.46 

160 

3.36 

3.44 

3.58 

3.72 

180 

3.56 

3.63 

3.75 

3.90 

200 

3.74 

3.81 

3.95 

4.12 

220 

3.91 

4.00 

4.13 

4.30 

240 

4.07 

4.15 

4.30 

4.48 

260 

4.23 

4.31 

4.47 

4.66 

280 

4.38 

4.46 

4.63 


300 

4.53 

4.62 

4.80 



l 


Key. Ft. 
.61 
.79 
.92 
1.03 
1.11 
1.30 

1.48 
1.64 

1.78 
1.90 
2.03 
2.25 
2.44 

2.78 
3.09 
3.35 
3.60 
3.87 
4.06 
4.29 

4.48 


Key. Ft. 
.64 
.83 
.97 
1.09 
1.18 

1.40 
1.59 
1.76 
1.91 
2.04 
2.18 

2.41 
2.62 

2 98 
3.32 

3 61 
3.87 
4.17 
4.38 


1 

To 


Key. Ft. 

.68 

.88 
1.03 
1.16 
1.26 
1.50 
1.70 
1.88 
2.04 
2 19 
2.33 
2.58 
2.80 
3.18 
3.55 
3.88 
4.15 


rid Art. 4. To proportion the abuts for an arch of stone or 
In l>rick, whether circular or elliptic. (Original.) 

The writer ventures to offer the following rule, in the belief that it will be found 
■Is o combine the requirements of theory with those of economy and ease of applica- 
i ion, to perhaps as great an extent as is attainable in an endeavor to reduce so com¬ 
plicated a subject, to a simple and reliable working- rule for prac- 
.3. tical bridge-builders. This is all that he claims for it. Notwithstanding its 
j s simplicity, it is the result of much labor on his part. It applies equally to the smallest 
1 1( mlvert, and to the largest bridge ; whatever may be the proportions of span and rise; 
,» uid to any height of abut whatever. It applies also to all the usual methods of filling 
1,1 ibove the arch ; whether with solid masonry to the level v f Fig 2, of the top of the 
l„i irch; or entirely with earth ; or partly with each, as represented in the fig: or with 
Jparallel spandrel-walls extending to the back of the abut, as in Fig 2%. Although 
, /the stability of an abut cannot remain precisely the same under all these conditions, 
Jyet the diff of thickness which would follow from a strict investigation of each par- 
r ,j ;icuiar case, is not sufficient to warrant us in embarrassing a rule intended for popu- 
ar use, by a multitude of exceptions and modifications which would defeat the very 
object for which it was designed. We shall not touch upon the theory of arches, 
ixcept in the way of incidental allusion to it. Theories for arches, and their abuts, 
'pel unit all consideration of passing loads ; and consequently are entirely inapplicable 
n practice when, as is frequently the case, (especially in railroad bridges of moderate 
spans,) the load bears a large ratio to the wt of the arch itself. Hence the theoretical 
ine of thrust has no place in such cases. Our rule is intended for common practice: 
ind we conceive that no error of practical importance will attend its application to 
iny case whatever ; whether the arch be circular or elliptic. 


It. gives a thickness of abut, which, without any backing 
r»f earth behind it. is safe in itself, and in all cases, against 
the pres, when the bridge is unloaded. Moreover, in very large arches, 
n which the greatest load likely to come upon them in practice is small in comparison 
with the wt of the arch itself, and the filling above it, our abuts would also be safe 
Tom the loaded bridge, without any dependence upon the earth behind them ; but 
is the arches become less, and consequently the wt of the load becomes greater in 
sroportion to that of the arch, and of the filling above it, we must depend more and 
nore upon the resistance of the earth behind the abuts, in order to avoid the neces- 
ijt.y of giving the latter an extravagant thickness. It will therefore be understood 
hroughoui that, except when parallel spandrel walls are used, our rules suppose that 
%f ter the bridge is finished, earth will be deposited behind the abuts, and to the height 
of the roadway, as usual. 


























698 


STONE BRIDGES, 


Iu small bridges and large culverts of first class railroads, subject to the jarring 
of heavy trains at high speeds, the comparative cheapness with which an excess of. 
strength can thus be given to important structures, has led, in many cases, to the 
use of abutments from one-fourth to one-half thicker thau by the following rule. 
If of roue'll rubble add 6 ins to insure full thickness in every part. 


Thicks o n of abut at spring 
in ft, when the height o s 
does not exceed 1 times the 
base 8 p 


Bad in ft 
5 


+ 


rise in ft 

10 


+ 2 ft. 


Mark the points n and y thus ascertained. Next, from the center i, of the span 
or chord eo, lay off ih, equal to ^ part of the span. Join ah; and through n, and 
parallel to ah, draw the indefinite line gnp of the abut. Do the same with the 
other abut. Make y m and ng each equal to half the entire height i t of the arch ; 
and from g draw a straight line g x, touching the back of the arch as high up as pos¬ 
sible; or still better, as shown at tm, with a rad d t or d m, (to be found by trial,) 
describe an arc t m. Then gx or t m will be the top of the masonry filling above the 
arch;* and this should be completed before striking the centers; before which, 
also, the embkt should be finished, at least up to y n. 



Now find by trial the point s, Fig 2, at which the thickness sp is equal to two- 



* Except whem 
1 


THE RISE 
BUT ABOUT iOF THB SPAM, OK 

cess; in which case carry the 
masonry up 


solid to the level 
vtf, of the top of the arch. Or 
if the arch is a large one, ex¬ 
ceeding say about 60 ft span ; 
and especially if its rise is 
greater than about -1 of its 
span, it is better to economize 
masonry by the use of parallel 
interior spandrel-walls, 11. Fig 
carried up to vtf. Fig 2. 
Indeed, such interior walls may 
often be advantageously intro¬ 
duced in much smaller arches. 
When high, they are steadied 
by occasional cross-walls, as 11, 
f ig Their feet should bt 

spread by offsets, as shown at o o o, so as to bear upon tbe whole surf of the hack of the arch : thus 
equalizing the pres upon it. On top of the walls flagstones may be laid, or small arehes may bt i 
turned from wall to wall, for supporting the ballast, &c, of the roadway. The spaces below are lefl 


hollow. In Fig 2J4 the dark part w is supposed to be a section across an abutment; but omitting * 
the second cross-wall, similar to 11. In R R bridges, put a spaudrel-wall, 11, under each rail. 



























































































STONE BRIDGES. 


699 


irds of the corresponding vert height os, and draw sp. Then will the thickness 
i or ey be that at the springing line of the given circular or elliptic arch of any 
' »e and span; and the line gp will be the back of the abut; provided its height os 
es not exceed 1% times sp\ or in other words, provided sp is not less than % of 
■ In practice, o s will rarely exceed this limit; and only in arches of considerable 
>e. But if it should, as for instance at o q, then make the base qu equal to sp, added 
one-fourth of the additional height sq; and draw the back u w, parallel to g p ; 
d extending to the same height, &c, as in Fig 2. If, however, this addition of x /£ 
s q should in any case give a base q u, less than one-half the total height o q, (which 
11 very rarely happen in practice,) then make qu equal to half said total height; 
l awing the back parallel to gp, and extending it to the same height as before. The 
i ditional thicknesses thus found below sp, have reference rather to the pres of the 
rth behind the abut, than to the thrust of the arch. In a very high abut, the inner 
' le g p would give a thickness too slight to sustain this earth safely. 

When the height ob. Fig 2, of the abut is less than the thickness on at spring, a 
' iall saving of masonry (not worth attending to, except in large flat arches) may be 
■i ected by reducing the thickness of the abut throughout, thus: Make ok equal to 
' i, and draw kl. Make oz equal to 3^ of on, and draw l z. Then, for any height 
l > i of abut less than on, draw b v, terminating in l z. This bv will be sufficient base, 
the foundations are firm. The back of the abut Mill be drawn upward from v, 
rallel to gp, and terminating at the same height as g or w. 

Rem. 1. All the abuts thus found will (with the provisions in Art 6) be safe, 

1 jthout any dependence upon the w'ing-walls; no matter how high the embkt may 
tend above the top of the arch. If the bridge is narrow, and the inner faces of 
e wing-walls are consequently brought so near together as to afford material as- 
itance to the abuts, the latter may be made thinner; but to what extent, must 
pend upon the judgment of the engineer. 

SV'e, however, caution the young practitioner to be careful how he adopts dimensions less than those 
reu by our rule. There are certain practical considerations, such as carelessness of workmanship: 
I wness of the mortar; danger of undue strains when removing the centers; liability of derange- 
•nt during the process of depositing the earth behind the abuts, and over the arch ; &c, which must 
» t be overlooked; although it is impossible to reduce them to calculation. 

j Whenever it can be done, the centers should remain in place until the embkt is finished; and for 
’ me time afterward, to allow the mortar to set well. But for more on this see Rem 4, p 713. 


Rkm. 2. A good deal of liberty is sometimes taken, in reducing the quantity of masonry above the 
ringing line of arches of considerable rise, and of moderate spans. M'hen care is taken to leave 
e centers standing until the earth filling is completed above the arch, and behind Us abuts, so thai 
may not be deranged by accident during that operation ; and when good cement is used instead of 
mmon mortar, such experiments may be tried with comparative safety ; especially with culvert 
ches, in which the depth of arch-stones is great in proportion to the span They must, however be 
t to the judgment of the engineer in charge ; as no specific rules can be laid down for (hem. They 
n hardly beregarded as legitimate practice, and we cannot recommend them. M e have known 
arly semicircular arches, of 30 to 40 ft span, to be thus built successfully, with scarcely a particle 
masonry above the springs to back them. Such arches, however, are apt to fall, if at any future 
riod the earth filling is removed, without taking the precaution to first build a center or some other 
pport for them. Even when the embkt can be finished before the centers are removed we canno 
commend (and that only in small spans) to do less than to make n g, h ig 2, equal to M of the total 
^ghtTt of the arch ; and from g so found, to draw a straight line touching the back of the arch as 
1 gh up as possible. 

Rem. 3. We have said nothing about battering' the faces of the abuts, 

tcause in the crossing of streams, the batter either diminishes the water-May : or 
quires a greater span of arch. Such a batter, however, to the extent of from 
\\/ 9 ins to a ft, is useful, like the offsets, for distributing the wt of the structure 
id its embkt, over a greater area of foundation ; especially when the last is not 
ituraliv very firm ; or when the embkt extends to a considerable height above the 
ch in our tables, Nos 3 and 5, of approximate quantities of masonry in semi- 
rcular bridges of from 2 to 50 ft span, the faces are supposed to be veit. 


Art 5 Abutmenbniers. When a bridge consists of several archesisus- 
dned bv piers of only the usual thickness, if one arch should by accident of ’flood, 
- otherwise be destroyed, the adjacent ones would overturn the piers; and arch 
Tier arch would then fall. To prevent this, it is usual in important bridges to make 
.me of the piers sufficiently thick to resist the pres of the adjacent arches, m case 
f such an accident; and thus preserve at least a portion of the bridge from luin. 
uch are called abutment-piers. 


Our formula 0 f ^ -f 2 ft, for the thickness at spring: with the back battering as before, 


—- 5 nr io 

the rate of JL of the span to the rise; face vert; will of itself (without any modification for groat 
, ! , av.nt.nier for anv unloaded bridge; and to any height whatever; due 

ift/Als) give a ^ 1 t0 ti e oon^idora 1 1 oV. alluded to in the next Art. Thus, for an abnt-pier 

T?f.*5r3rrr- SKS3K l.*: ^ IK S2SM*S:*J5.S 

KSdVuSToi s? as u».!«*•*» 






700 


STONE BRIDGES. 




may be secure from the pres of the earth behind them; as well as from the pre 9 of the arch ; a eo 
sideratiou which does not apply to abut-piers; in which only the pres of the arch is to be resisted. 

But although the abut-pier thus found by our formula, would be abundant 
safe, yet its shape a b c o , Fig 3, is inadmissible. In practice it would 
changed to oue somewhat like that shown by the dotted lines ; having an eqt 
degree of batter on both faces. This of course requires more masonry, wi 
but little increase of stability; but that cannot be avoided. 

When an abut-pier is built in deep water, or in a shallow stream st 
ject to high freshets, care must be taken that water cannot find its way wad 
the pier, and thus produce an upward pres, which will either diminish, 
entirely counteract its efficiency as an abut. See Remark 2, Art 4, of H 
drostatics. 

Art. 6. Inclination of tlic courses of masonr 
below the springs of an arch. Although our foi 

going rule gives a thickness of abut which cannot be overturns 
or upset , by the pres of the arch, yet if the arch be of large spa 
and small rise, its great hor thrust may produce a sliding oi 
ward of the masonry near the level of the springs, if the ston 
are laid in hor courses; especially if the mortar has not 6et wel 

This danger, it is true, could be avoided by confining the courses togett 
by iron bolts and cramps; or by increasing considerably the thickness of t j 
abuts; but the expense of doing either of these, leads to the cheaper expedie! 
of inclining the masonry, as shown between oand n. Fig 4; the courses nea j 
being steeper; and gradually becoming less steep near ! 
By this process the arch is virtually prolonged into the bo 
of the abut, so far that when the inclination of the low 
masonry ceases, as at n, the direction of the theoretic ] 
line of thrust , or of pres of the arch, (rudely represent! 
by the dotted curved line o n ) is nearly at right angles 
the joints of the hor masonry below n ; and consequent 
said thrust is unable to produce sliding at that point. I 
tween o and n, the line of pres is everywhere so nearly 
right angles to the variously inclined joints, as to preefu 
the possibility of sliding in that interval also. See Art 
of force in Rigid Bodies.* The abut being thus s: j 
throughout from both overturning and sliding, can f 
only from defective foundations; or from the inferior' 
of the stone of which it is built; aud which, if soft, m 
be crushed. 

This inclination of the masonry is as nec( 
sary in an elliptic arch, Fig 434, as in a circul 
one. 


* TTns curved line of pressures is found in the manner directed at Rem 1 p 360 and at Fie 93 

231 Rankine, Moseley, and others call it the line of resist an ee uf.lu 

ply line of pressures to another line which need not be introduced in a r a . ' 

abutments, walls, dams. <fcc. Thev however e .n me 1D Poetical consideration 

because at any part of the height o'f the abut such pofnf show S P wher'e'Vn "!l 1 “ “ a CeM<er of P re8gu 
for many purposes be assumed to be concent rater] The r, 8 - e al l tbe pressure or thrust m 

ZSZ ‘S “ ‘XTJ'S'Wr'Wi." 

dined bed-joints, we shall always be far within the limit of ano Unlb ,i 01 ? the n,!xt P age for draw ing j 
0„r r„,„ rto o.„ tor UU. Jj. 






























701 


STONE BRIDGES. 


tb Jsnri.'urs tl «n f0rn ? 13 f’ ,ftin, - v unf ‘ lvorable for uniting the arch-stones with the inclined masonry near 
narv P case S U^ dlProperly ; or about at right angles to its resultant. Itfordi- 
1 V s dlfhc “'v ma> be overcome by making the joints of only the outside or showing arch- 
.. ° !0 h “ f ® rl “‘to the elhptic curve; as between e and «; while the joints of the inner or hidden 
will r»r»h r, aV the directions shown between g and u , nearly at right angles to the line of thrust It 
^ I'fPP® 0 ' however, that the young engineer will have to construct elliptic arches of suffi- 

nieut magnitude to require either this, or any equivalent expedient, For spans less than 50 ft, with 

rises not less than about A of the span, nothing of the kind is actually necessary, if the mortar is 
good, and has time to harden, t 

In order to incline the masonry of any abut with sufficient accuracy, it would 

Art n -9 ofv y first f v> - tla ', C V, tl , ie curved llne of P res of the given arch, as directed in 
Art oi force in Rigid Bodies, so as to arrange the bed joints about at rightangles 
to it at eveiy point ot its course; butwe offer the following processas sufficing for all 
ordinary practical purposes; while its simplicity places it within the reach of the com¬ 
mon mason. In actual bridges the direction of the actual thrust changes as the load 
is passing; therefore, in practice no given degree of inclination of the abut masonry 
:an conform to it precisely during the entire passage. Consequently, anv excess of 
renuement in this particular, becomes simply ridiculous; especially in small spans. 

Rule for Inclining’ the beds of the masonry in the abuts. 

Add together the rad cm, Fig 4; and the span of the arch. Div the sum by 5. To 
the quot add 3 ft. Make o t, on the rad, equal to the last sum. Then is t a central 
point, toward which to draw the directions of the beds, as in the fig. Draw t s lior 
and from t as a center, describe the arc oy, o being the center of the depth of the’ 
springers. From y lay off on the arc the dist yn, equal to one-sixth part of t y: draw 
tn a. It will never be necessary to incline the masonry below this t n a. Neither 
need the inclination extend entirely to the face m i of the abut; but may stop at e, 
about half-way between i and n. From e upward, the iucliuation may extend for¬ 
ward to the line e m. 


t The feet of both elliptic anil semicircular arches are always made hor ; but it is plain from Fig 
4^. that this practice is at variance with correct principles of stability in the case of the ellipse Tt 
is the same in the semicircle. In ordinary bridges of the latter form, the vert pres, or weight resting 
on each skewback, is (roughly speaking) usually about from 3 14 to 4 times the hor pres on the same ; 
and the total pres is about 4 times as great as the pres on the keystone. Therefore, theoretically the 
skewback should usually be about 4 times as deep as the keystone ; and its bed, instead of being iior 
should be inclined at the rate of about 1 vert to 4 hor. ' 







702 


STONE BRIDGES, 


When the arch 



is flat, this inclination may become so steep, especially in the upper parts, that 
struts, or shores of some kind, must be used for preventing the ma¬ 
sonry from sliding down, until the completion of the arch secures it 
from doing so. The hor courses between the face m i, aud the line 
o e , will aid somewhat in this respect. 

This method should be applied to all very large arches whose 
rise is one third, or less, of the span. As before remarked, it 
is not actually necessary in arches not exceeding about 50 ft span, 

and not flatter than -t- of the span. Indeed, if the earth filling can 
be deposited before the centers are removed, these limits may be con¬ 
siderably extended without danger. Still, since a certain degree of 
inclination is attended with very little trouble or expense, we would 
recommeud for even such arches, a process somewhat like the follow¬ 
ing: From half the span take the rise. Div the rein by 3. Make o f. 
Fig 5, equal to the quot. Draw t n, aud o m, hor. Div the angle 
8 o m into two equal parts, by the line o a. Iucliue the masoury so 
as to be parallel to o a, as far down as f n. The inclined courses 
may extend out to the face o t, or not, at pleasure. 


i Ji cl 



J^t|7 


Rem. 1. To find the length ( ab , Fig 7) 
from face to face of a culvert. From 

the height ht of the embkt, take the above ground height n a 
of the culvert; the reiu will be the height h o of the embkt 
above the culvert. Then the rcqd length a 6 is plainly equal 
to the top width id of the embkt, added to the two disfs as, 
eft, which correspond to its steepness of side-slopes. Thus, if 
the side-slope is, as usual, to 1, then as and c 6 will each be 
equal to 1U, times o ft; or the two together will be 3 times o ft. 
So that if the width i d is 14 ft, and A o 5 ft, the length a ft w ill be 


14 + (5 X 8) = 14 + 15 =r 29 ft. 



Art. 7. The following tables, 3. 4, and 5. of quantities, will 

be found useful for expediting preliminary estimates; for which purpose chiefly they 
are intended ; hence no pains have been taken to make them scrupulously correct, 
but rather a little in excess of the truth. The first column of Table 3 contains the 

total vert height oc , Fig 6, from the 
crown n of a semicircular arch, to 
the foundation or base g m of its 
abut. The other columns give ap¬ 
proximately the number of cub yds 
contained in each running foot, or 
foot in length of the culvert or 
bridge, measured from end to end 
(face to face) of the arch proper; 
and including only the arch and its 
abuts, as shown in Fig 1; or in the 
half section nprngy in Fig 6; in¬ 
cluding footings to the abuts, but 
omitting the wing-walls (tew), and 
the spandrel-walls (*), Figs 6 and 
2%. At the foot of each column is the approximate content in cub yds of the two 
spandrel-walls by themselves; one over each face of the arch. 


These spandrel-walls are calculated on the supposition that their thickness at base, at their junc¬ 
tion with the wing-walls, where their height is greatest, is equal to of their height at that point: 
exo“pt where that proportion gives a less thickness at top than 2^ ft; and that thev extend 2 ft (o a) 
above the top o of the aroh. At the top of the arch, they are all supposed to be 2t<j ft thick at top; 
that beiug assumed to be about the least thiokness admissible in a rubble wall in such a position. 
Both the back and the face are supposed to be vert. The contents of these spandrel-walls will vary 
somewhat, however, even in the same span, with the height of the abut and the arrangement of the 
wings. They, however, constitute so small a proportion of the entire oontents given in Table 5, that 
this consideration may he neglected in preliminary estimates. They are so firmly bonded into the 
masonry of the wings at their highest points, and so strongly connected by mortar with the backing 
of the aroh at their bases, that they require uo greater thiokness however high the omb taay be. 

The contents of the four wins;-walls, of which tijw h , Fig 6, is one 

will be found in a table (No. 4) immediately following that for the body of the cul¬ 
vert. We have also added a table (No. 5) for complete semicircular culverts of 
various lengths, including their spandrel and wing walls. 




































STONE BRIDGES, 


703 


Rem. 1. Although the thickness of wing-walls increases in all parts with their 
height, they are not made to show thicker at nj than at ti, Fig 6; hut (as seen in the 
fig) are oftsetted at their back tn, a little below their slanting upper surf ij t so as 
to give a uniform width for the steps or flagstones, as the case may be, with which 
they are covered. In the fig the covering is supposed to be of flagstones; but steps 
are preferable, being less liable to derangement. To prevent the flagstones from 
sliding down the inclined planey the lower stone i should be deep and large, and 
laid with a hor bed. The flags are sometimes cramped together with iron, and bolted 
dow n to the wall. Steps require nothing of that kind, as seen at s , Fig 11. 

Rem. 2. The tables show the inexpediency of too vnnch con¬ 
tracting' the width of waler-way, with a view to economy, by adopting 
a small span of arch, when a culvert of greater span can be made, of the same total 
height. 

For the wings must be the same, whether the span be great or small, provided the total height is 
the same in both cases ; and since the wings constitute a large proportion of the entire quantity of 
masonry, in culverts of ordinary length, the span itself, within moderate limits, has comparatively 
little effect upon it. Thus, the total masonry in a semicircular culvert of 3 ft span, 8 ft total height, 
and 60 ft long between the faces of the arch, is, by Table 5, 151cub yds ; while that of a 5 ft span, 
of the same height and length, is 152.4. A semicircular bridge of 25 ft span, 24 ft total height, and 
40 ft between the faces of the arch, contains 1031 cub yds; while one of 35 ft span, of the same height 
and length, contains 1134 yds; so that in this case we may add nearly 50 per cent to the water-way, 

by increasing the masonry of the bridge but -jqjth part. 

Rem. 3. Partly for the same reason, and partly because the culverts for a 
double-track road are not twice as long as those for a single- 
track one, the quantity of culvert masonry for the former will not average more 
than about from % to % part more than that for the latter; so that it frequently 
becomes expedient to finish, the culverts at once to the full length required for a 
double track, although the embkts may at first be made wide enough for only a 
single one, with the intention of inci'easing them at a future time for a double one. 

Thus, the average size of culverts for a single track may be roughly taken at 6 ft span, 30 ft long 
from face to face, and 10 ft total height; and such a one contains, by Table 5, 140 cub yds. For a 
double track, it would require to be about 12 feet longer; and we see by Table 3 that this will add 
2.67 X 12 32 cub yds; making a total of 172 yds instead of 140; thus adding rather less than ^ 

part. When the culverts are under very high embkts, and consequently much longer, the addition 
for a double track becomes comparatively quite trifling. 

Table 3, of approximate numbers of cub yds of masonry 
per foot run, contained in the arches and abutments only, as 

shown in Fig 1 (omitting wings, and the spandrel-walls over the faces of the arches) 
of semicircular culverts and bridges, of from 2 to 50 ft span, and of different total 
heights, ht, Fig 1, oroc, Fig 6. It w ill be seen that in many cases, a bridge of larger 
span contains less masonry than one of smaller span, when their total heights are the 
same. There is a liberal allowance for footings or offsets at the bases of the abuts. 


TABhE 3. (Original.) 


Total 

j Span 

Span 

Span 

Span 

Span 

Span 

Span 

Span 

Span 

Height. 

2 ft. 

3 ft. 

4 ft. 

5 ft. 

6 ft. 

8 ft. 

10 ft. 

12 ft. 

15 ft. 

Feet. 

Cub. y. 
42 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 



63 

67 







4 

.79 

.83 

.87 

.92 

.97 





5 

.99 

1.04 

1.08 

1.15 

1.21 





6 

1.28 

1.28 

1.28 

1.37 

1.46 

1.58 

1.69 



7 

1.62 

1.59 

1.55 

1.64 

1.72 

1 85 

1.97 

2.12 


8 

2.01 

1.96 

1.91 

1.95 

1.99 

2.13 

2.26 

2.38 


9 

2.45 

2.38 

2.31 

2.29 

2.27 

2.42 

2.56 

2.65 

3.02 

10 

2.94 

2.85 

2.76 

2.72 

2.67 

2.77 

2.87 

2.93 

3.34 

11 


3.38 

3.26 

3.19 

3.12 

3.16 

3.19 

3.23 

3.67 

12 


3.98 

3.82 

3.72 

3.62 

3.57 

3.52 

3.55 

4.01 

13 



4.42 

4.29 

4.17 

4.10 

4.02 

3.86 

4.36 

14 



5.08 

4.90 

4.77 

4.67 

4.57 

4.41 

4.72 

15 




5.57 

5.42 

5.30 

5.17 

5-01 

5.09 

lti 




6 30 

6.12 

5.97 

5.82 

5.56 

5.69 

] 7 





6.87 

6.70 

6 52 

6.26 

6.34 

18 

1 



7.69 

7.48 

7.27 

7.01 

7.04 

19 






8.32 

8.07 

7.71 

7.69 

20 






9.20 

8.92 

8.56 

8.49 







9.82 

9.46 

9.34 








10.8 

10.3 

10.2 









11.3 

11.1 









12.3 

12.1 










13.2 










14.2 

26 , 




_ 


— 

— 




Contents of the two spandrel-avails, over the two ends of the arch, in cub yds. 


| 2.9 | 3.7 | 44 | 5.2 | 5.8 I 7.9 | 9.8 | 12. | 16. 
































































704 


STONE BRIDGES. 


TABLE 3. (Continued.) 


Total 

Spau 

Span 

Total 

Spau 


Total 

Span 

Height. 

20 ft. 

25 ft. 

Height. 

35 ft. 


Height. 

50 ft. 

Feet. 

Cub. y. 

Cub. y. 

Feet. 

Cub. y. 


Feet. 

Cub. y. 

12 

4 60 


20 

10.5 


27 

18.0 

13 

4.98 


21 

11.0 


28 

18.7 

14 

5.37 

6.10 

22 

11.6 


29 

19.4 

15 

5.77 

6.41 

23 

12.2 


30 

20.1 

16 

6.18 

6.76 

24 

12.7 


31 

20.9 

17 

6.60 

7.16 

25 

13.3 


32 

21.6 

18 

7.03 

7.61 

26 

13.8 


33 

22.4 

19 

7.47 

8.10 

27 

14.5 


34 

23.1 

20 

8.12 

8.60 

28 

15.1 


35 

23.9 

21 

8.82 

9.02 

29 

15.7 


36 

24.7 

22 

9.57 

9.72 

30 

16.3 


37 

25.5 

23 

10.4 

10.4 

31 

17.0 


38 

26.3 

24 

113 

11.2 

32 

18.1 


39 

27.1 

25 

12.2 

12.1 

33 

19.2 


40 

28.0 

26 

13.1 

13.0 

34 

20.4 


41 

28.8 

27 

14.1 

14.0 

35 

21.7 


42 

30.0 

28 

15.2 

15.0 

36 

23.0 


43 

31.5 

29 

i6.3 

16.1 

37 

24 3 


44 

33.0 

30 

17.4 

17.2 

38 

25.7 


45 

34.6 

31 

18.6 

18.4 

39 

27.2 


46 

36.3 

32 

19.9 

19.6 

40 

28.7 


47 

38.1 

33 

21.2 

20.9 

41 

30.2 


48 

39.8 

14 

22.6 

22.2 

42 

31.8 


49 

41.6 

&:> 

24.0 

23.6 

43 

33.5 


50 

43.6 

36 

25.4 

25.0 

44 

35.2 


51 

45.5 

37 

26.9 

26.5 

45 

36.9 


52 

47.4 

38 

28 5 

28.0 

46 

38.7 


53 

49.4 

39 

30.1 

29.5 

47 

40.6 


54 

51.6 

40 

31.7 

31.2 

48 

42.5 


55 

53.7 

41 


32.8 

49 

44.4 


56 

55.9 

42 


34.5 

50 

46.4 


57 

58.1 

43 


36.3 




58 

60.4 

44 


38.1 




59 

62.7 

45 


40.0 




60 

65.1 


Contents of the two spandrel-walls, over th* two ends of the arch, In cub yds. 


_28._I 4'->. II_|_8 5. II | 195. _ 

Art. 8 . The following; table of contents of wing-walls, or wings, will, 
like the preceding one, be useful in making preliminary estimates. The wings 
no, no, shown in plan at Fig 8, are supposed to form an angle aoc, of 120°, with the 
face, or end o o of the culvert. Their outer or small ends n n, are all assumed to he of 
the dimensions shown on a larger scale at E. Thickness at base at every part equal 
to of the height of the wall at said part; except when that proportion becomes 
too small to allow the width or thickness at top to be 2.5 ft; in which case it is en¬ 
larged at such parts sufficiently for that purpose. See Remark 2 . This happens only 



when the height mm, Fig E, of the wing, becomes less than 9 ft. Batter of face Vt 
ins to a ft; or 1 in 8 . Back vert; but olfsetted, if necessary, for a short dist below 
the top so as to give a uniform showing top thickness of 2 ]/ 2 ft. The masonry is 
supposed to be good well-scabbled mortar rubble. The height given in the first 
u> umn is the greatest one; or that at o o, (or wj. Fig 6 ,) where the wing joins the 
“ e , of ‘ he c " 1 )' ert - . In the table no allowance is made for footings (offsets or steps) 
at the base of the wings; as these are frequently omitted in wings on good founda- 
















































oTOiNE BRIDGES 


TUT) 


oii8. In taking out quantities from the table, bear in mind that the height of the 
ings is usually a little greater than that of the culvert itself. 

Table 4, of approximate contents, in cub yds, of the four 
wing-walls of a culvert, or bridge. (Original.) 

The heights are taken where greatest; as at j w, Fig 6 


Height 

of 

wing. 

Length 

of 

one wing. 

Cub. yds. 
in 

4 wings. 


Height 

of 

wing. 

Length 

ot 

one wing. 

Cub. yds. 
in 

4 wings. 

Feet. 

6 

Feet. 

1.73 

4.04 


Feet. 

30 

Feet. 

43.3 

818 

7 

3.46 

8.85 


32 

46.8 

997 

8 

5.20 

14.6 


34 

50.3 

1192 

§ 

6.93 

21.5 


36 

53.7 

1414 

10 

8.66 

30.2 


38 

57.2 

1661 

11 

10.4 ■ 

40.9 


40 

60 7 

1928 

12 

12.1 

53.7 


42 

64.2 

2220 

14 

15.6 

85.2 


44 

67.6 

2552 

16 

19.1 

128 


46 

71.1 

2912 

18 

22.5 

183 


48 

74.6 

3306 

20 

26.0 

247 


50 

78.0 

3741 

22 

29.5 

329 


55 

86.7 

4942 

24 

32.9 

426 


60 

95.3 

6404 

26 

36.4 

541 


65 

104 

8131 

28 

39.8 

672 


70 

113 

10155 


To reduce cub yds to perches of 25 cub ft, mult by 1.080. 
To reduce perches to cub yds, mult by .926, or div by 1.08. 


The contents for heights intermediate of those in the table may be found approximately by simple 
roportion. 

Rem. 1. It is not recommended to actually prolong all wings until their dimen- 
oii8 become as small as shown at E, in Fig 8. In large ones it will generally be 
lore economical to increase their end height m m, a few feet. The contents, liow- 
ver, may be readily found by the table in that case also. Thus suppose the height 
f the wings at one end to be 30 ft, and at the other end 8 ft; we have only to sub- 
ract the tabular content for 8 ft high, from that for 30 ft high. Thus, 818 —14.6 = 
,03.4 cub yds required content. 

Rem. 2. It might be supposed that inasmuch as the wings of arches often have to 
nstain the pressure from embankments reaching far above their tops, they 'hould, 
ke ordinary retaining-walls, be made much thicker in that case. But the c/ct that 
hey derive great additional stability from being united at their high enas to the 
ody of the bridge or culvert, renders such increase unnecessary when proportioned 
iy our rule; no matter how far the earth may extend above them; as shown by 
bundant experience. 

Relying upon this aid. we may indeed, when the earth does not extend above the top, reduce the 
rase at o to one third of the ht, as shown at of; and by dotted line t s. Experience shows that we 
say also do the same even when the earth reaches to a great height above the top; provided that 
he wings, instead of being splayed or flared out, as at o n, o n, merely form straight prolongations 
f the abutments of the arch, as "shown by the dotted lines at o g w. In this case the pressure of the 
arlh against the wiugs is less than when they are splayed. We have known the thickness at o 
o be reduced in such cases to less than one-third the height, when the wings were 15 ft high, aud 
he height of the embankment above their tops 16 feet in one case, and 56 ft in another. In another 
istance, similar wings 25J^ ft high, and with 29 ft of embankment above their top, had their bases 
t o rather less than of the height. In all these cases, the uniform thickness at top was 2.5 feet; 
acks vertical. We mention them because this particular subject does not seem to be reducible to 
ny practical rule. The last wall appears to us to be too thin ; especially if the earth is not deposited 
i layers; and after allowing the mortar full time to set. The labor, however, required in compnet- 
ig the earth carefully in layers, may cost more than is thereby saved in the masonry. The young 
ractitioner must’bear this in mind when he wishes to economize masonry by such means: and also 
hat the thin wall may bulge, or fail entirely, if the earth backing is deposited while the mortar is 
mperfectly set. 




















nr\ * 


i uo 


STONE BRIDGES, 


Table 5. Approximate contents in cubic yards, of coi 
plete semicircular culverts and bridges of from 2 to 50 fe 
span; including the 2 spandrel walls; and the 4 wings; all proportioned by l 
foregoing directions; and taken from the two preceding tables. The height in I 
second column, is from the top of the keystone to the bottom of the foundation. 'J 
wings are calculated as being 2 ft higher than this, including the thickness of t 
coping. The wings are frequently carried only to the height of the top of the an 
thus saving a good deal of masonry. Table 4, of wings alone, will serve to make i 
proper deduction in this case. 

The several lengths are from end to end, or from face to face, of the arch pro) 
The contents for intermediate lengths maybe found exactly; and those for int 
mediate heights, quite approximately, by simple proportion. In this table, as 
No. 3, it will be observed that when the heights are the same, in both cases, a lar; 
span frequently contains less masonry than a smaller one. A semicircular culv 
or bridge contains less masonry than a flatter one, when the total height is the sa 
in both cases; therefore, the first is the most economical as regards cost; but it d 
not afford as much area of water-way ; or width of headway. 

(Original.) 


Span. 

Height. 

5 J 

•C . 

“o 

2* 

Length. 

30 Ft. 

Length. 

40 Ft. 

Length. 

60 Ft. 

Length. 

80 Ft. 

Length. 

100 Ft. 

Length. 

120 Ft. 

Length. 

140 Ft. 

•*5 

£§ 

Length. 

180 Ft. 

Length. 

200 Ft. 

Ft. 

2 

Ft. 

5 

6 

7 

8 
10 

Cub Y. 

27 

37 

49 

63 

101 

Cub Y. 

32 

43 

57 

73 

116 

Cub Y. 

42 

56 

73 

93 

145 

Cub. Y. 

52 

69 

89 

113 

175 

Cub. Y. 

72 

94 

122 

153 

234 

Cub. Y. 

92 

120 

154 

193 

291 

Cub. Y. 

112 

146 

187 

233 

351 

Cub. Y. 

132 

171 

219 

273 

410 

Cub Y. 

152 

197 

251 

313 

469 

Cub.Y 

172 

222 

284 

353 

527 

Cub.Y. 

192 

248 

316 

393 

586 

Cub.Y. 

212 

274 

349 

433 

645 

3 

5 

6 

7 

8 

10 

12 

28 

38 

49 

63 

101 

149 

34 

44 

57 

73 

115 

169 

44 

57 

73 

93 

143 

208 

54 

70 

89 

112 

172 

248 

75 

95 

121 

152 

229 

328 

96 

121 

153 

191 

286 

407 

117 

146 

184 

230 

343 

487 

138 

172 

216 

269 

400 

567 

158 

198 

247 

308 

457 

646 

179 

223 

280 

348 

514 

726 

200 

249 

312 

387 

571 

806 

22 r 

275 

343 

426 

628 

885 

4 

5 

6 

7 

8 
10 
12 
14 

30 

38 

49 

63 

100 

147 

209 

35 

45 

57 

73 

114 

166 

234 

46 

58 

73 

92 

141 

204 

285 

57 

70 

88 

111 

169 

243 

336 

78 

96 

119 

149 

224 

319 

437 

100 

122 

150 

188 

279 

395 

539 

122 

147 

181 

226 

335 

472 

641 

143 

173 

212 

264 

390 

548 

742 

165 

198 

243 

302 

445 

625 

844 

186 

224 

274 

340 

500 

701 

945 

208 

250 

305 

379 

555 

777 

1047 

229 

275 

336 

417 

611 

854 

1149 

5 

6 

7 

8 
10 
12 
14 

41 

52 

65 

100 

146 

207 

47 

60 

75 

114 

165 

231 

61 

76 

94 

141 

202 

280 

75 

93 

114 

168 

239 

329 

102 

125 

153 

223 

314 

427 

130 

158 

192 

277 

388 

525 

157 

191 

231 

331 

463 

623 

184 

224 

270 

386 

537 

721 

212 

257 

309 

440 

611 

819 

239 

289 

348 

495 

686 

917 

267 

322 

387 

549 

760 

1015 

294 

355 

426 

603 

835 

1113 

6 

7 

8 

10 

12 

14 

16 

53 

66 

100 

146 

206 

281 

62 

76 

113 

164 

219 

311 

79 

96 

140 

200 

277 

373 

96 

116 

167 

236 

325 

434 

131 

156 

220 

308 

420 

556 

165 

196 

274 

381 

516 

679 

200 

236 

327 

453 

611 

801 

234 

276 

380 

526 

706 

923 

268 

316 

434 

598 

802 

1046 

303 

356 

487 

670 

897 

1168 

337 

396 

541 

743 

993 

1291 

372 

436 

594 

815 

1088 

1413 

oo 

7 

8 
10 
12 
14 
16 
18 

57 

70 

101 

147 

206 

281 

367 

67 

81 

118 

165 

230 

310 

405 

85 

102 

145 

200 

276 

370 

480 

104 

124 

173 

236 

323 

430 

554 

141 

166 

228 

308 

416 

549 

704 

178 

209 

284 

379 

510 

669 

854 

215 

25! 

339 

450 

603 

788 

1003 

252 

294 

395 

522 

696 

908 

1153 

289 

337 

450 

593 

790 

1027 

1302 

326 

379 

505 

664 

883 

1146 

1452 

363 

422 

56) 

736 

977 

1,266 

1602 

400 

464 

616 

807 

1070 

1385 

1751 

10 

8 

10 

12 

14 

16 

18 

74 

107 

148 

207 

280 

366 

110 

151 

206 

279 

364 

470 

85 

121 

166 

229 

309 

402 

108 

150 

201 

275 

368 

475 

131 

179 

236 

321 

426 

548 

176 

236 

306 

412 

542 

693 

221 
. 294 
377 
50.4 
, 659 
839 

266 

351 

447 

595 

775 

984 

311 

408 

518 

686 

891 

1129 

357 

466 

588 

778 

1008 

1275 

402 

523 

658 

869 

1124 

1420 

447 

581 

729 

961 

1241 

1565 

492 

638 

799 

1052 

1357 

1711 

12 

10 

12 

14 

16 

18 

20 

125 

168 

228 

306 

399 

512 

154 

204 

272 

362 

469 

598 

183 

239 

317 

418 

540 

684 

242 

310 

405 

529 

680 

855 

301 

381 

493 

640 

820 

1026 

359 

452 

581 

751 

960 

1197 

418 

523 

669 

862 

1100 

1368 

476 

594 

758 

974 

1241 

1540 

535 

665 

846 

1085 

1381 

1711 

594 

736 

934 

1196 

1521 

1882 

652 

807 

1022 

1307 

1661 

2053 
























































































































STONE BRIDGES, 


707 


fe< 

Fli¬ 

nt! 

I 

ft! 

ire 

et, 

up 

■ « 

lift 

sat 

do 


! 


Table 5 — (Continued.) (Original.) 


Span. 

Height. 

Length. 

15 Ft. 

Length. 

20 Ft. 

Leneth. 

30 Ft. 

•5 - 
tic* 

° © 
a* 

j Length. 

60 Ft. 

1 Length. 

80 Ft. 

! 

Length. 

100 Ft. 

Length. 

120 Ft. 

-C 

s o 
o 

e-s r “* 

tl* 

e o 
o SO 

Length. 

180 Ft. 

.c 

s§ 

Ft. 

15 

Ft. 

12 

U 

16 

IS 

20 

22 

Cub.Y. 

162 

215 

285 

360 

473 

595 

Cub. Y. 

182 

239 

313 

404 

515 

646 

Cub. Y. 

222 

286 

370 

474 

600 

748 

Cub.Y. 

262 

333 

427 

545 

685 

850 

Cub.Y. 

342 

427 

541 

686 

855 

1054 

Cub.Y. 

422 

522 

654 

826 

1024 

1258 

Cub.Y. 

502 

616 

768 

967 

1194 

1462 

Cub.Y. 

583 

711 

882 

1108 

1364 

1666 

Cub.Y. 

663 

805 

996 

1249 

1534 

1870 

Cub.Y. 

743 

899 

1110 

1390 

1704 

2074 

Cub.Y. 

823 

994 

1223 

1530 

1873 

2278 

Cub.Y. 

903 

1088 

1337 

1671 

2043 

2483 


14 

237 

264 

317 

371 

478 

586 

693 

801 

908 

1015 

1123 

1230 


16 

304 

335 

397 

458 

582 

706 

82!# 

953 

1076 

1200 

1324 

1447 

20 

18 

381 

416 

486 

556 

697 

838 

978 

1119 

1259 

1400 

1541 

1681 

20 

479 

520 

601 

682 

844 

1007 

1169 

1332 

1494 

1656 

1819 

1981 


22 

598 

646 

741 

837 

1028 

1220 

1411 

1603 

1794 

1985 

2177 

2368 


24 

739 

795 

908 

1021 

1247 

1473 

1699 

1925 

2151 

2377 

2603 

2829 


16 

327 

360 

428 

496 

631 

766 

901 

1036 

1172 

1307 

1442 

1577 


18 

403 

441 

517 

594 

746 

898 

1050 

1202 

1355 

1507 

1659 

1811 

o r. 

20 

500 

543 

629 

715 

887 

1059 

1231 

1403 

1575 

1747 

1919 

2091 

25 

22 

614 

66 ! 

760 

857 

1051 

1246 

1440 

1635 

1829 

2023 

2218 

2412 


24 

751 

807 

919 

1031 

1255 

1479 

1703 

1927 

2151 

2375 

2599 

2823 


26 

909 

974 

1104 

1234 

1494 

1754 

2014 

2274 

2534 

2794 

3054 

3314 


28 

1085 

1160 

1310 

1460 

1760 

2060 

2360 

2660 

2960 

3260 

3560 

3860 


22 

685 

743 

859 

975 

1207 

1439 

1671 

1903 

2135 

2367 

2599 

2831 


24 

817 

880 

1007 

1134 

1388 

1642 

1896 

2150 

2404 

2658 

2912 

3166 

35 

26 

969 

1033 

1181 

1309 

1585 

1861 

2137 

2413 

2689 

2965 

3241 

3517 

28 

1130 

1205 

1356 

1507 

1809 

2111 

2413 

2715 

3017 

3319 

3621 

3923 


30 

1327 

1408 

1571 

1734 

2060 

2386 

2712 

3038 

3364 

3690 

4016 

4342 


32 

1549 

1639 

1820 

2001 

2363 

2725 

3087 

3449 

3811 

4173 

4535 

4897 


35 

1946 

2054 

2271 

2488 

2922 

3356 

3790 

4224 

4658 

5092 

5526 

5960 


30 

1494 

1594 

1795 

1996 

2398 

2800 

3202 

3604 

4006 

4408 

4810 

5212 

l 

32 

1711 

1819 

2035 

2251 

2683 

3115 

3547 

3979 

4411 

4843 

5275 

5707 


34 

1956 

2071 

2302 

2533 

2995 

3457 

3919 

4381 

4843 

5305 

5767 

6229 

50 

36 

2228 

2350 

2597 

2844 

3:338 

3832 

4326 

4820 

5314 

5808 

6302 

6796 

38 

2519 

2650 

2913 

3176 

3702 

4228 

4754 

5280 

5806 

6332 

6858 

7384 


40 

2835 

2975 

3255 

3535 

4095 

4655 

5215 

5775 

6335 

6895 

7455 

8015 


42 

3197 

3347 

3647 

3947 

4547 

5147 

5747 

6347 

6947 

7547 

8147 

8747 


45 

3818 

3991 

4337 

4683 

5375 

6067 

6759 

7451 

8143 

8835 

9527 

10219 


50 

5063 

5281 

5717 

6153 

7025 

7897 

8769 

9641 

10513 

11385 

12257 

13129 


4" 

9 


I 




i 

I 


Art. 9. Especial pains should he taken to secure an nnyielding* foun¬ 
iat ion for culverts an<l drains under high eiiibkts; otherwise 
lie superincumbent weight, especially under the middle of the embkt, may squeeze 
hem into the soil below, if soft or marshy; and thus diminish the area of water¬ 
way, or at least cause an ugly settlement at the midlength of the culvert. Also, in 
oft ground, the embkt may press the side walls closer together, narrowing the 
hannel. This may be prevented by an inverted arch, or a bed of masonry, between 
he walls. A stratum from 3 to 6 ft thick, of gravel, sand, or stone broken to turn¬ 
like size, will generally give a sufficient foundation for culverts in treacherous 
narshy ground ; or quicksand, with but a moderate height of embkt. It should ex¬ 
end a few feet beyond the masonry in every direction, and should be rammed; the 
and or gravel being thoroughly wet, if possible, to assist the consolidation. Piling 
vKl sometimes be necessary. If the masonry is built upon timber platforms, or a 
mooth surface of rock, care must be taken to prevent it from sliding, from the pres 
if the earth behind it. This same pres may even overthrow the piles, if they are 
tot properly secured against it. 


Art lO. Drains. 

trains of the dimen- 
ions in Fig 11, con- 
ain 1 perch, of 2b 
ub ft; or .926 of a 
ub yd, per ft run. 

They are frequently 
tilt of dry scabbled 
ibble. and paved with 
lawls. When there is 
iueh wash through 
iem. with a consider- 
ble slope, it is better to 
mtinue the foundation 



49 















































































































708 


STONE BRIDGES, 


.olid clear across. This is often done without W 

length, may he in- 

SSSrSSSS L :-.-,s;f.= 

mmmm-wjmwmtf* 

Art 11 The drainage of the roadways of stone bridges of severs 
arches is generally effected by means of open gutters, which descend slightly fioi 
the crowns of the arches, each way, until they reach to near the ends of the re 

pnective spans. 

caaisfAt sjswsKtt i a.—. 

the water to fall freely through the air from that height. 

TaHl<> 6 of annroxiinate contents, in cnh yds, of a soli< 
nicr of masonry, 6 ft by 22 ft on top; and battering 1 inch to a ft on each o 

Ps t faces The contents of masonry of such forms must be calculated by the prismoldal fotmul. 

ro S r s trengtheuing 

bauer U generally reduced to * inch or less to a foot. Hollow piers require good well-bedded m 
soury. (Original.) 


Ht. 

Ft. 

Lgth 

at 

base. . 

Bdth f 
at 

b tse. J 

lubic 
tardt I 

lit. 

Ft. 

Lgth 

at 

base. 

Bdth 

Mt 

base. 

Cubic 

yards 

Ht. 

Ft. 

Lgth 

at 

base. 

Bdth 

at 

base. 

Cub 

yarc 

(5 

23. | 

7. 

32.5 

52 

30 67 

14.67 

537 

128 

43 33 

27.33 

275 

7 

.17 

.17 

38.6 

54 

31. 

15. 

570 

130 

.67 

-67 

284 

g 

35 

.33 

44.9 

56 

.33 

.33 

f05 

132 

44. 

28. 

294 

9 

10 

23.5 

•>7 

7.5 

.f>7 

51.3 

58. 

58 

60 

.67 

32. 

-.67 

16. 

641 

679 

134 

136 

.33 

.67 

.33 

.67 

303 

312 

11 

.83 

.83 

G4.8 I 

62 

.33 

.33 

717 

138 

45. 

29. 

322 

12 

13 

24. 

.17 

8 . 

.17 

71.7 

79. 

64 

66 

.67 

33. 

.67 

17. 

757 

798 

140 

142 

.33 

.67 

.33 

.67 

00*2 

34: 

ll 

33 

.33 

86.4 

68 

.33 

.33 

810 

144 

46. 

30. 

3aS 

15 

24.5 

8.5 

94. 

70 

.67 

.67 

8S4 

146 

.33 

.33 

36S 

10 

.07 

.67 

102 

72 

34. 

18. 

928 

148 

.67 

.67 

£7: 

17 

.83 

.83 

110 

74 

.33 

.33 

973 

150 

47. 

31. 

38i 

18 

25. 

9. 

118 

76 

.67 

.67 

1021 

152 

.33 

.33 

39 

19 

.17 

.17 

127 

78 

35. 

19. 

1070 

154 

.67 

.67 

40. 

20 

.33 

.33 

135 

80 

.33 

.33 

1120 

156 

48. 

32. 

41< 

21 

25.5 

9.5 

144 I 

82 

.67 

.67 

1171 

158 

.33 

.33 

42! 

44< 

22 

.67 

.67 

153 

84 

36. 

20 . 

1224 

160 

.67 

.67 

25 

.83 

.83 

165 

86 

.33 

.33 

1278 

162 

49. 

33. 

4;>* 

24 

26. 

10 . 

172 

88 

.67 

.67 

1334 

It 4 

.33 

.33 

46 

25 

.17 

.17 

182 

| 90 

37. 

21 . 

1392 

1 G 6 

.67 

.61 

47 

20 

.33 

.33 

1*2 

9 i 

.33 

.33 

1 Ini 

168 

50. 

34. 

48 

27 

26.5 

10 5 

202 

94 

.67 

.67 

1510 

170 

.33 

.33 

50 

28 

.67 

.67 

212 

96 

38. 

22 . 

1 ’ 69 

172 

.67 

.67 

51 

29 

.83 

.83 

221 

98 

.33 

.33 

1631 

174 

51. 

35. 

5*2 

30 

27. 

11 . 

234 

loo 

.67 

.67 

1695 

176 

.33 

.33 

54 

31 

.17 

.17 

245 

102 

39. 

23. 

1761 

178 

.67 

.67 

65 

32 

.33 

.33 

256 

104 

.33 

.33 

1829 

180 

52. 

36. 

5t 

3:3 

27.5 

11.5 

268 

106 

.67 

.67 

1899 

182 

.33 

.33 


34 

.67 

.67 

280 

108 

40. 

24. 

1968 

184 

.67 

.67 


35 

.83 

.83 

292 

110 

.33 

.33 

2041 

186 

53. 

37. 

61 

36 

28. 

12 . 

304 

112 

.67 

.67 

2115 

188 

.33 

.33 

6 ‘. 

38 

.33 

.33 

329 

H4 

41. 

25. 

2191 

190 

.67 

.67 

6 

40 

.67 

.67 

356 

116 

.33 

.33 

2269 

192 

54. 

38. 

6 

42 

29. 

13. 

383 

118 

.67 

.67 

2346 

194 

.33 

.33 

6 ' 

- 44 

- - .33 

.33 

411 

120 

42. 

26. 

2424 

196 

.67 

.67 

6 

40 

.67 

.67 

441 

122 

.33 

.33 

2504 

198 

55. 

39. 

7' 

48 

30. 

14. 

472 

124 

.67 

.67 

2587 

200 

.33 

.83 

7 

50 

.33 

.33 

504 

126 

4, 

27. 

2672 

202 

.67 

.67 

7 





































































BRICK ARCHES. 


709 


. A**5* Brick Arches. Since even good brick fit for large arches lias 
ai less crushing strength than good granite or limestone, and is inferior even to 
S +i n * ? ne ’ w hile its weight does not differ very materially from stone, it is 
plain that it cannot be used in arches of as great span as stone can. Some of 

■fh cient^f d ' V f f U1 f V W hl f o tia , ve stood for inan Y years, have a theoretical co- 
dhcient of safety of but about 3; whereas the authorities direct us not to trust even 

“ 0r v tban one-twentieth of its crushing load. This last, however, ap- 
H* 6 wn „ ter . to h pone of those hasty assumptions which, when once ad- 
J!!hr! U }° P rofe ssi o nal books, are difficult to be got rid of. It is his opinion that 
eement, and proper care in striking the centers, one-tenth of the ulti- 
n‘l te !:r. engtb 1S sufficiently secure against even the abnormal strains caused by 
ne settling at crown, and rising at the haunches when the centers are struck. It 
s useless to attempt to fix limits of safety for bad materials poorly put together. 

Kem. 1. The common practice of building brick arches in a series of con¬ 
centric ring’s, as at acee, big 12, with no other bond between them than 

that afforded l»y the mortar, is censured by 
authorities, on the ground that the line of 
pressure in passing from the extrados to 
the intrados tends to separate the rings, 
and thus weaken the arch by, as it were, 
splitting it longitudinally. The reason 
for using these rings, instead of making 
the radial joints continuous throughout 
the depth m n of the arch, as at b, is to 
avoid the thick mortar-joints at the back of 
the arch, and shown in the Fig. If the 
center of an arch built as at b be struck 
too soon, the soft mortar in these thick 
to cause great settlement at the crown, 



pints will be so much compressed as_ 1V , fll ,, 11C tlUYVIi 

“browing the arch out of shape, and creating such inequality of pressure as 
Jnght even lead to its fall, especially if flat. As a compromise between rings 
ijnd continuous joints, they are sometimes employed together, so as to get. rid of 
of the long radial joints ; and at the same time to break at intervals 
ie continuity of the rings. Thus in Fig 12. which is supposed to be brick-and- 
f half deep, beginning at the abutment a, we may lay half-brick rings as far as 
qiy to e o e; then cutting away the brick o to the line e e, we may lay from 
\e tomna block of bricks with continuous radial joints, the same as at b; and 
jien start again with three rings; and so on alternately. A still better, but 
i] ore expensive, mode would he to fill e e, m n with a regular cut-stone voussoir. 

I The proper intervals for changing from rings to blocks will depend upon the 
lumber of the rings and the depth c a of the arch ; reference being also had to 
Inducing the amount of brick cutting as much as possible, 
j These points can be best decided on from a drawing of a portion of the arch 
n a scale of 3 or 4 ins to a foot. Generally the rings are made only half-brick, or 
>out 4 to 4.5 ins thick, as at a c; and in Brunei’s Maidenhead viaduct of two ellip- 
“c brick arches of 128 ft span, and 24.25 ft rise; the boldest brick arches yet at- 
mpted; but which have been estimated to have a co-efficient of safety of but 
ires against crushing at the crown. 

So many others of from 70 to 100 ft span have been successfully built entirely in 
tigs of either half or whole brick thick, as to justify us in attaching but little weight 
the above theoretical objection, provided first class cement Ue used, and time 
lowed it to become nearly or quite as hard as the bricks themselves, before 
riking the centers. Under such circumstances we should not object to a series 
rings even 1.5 bricks thick, laid alternately header and stretcher, as at b. 

If tile bricks were voussoir-shnped, that is, a little thicker at one 
d than the other, then rings a whole-brick thick could be used without any in¬ 
ease in thickness of mortar-joint at the back of each ring. Still with more 
an one ring, the radial joints would not be continuous, as at b, but broken as at 
. Such bricks however would be more expensive to make; and moreover, in 
der fully to answer the intended purpose, they would have to be made of many 
tterns, so as to conform to the many radii used in arches; and even to the 
dii of the different rings, when the depth of the arch required several of them. 


See foot-note, p 671. 








710 


BRICK ARCHES. 


Item. 2. Wet the hriehs before laying. See last paragraph of p 6*0. 

Rem. 3. When the ends or faces of a brick arch are to be finished with Cllt- 
stone voussoirs, these had better not be iuserted nntil some time after, tlie. 
completion of the brickwork, the hardening of the mortar and a partial easing 
of the centers; lest they be cracked or spawled by the unequal settlements ot then 
selves and the bricks. 


jlem. Brich arches, from their great number of joints are apt to settle 
much more than cut stone ones when the centers are removed, and thereby tc 
derange the shape of the arch, and at times, without due care, even to endangei 
its safety, especially if it be large and flat. When the span exceeds about do to .k 
ft and particularly if flat, use only brick of superior quality in good cement 
mortar With even best materials and work we advise the young engineer n<r 
to attempt brick arches for railroad bridges of greater spans than about the iol 
lowing. Considerably larger ones than some of them have been built, and hav< 
stood; but their coefs of safety are not in all cases sat isfactory. In this table tn» 
rise is in parts of the span. 


!' 


R. 

S. 

R. 

s. 

R. 

S. 

R. 

s. 

R. 

H. 

.5 

100 

l A 

88 

.225 

68 

% 

50 

.134 

35 

.4 

97 

.29 

82 

1 

K 

60 

.155 

45 

\-y% 

30 

.36 

93 

Va 

75 

.183 

55 

* 7 

40 




On the Filbert Street Extension of the Penns R R, in Phila 

arc four brick arches of 50 ft 1 inch span, and with the very low rise of 7 it. The, 
are 2 ft 6 ins thick, except on their showing faces, where they are but 2 ft. Th 


joints are in common mortar, and about x /± inch thick. These four arches, abou 
200 yards apart, with a large number of others of 20 ft span, form a viaduct. Th 
piers between the short spans are 4 ft 3 ins thick. Those at the ends of the 50-1 


piers between tne snort sp: 
spans, 18 ft 6 ins. The springing lines of all the arches are about 6 to 8 ft above tb 
ground. One of the 60-ft arches settled 3 ins upon prematurely striking th 
centers; but no further settlement has been observed, although the viaduct ha: 
since built (1880) had a very heavy freight and passenger traffic, at from 10 to ‘1 
miles per hour. Roadbed, about 100 ft wide, giving room for 9 or 10 tracks. 

































CENTERS FOR ARCHES. 


711 


CENTERS FOR ARCHES. 


Art. 1. A center is a temporary wooden structure (built lying flat, on a full 
size drawing, on a fixed platform, under cover or not) for supporting an arch 
while it is being built. It consists of a number of trusses or frames,/, ft Fig. 1» 
placed from 1 to 6 ft apart from cen to cen, and covered with a flooring l, l, of 
rough boards or planks, usually laid close, and called the sheeting or lag¬ 
ging, immediately upon which the archstones are laid. In Fig 3, the lag¬ 
ging is not laid close. There is no great economy in placing the frames very 
far apart, on account of the greater required amount of lagging, the thickness 
3f which increases rapidly. For the thickness of lagging see Rem 9, p 719. The 
frames are of many designs. Thus Figs 7 and 9, pp 4, 550, are often used for 
small spans (say 15 to 25 ft), their upper timbers supporting throughout their 
length planks on edge, with their upper edges trimmed to conform to the curve 
}f the arch. Fig 14, p 570, cov- 
jred in the same way, is some¬ 
times used for still longer spans, 
jay 25 to 40 feet; also Fig 28, p 
‘>94; Fig 31, p 595 ; and Fig 35, p 
>98, for still longer ones. 



The centers rest by the ends of 
heir chords, c, upon wooden 
itriking wedges w, Fig 1, 
upported by standards com- 
>osed of posts p, whose tops are 
:onnected by cap-pieces o; 
tnd whose feet rest on string¬ 
ers s; the whole being braced 
'Jliagonally as shown. 

,' p If the ground is very firm, and 
, c he arch light, the standards may 
est on it, with the interposition 
mil adjusting-blocks, n, be- 
ls ow the stringer, to accommodate 

^regularities of the surface of the ground, as in the Fig. These blocks should 
^ >e somewhat double-wedge-shaped, so that by driving them the standard may 
!,j>e raised at any point in case it should settle a little into the ground. But for 
' leavy arches the standards must rest on a much firmer foundation, such as short 
docks of brickwork sunk a few feet into the ground, or some other device 
idapted to the case. Frequently projecting offsets or footings, or at times re- 
esses. are provided in the masonry of the abutments and piers for this express 
Purpose; and with a view to this it is well to design the center at the same time 
a the arch. Knowing the wt of the arch the 
•roper dimensions of the posts may readily be 
bund by table p 459, etc. Up to spans of 50 or 60 
t a single row of posts (one under each end of 
;ach frame) will suffice; but for much larger ones 
wo or three rows, 2 or more feet apart may be- 
ome expedient, as in the lower Fig 2. 

The striking or lowering-wedges 

.efore alluded to are for striking or lowering the 
enter after the completion of the arch. They 
onsist of pairs of wedge-shaped blocks, w w , at A, 

'igs 2, of hard wood, from 1 to 2 ft long, about half 
s wide, and a quarter or more as thick, (sufficient 
o lower the center from say 2 to 6 or more inches, 

.ccording to span and other circumstances,) rest- 
ng on the cap o, of the standard, while the chord 
of the frame rests on them. When the end of a 
rame is supported by two or more posts p, as at B, 
ig 2, instead of upon one, the striking-wedges are 
oinetimes made sis there shown; and where B v 
s one long wedge at right angles to the abutment, 
nd acting as four wedges which may all be low- 
red together by blows against the end B. 

Up to spans of 60 or 80 ft, all the frames may rest on but two wedges like B w, 

















































712 


CENTERS FOR ARCHES. 


, * a -i. troimveroelv across tho entire arch. Then all the 

each so long as to reach tr»nn\er »<- b d „e a r end of Art 9. 
frames can he lowered at one dry wood, the taper 

If we had to consider only the faction of y w 7 thout any danger of their 

of these wedges might be as steep> a* 1 1 t 1 j ^ would then require very 

sliding upon each other of their own actor 1, them , when the center 

moderate blows to start them, oi utmost importance, especially in large 

ssi. isSs* 

IwnTtT^rCS'ut^iblJ •*<*^“^J^rthlnat^iineorSfo, 

lines at equal .lists apart should bedrawn ®>* «»'““this should not ex 
guide for lowering them all to the same ex tent a out ’ an e i K hth of an inch, fo 

ceed in all about half an inch* * n . h ner dav in all for spans over 100 ft 

50 ft spans; or about .1 to .2o of a " int , h .P , • ., u » relies, not only becaus 

Slowness is especially to be reconnnen c creater derangement of shape 

their greater lesThan K average entshing strengtl 

oac^ a f^trite, n iltt»esto > n!vor 1 sandstoue,^nd^thereforas wlie^the strahis^ar 

ss “ zti zsz ^ 

arches, see p < 09. Finland of first class cut stone, span 150 ft, ris 

At Gloucester Bridge, England oi nniiM hours ; an 

35 ft, the centers were entirely ftrnck ^tbin the vtr^n<)ry Englan( 

the crown of the arch descended 10 . A care was taken in easing tl 

of first class cut stone, span -00 ft “ j. .. 5 j This case however w: 

centers that the crown of the arch settled but 2.5 to this favorab 

marked by two or three peculiarities, a ieg 0 f frames supported as usu; 

result. Namely, the center instead of g * . , , a i t hou< r h small, degree < 

by their ends, and of course involving an . pj *^1®’ or settlement, consist* 

.*.n_ .. .. ,-v «. ♦ inn I <n wl 1 



O dp, fA 1 11 O V4 WVl/ 1 . 

essentially of vertical and u 
dined posts or struts, see Fig 
footing on four temporary pie 
of masonry, 7 or 8 feet thick, bu 
in the river, parallel to the abv 
ments, and as long as they. The 
piers supported six frames ( 
rather six series') about 7 ft apa 
ceil to cen,of such struts, tooth 
on cast iron shoes. Fig 3 sh<> 
half of one series. Each frai 
or series consisted of four fan-1 
setsof posts, all in the same v. 


tical plane. The long horizontal pieces seen extending from side to side of t 
arch were bolted to the struts to increase their stiffness; and other pieces lor t 
same purpose united the six series transversely. Here each strut sustains its o; 
share of the weight of the archstones, and transfers it directly to the unj iddi 
foundation of the pier; whereas in the usual trussed centers, the entire load re 
upon the frames, and is finally transferred to the comparatively unstable supp 

Ui The tops^o/the posts of a series varied about from 5 to 8 ft apart cen to c* 
and were^onnected^by acontinuouscurved rib,.-r,o two th.ckne^of 4 



‘tKSLapai SIKof thS ar'chstones, an. On these wedges and 

tendliTg over all six of the frames, were the lagging pieces l, 4.5 ms thick 
This peculiar arrangement of the striking-wedges and 1 
eine has in large spans, great advantages over the usual one ot placing them o 
at the ends of the frames. In the last the entire center and the entire arch 
lowered together, without giving an opportunity to rectify any slight derai 
ments of shape or inequality of bearing ttiat may have occurred in the arch dui 
its construction, This center, designed by Mr. frubshaw, admits of low > 
either the whole equally, or any one part a little more or less than tbe ot > 
He had much experience in large arches, and stated that during the striking 
found that he had an arch under better control, or could humor it better, by k< 
ing the haunches a little down, and the crown a little up, until near the end 

the operation. 













CENTERS FOR ARCHES. 


713 


Rem. 1. Instead of piers of masonry for supporting the feet of the 
posts, wooden cribs or piles may often be used if the arch is over water. 

Tl»e principle of supporting' even trussed frames by struts 

at points of the chord as far from the abutments as circumstances will admit of 
in addition to those at the very ends) should always be applied when possible, 
in order to reduce their sagging to a minimum. Steps or offsets in the 
masonry of the abutments and piers may be provided for receiving the feet 
of such struts, when they are inclined. 

Rem. 2. Screws may be used instead of wedges for lowering centers. At 
.he Pont d’Alma, Paris, ellipse of 141.4 ft span, and 28.2 ft rise, the frames were sup- 
aorted by wooden pistons or plungers, the feet of which rested on sand con- 
lined in plate-iron cylinders 1 ft in diam and height, and having near 
he bottom of each a plug which could be withdrawn and replaced at pleasure, 
bus regulating the outflow of the sand and the descent of the center. This de¬ 
vice succeeded perfectly, and is well worthy of adoption under arches exceeding 
ibout 60 ft span. When much larger than this the driving of the wedges on 
itriking requires heavy blows, and becomes a somewhat awkward operation, re¬ 
tiring at times a battering-ram, even when the wedges are lubricated. In rail- 
•oad cuttings crossed by bridges, the earth under the arch has been 
nade to serve as a center, by dressing its surface to the proper curve, and then 
imbedding in it curved timbers a few feet apart, and extending from abut to abut, 
' or supporting the close plank lagging. 

, Rem. 3. All centers must yield or settle more or less under the wt 
;i*f the arch, especially when supported only near their ends; and since the arch 
y tself also settles somewhat not only when the centers are struck, but for some 
,j ime after, it is advisable to make them at first, a little higher than the finished 
i, irch is intended to be. This extra height, when the supports are at the ends, 
; ,jnay be from 2 to 4 ins per 100 ft of span for cut stone arches (according to time 
ill >f striking, character of masonry, workmanship, etc.), and about twice as much in 
irick ones. 

Rem. 4. The proper time for striking centers is a disputed 
t<? >oint among engineers, some contending that it should be done as soon as the 
it.rch is finished and sufficiently backed up; and others that the mortar should 
: irst be given time to harden. It is the writer’s opinion that inasmuch as in 
srut-stone arches the mortar joints should be very thin ; and since, in such, the 
ui uortar is at best of very little service, it is of no importance when they are struck; 
Provided the masonry backing, and the embkt up to y n Fig 2, p 698, have been com¬ 
pleted ; but that in brick or rubble, the numerous joints of both of which require 
launch mortar, (which for hardness should consist largely of cement,) 3 or 4 months, 
J,r longer, if possible, should be allowed it to harden sufficiently to prevent undue 
in ompression and consequent settlement when the centers are struck. The con¬ 
tinuance of the centers need not interfere with traffic over the bridge. 

if Art. 2. The pressure of archstones against a center is very trifling until after 
'he arch is built up so far on each side that the joints form angles of 25° or 30° 
l /ith the horizontal. Theoretical discussions on this pressure make no allowance 
a or accidental jarrings in laying the archstones, or by the accumulation of material 
vi eady for use, laborers working on it, &c. Without going into any detail, we merely 
idvise on the score of safety not to assume it at less than about the following pro¬ 
portions or ratios to the weight of the entire arch, namely, in a semicircular arch 
17 • rise .35 span, .61; rise .25 span, .79; rise .2 span, .86; rise .167 span, or less, 1, 
r equal to the wt of the arch. This gives the pressure of a semicircular arch 
pon its centers rather less than halt its wt. The wt of tke centers 
heinselvcs when supported only near the ends must be considered as part 
f the load borne by them. 

Art. 3. We have seen that as an arch a a a is being gradually built upward on 
oth sides, after passing the points e,e, Fig 4, where its joints form angles of 
bout 30° with the horizontal a a, the arch begins to press more and more 
non the centers ; thereby tending to flatten them at the haunches, as shown at h 
n the dotted line; and consequently to raise them at the crown, as shown at c. 
hit as the building goes on still higher, the added stones press much more heavily 
non the centers than those below had done, and thereby tend to a final derange- 
aent of the centers just the reverse of that caused by the lower ones; namely to 
epress them at the crown a, as at o; and consequently to raise the haunches as 
t n • and this the more because the upper stones actually tend to lift or ease the 
nver ones from the lagging. In some cases where this tendency hasi becii, in- 
reased bv forciiifiT the keystones into place by too hard driving, the lagging 
nder the haunches could be drawn out without any trouble before the centers 
/ere eased at all. On striking the centers this tendency to sink at crown and 









CENTERS FOR ARCHES. 


714 


at haunches is very apt to exhibit itself more or less dangerously in the arch- 
Soncs themselves as hi fig 5, causing those near the crown to press very hard 
together at the extrados, and to separate from each other at the intrados, \*hile 
S the haunch^ the reverse takes place. Hence the angles of the stones are 
frequently split and spawled off near c and h by this unequal pressure. These 



derangements are of course much more likely to be serious in high arches thaii 
In flat ones, especially if their spandrels are not sufficiently built up befor : 


lowering the centers. „ , . , 

In the Grosvenor bridge, before alluded to, of 200 ft span, this dangerous exces 1 
of pressure near c and h was prevented by covering the skewback joint of th i 
8prin<dn<* course at each abutment with a wedge of lead 1.5 ins thick at the in t 
trados of the arch, and running out to nothing at the extrados. Beside this f 
strip 9 ins wide of sheet lead was laid along the intrados edge of every joint unt 8 
reaching that point at which it was judged that the line of pressure would pa> t 
from the intrados to the extrados; after which similar strips were laid along th| 
extrados edges of the joints, up to the crown. Hence when the centers weij 
struck this excess of pressure merely compressed the lead, and was thus enable fc 

. i! i*_1J* ..vnnlir nrnr tlm OlltifO (Ipillh fit' t.hft ioint.S_ See Trai 1 


to distribute itself more evenly over the entire depth of the joints. 

Inst Civ Eng London, vol i. 

At the bridge at Neuilly. France (of 5 elliptic arches of 120 ft spa? 
and 80 ft rise), the centers were so radically defective in design that the arch 
sank 13.25 ins at crown during the time ot bulldiug; and 10.5 ins more durii ll 
and immediately after the striking; or say 2 ft In all. Their construction matt 
the striking very tedious and hazardous; greatly endangering the lives of tl 
workmen and the existence of the arches. Some of the joints at the extrad 
at the haunches opened an inch each; and those at the intrados of the crown . 
of an inch. By the exercise of great care and humoring in lowering the centei 
these openings were much reduced. 

Item. 1. Chamfering the edges of the archstones diminish 

the danger of their spawllng off from unequal pressure; as does also the sera 
ing out of the mortar of th© joints for an inch or two in depth 

fore striking the centers. 

Item. 2. It is evident that in order to prevent, or at least to diminish t 
alternate derangements of the center, those of its web members which at fi 
acted as struts near the haunches, Fig. 4, to prevent them from sinking as 
h, must afterwards act as ties to prevent them from rising as at n; while th< 

which at first acted as ties n( 



the crown a, to prevent it fr< i 
rising as at c, must afterwa: 
act as struts to prevent it fr< 
sinkingasato. In other wor 
the principle of count! 
bracing must be atteiu 
to as well in a frame or tr 
for a center, as in one fo 
bridge. If the web memb 
are on the Warren or sim 
triangle system, as in Fig 
p 5S9, this may be effected 
making each member a 
strut; or the Pratt, or 
Howe system, Fig 35, p i 


may be used. 

Art. 4. From the foregoing it is plain that a simple unbraced wood 























CENTERS FOR ARCHES. 


715 


l CU , rved " b 1S > on aec o«nt of its great flexibility, about as unfit a form 
ioSSlrtpnth 11 ? 81e " for a center, except for very small spans, where a great propor- 
tonal depth of rib can be readily secured. Still the writer has seen it used for a 


ional depth of rib can be readily secured, „„„* ffurer , Kls see] 

ni rih 116 se ™ i cn' cu lar arch of 35 ft span, with archstones 2 ft deep. Fig 6 shows 
„ r . lb and the arch, a a, drawn to a scale. Each rib consisted of two thicknesses 

l V g ch £h k nippe^f th l of . about ®- 5 . ft « treenailed together so as to break joint, 
J *5 .J' l P Ce of Plank was 12 ins deep at middle, and 8 ins at each end 

ie top edge being cut to suit the curve of the arch. The treenails were 1 25 ins 
ot them , sh ? w cd to each length. These ribs were placed 17 ins 


fs a ioni° afe C e a ^h°S ^ d th stea ? ie J together by a bridging piece of inch board,'13 


f'Sf. U "t eT the arch ’ *5^ T et f no ch ' urds ‘ u'mtVtS .Sposite feel 

f the ribs. The ribs were covered with close board lagging which also assisted 

Iird^ a o/it n s g b? e hJ togeth , er transversely As the arch approached about two- 
* a tS f he ^ ht on each side, the ribs began to sink at the haunches, as at h, 
lg 4, and to rise at the crown, as at c. This was rectified by loading the crown 
- lth stone to be used in completing the arch; which was then finished without 
I irtuer trouble. 

J A still more striking: example of the use of a simple unbraced 
if ooden rib, was in the old National Turnpike bridge over Wills Creek, Virginia. 
jr his bridge, of which one arch with ’ 5““*- 

s center is shown in Fig 7 drawn 
» a scale, consisted of two elliptic 
it stone arches 26.5 ft wide across 




u 


>ai 


III' 


•adway. and of 60 ft span, and 15 
rise. The archstones were 3 ft 
iep at crown, and 4 ft deep at 
:ewbacks. Each frame of 
hie center was a simple rib 6 
s thick, composed of three thick- 



!sses ot 2 inch oak plank in different lengths (about 7 to 15 ft) to suit the curve 
id at the same time to preserve a width of about 16 ins at the middle of each 
ngth, and 12 ins at each of its ends. The thicknesses were well treenailed to¬ 
iler, breaking joint and showing from 10 to 16 treenails to a length, 
fcej Here, as in Fig 6, there were no chords, owing to the violence of the floods in 
ie creek These ribs were placed 18 ins from cen to cen, and steadied against 
ie another by a board bridging-piece 1 ft long, at every 5 ft. These were of 
tb >urse assisted by the lagging. 

id When the archstones had approached to within about 12 ft of each other near 
iiie middle of the span, the sinking at the crown, and the rising at the haunches 
leind become so alarming that pieces of 12 X 12 oak, 00, were hastily inserted at 
tervals, and well wedged against the archstones at their ends. The arch was 
en finished in sections between these timbers, which were removed one by one 
this was done. 


Rem. I. Such instances of partial failure are very instructive 

la Vnr on * 1, r. ± V. --_ l l _ l , • , . . " .. * 


fir t. 


. .—, -- ---— 77 , , *■■■> iniiurv aio very lUSiruCUVe. 

is indeed by such, rather than by theoretical deductions, that the proper dimen- 

I 11 Q Q VA Q1T1 UPll of 1 11 O T7 oof r> *1 n r~t f in ^ t 1 . 1 . _ • • 


ms are arrived at in a vast number of cases pertaining to engineering, ma- 
unery, &c.* Thus we might with entire confidence of no serious mishap apply 
bs of the foregoing dimensions to spans only half as great. 

ho Kem. 2. Assuming the rib-planks to be 12 ins wide, it would, as a matter of 
tail, be better to make them about 10 ins wide at the ends instead of the 8 ins 
Fig 6 making top curve 2 ins. To secure this, their lengths, depending on the 
r-Mlius of the rib, must not exceed those in the following table: 


Rad 

of Arch. 

Greatest Length. 

Rad 

of Arch. 

Greatest Length. 

Feet. 

Feet and Ins. 

Feet. 

Feet and Ins. 

5 

2 “ 5 

30 

6 “ 4 

10 

3 “ 4 

35 

7 “ 0 

15 

4 “ 2 

40 

7 “ 6 

20 

5 “ 0 

45 

7 “ 10 

25 

5 “ 9 

50 

8 “ 2 


1 The young engineer should make and preserve Tull notes in detail of all such as may fall within 
l<] notice; and if the professional journals would do the same thing in regard to failures which are 
istantly occurring, they would greatly increase the value of their papers. 

































716 


CENTERS FOR ARCHES. 


Art. 5. 

during the 


2 


a 

=f= 


a 

=t= 


r 


If cut \]A times as long as this table, they will be very approximately 8 i 
wide at ends; or each will on top curve 4 ins. 

I 11 oases where all possible headway Is essential 

building of the arch, as in the two foregoing ones, the writer would 

suggest the expedient rudely illustrated by 
Fig 8; namely to place the centers 
above the arch, instead of below 
it; and after the arch is completed in sec¬ 
tions, a a, instead of lowering- the ceil 
ters, to take them apart. The cen 
ters might resemble in principle Fig 35*4 
p 598. 

Fig 8 is a transverse section through par 
of the center, and of the arch a a. Hen 
rc,rc,rc, are frames of the center say 5 or 
ft apart; and of any depth and construct 101 


2 


Fig 8. 


c 


whatever that may be necessarv to insure absolute safety, and l 1 is the lagging 
Having built the arch from abutment to abutment in a senes of sections a, a, a, ne 
cessarily separated sav a foot or more by the deep frames, we may take the center 
apart and then till in the narrow intermediate sections upon a lagging suspende 
by iron rods from the already completed sections. Good concrete might be use 
for these narrow sections. In some cases it might be well to use deep plate 
iron ribs of I section, resting the lagging on the lower flange. Part ot th 
web might be left remaining embedded in the masonry; and the upper part an 
both flanges removed after the arch is finished, 


Art. 6. Centers with hor chords c c Fig 9 are objectionable (notwitl 

" rge spans of great rise, as on right side ot the Fig, o 


standing their strength) in large spans of great 



account of the excessive lengt 
required for the web member 
and hence it will in such cas< 
usually be found expedient ( 
adopt * something analogous 1 
what is shown on the left hau 
of the Fig. Here a truss/, short 
and shallower than that on tl 
right hand, is substituted for tl 
latter. At its ends provision mu 
be made for supporting not on 
itself, but the archstones belo ! ; 
it. As the pressure of these loi ' 
er archstones is comparative 
small, this may usually De effect' 
by resting the end of the frai ! 
may in large spans be aided 


f upon another and shallower frame 0 a. This —-„ 

either inclined or vertical struts, either single or braced together; or asthetresti 
on p 755. Sometimes one shallow truss like / is sustained upon another tru'j 
throughout its entire length. The striking-wedges for these various supports m 
be placed at either their tops or their feet, as may be most convenient. 

Art. 7. For flat arches of 10 feet clear span, a mere board . 
Fig 10 12 ins deep, by 1.5 ins thick, with another piece c of the same tliiekm 

on top of it, trimmed to the curve, and ec 
fined to 0 0 by nailing on two cleats of n; 
row- board, will answer every purpose, wile 
intervals of 18 ins from cen to cen. If t i 
upper piece also is as much as 12 ins deep it 
its center, the clear span may be extend in 
to 15 ft. fa 

For spans of 10 to 15 ft, and of a 1 
rise, two thicknesses of plank from 1 to 2 
thick according to span; 8 to 12 ins wide 
middle of each piece, in lengths as per table, Rem 2, Art 4, well nailed 



«piked together, according to span, breaking joint as in Fig 6, will answer 
listances of 2 to 3 ft apart cen to cen. For greater dists apart increase the thF 


s_ 

distances of 2 to 3 ft apart 
ness of the planks proportionally. 

If the centers have to be moved from place to place, to se j!i 

for other arches, then, to preserve them from injury in handling, their feet sho ill 
be united by nailing on one or both sides of each frame a chord piece of aboi ft 


























































CENTERS FOR ARCHES. 


717 


eh board ; and also a vertical piece or pieces of the same size from the center 
the chord to the top of the frame. 

when they sire not to be moved, the chord pieces are useful 
en in so small spans, inasmuch as they render the striking easier, by not allow- 
g the feet ot the ribs to give trouble by spreading outward and pressing against 
e, abutments. 



For spans of 15 to 30 ft, and for any rise not less than one sixth of the 
an, the following dimensions, varying with the span, may be used for distances 
art of 3 ft from een to cen. 
e Fig 11. For the bow b, 
o thicknesses of 1 to 2 inch 
ank from 9 to 12 ins wide 
the middle; and from 7 to 
ins at each end, well spiked 
gether breaking joint as at B, 
g6. For the chord c, two 
lcknesses of plank of same 
:e as the bow at its middle; 
aced on outsides of bow, and 
dl spiked to its ends. A 
?rtical v, in one piece as 
de as a bow plank, and twice 

thick. Its top is placed under the bow, and is confined to it by two pieces, o, o, 
bow plank twice as long as the bow plank is deep, and spiked to both v and the 
w. The foot of v passes between the two thicknesses of the chord c, and is 
iked to them. Two oblique tie-struts, s, each of two pieces of bow 
ink, outside of the bow and vertical v; footing against each other; and spiked 
bow and v. These with v divide the bow into 4 parts. 

Item. 1. The above dimensions are suitable to a rise of one sixth. If the 
se is one fourth, the t hickness only of the planks may be reduced one third 
rt; and for a rise of one third or more, we may reduce to one half. 

Rem. 2. If in the larger of these spans the struts s should show any incli- 
tion to bend sideways, nail on some pieces t from frame to frame. Also in the 
'ger ones with rises exceeding one third, insert four double struts s, instead 
two; thus dividing the bow into 6 parts, as at. left side of Fig. 11. For spans of 
to 35 ft, add also two struts like a a, of same size as v. 


Art. 8. For spans greater than about 30 ft. the writer believes 
at as a general rule (liable to modifications according to the judgment of the 
gineer in charge) the following ideas will lead to safe practice. Namely, to 
opt a bowstring truss with a simple Warren or triangular web, as at / on the 
t side of Fig 9. The bow to rest on the chord, and each to be of a single thick- 
ss. The web members (especially in large spans) to be also of single thickness, 
d placed below the bow, resting on the chords, and well strapped to both, so as 
act as either ties or struts. In smaller spans the web members may each be in 
) o thicknesses, one bolted or treenailed to each side of the bow and chord. Other 
ides will suggest themselves; but we have not space for such details. 

11 Dr a web of the Howe, or of the Pratt system, as on the right side of Fig 9 may be 
ed. But in reference to both of these it may be remarked that the use of 
>ng iron rods in centers of large spans is highly objectionable, owing 
j the different rates of expansion between iron and wood. Therefore if these 
( stems are used, all the members should be of wood. The lattice may be used. 

, Even when the rise of the arch exceeds .25 of the span, it is better not to let 
j at of the centers exceed that limit; but adopt the expedient shown at 
il e left side of Fig 9, with a rise oP about one sixth of the span. 

I Rem. 1. To fix on the number of web triangles in a Warren 
iss or frame for a center, find the square root of the span, and to it add one 
i ith of the span. Divide their sum by 2, and call the quotient n. Divide the 
in by n. If this quotient is a whole number use it; or if the quotient is partly 
cimal, use the whole number nearest to it, as a distance in feet to be stepped olf 
>ng the chord; thus dividing the chord into a number of equal parts. All the 
ints thus found on the chord, are the places for the feet of the triangles, 
xt, from half way between each two of these points, draw vertical lines to the 
w. The points thus found along the bow, are the places of the tops of the 
angles. This rule will be used in connection with the following Table of Areas 
Bows, as the two are dependent on each other. 

n large arches the timber of the bow should not be wasted by 

milling its upper edges to the curve of the arch, but should be left straight; and 
>arate pieces so trimmed, like c in Fig. 10, should be spiked on top of them. 












718 


CENTERS FOR ARCHES. 


The transverse area of the bow, in square inches, may be taken from 

the following table; and may in practice be assumed to be unitorm throughout 
its entire length ; which in tact it is quite approximately. See Rein 2. 


table for bowstring centers. 

Table of areas in square inches at the crown of each Bow, of properlj 
trussed Bowstring frames for centers of stone or brick arches. The frames t( 
be placed 5 feet apart from cen to cen. With these areas, the combined weight 
of arch, center (of oak), and lagging, will in no case in the table strain the Bov 
at crown of the greatest spans quite 1000 lbs per square inch ; diminishing gra« 
uallv to 600 or 700 lbs in the smallest spans, which are more liable to casualty 
The depths of the archstones may be taken fully equal to those in our table,. 
697. Although centers of moderate span are usually made of-white or yello 
pine, spruce, or hemlock, all of which are considerably lighter than oak, we hav 
for safety assumed them to be of oak, in preparing our table. 

For spans of from 10 to 20 feet use the same sizes as for 20 feet. 










Origina 




Rise in parts of the Span. 
.35 .3 .25 .2 

.15 


.5 

.4 

.1 

Span 

in feet. 


Areas of transverse section of 

in square inches. 

Bow, 

59 

20 

14 

17 

19 

21 

24 

29 

38 

25 

18 

22 

25 

28 

33 

40 

53 

80 

30 

23 

28 

32 

37 

43 

51 

71 

103 

35 

28 

34 

40 

45 

54 

64 

87 

125 

40 

34 

41 

48 

55 

65 

77 

106 

150 

45 

40 

49 

57 

65 

76 

92 

126 

175 

50 

47 

57 

66 

76 

89 

107 

146 

203 

55 

53 

64 

75 

87 

102 

121 

166 

233 

CO 

60 

73 

85 

99 

115 

135 

187 

263 

65 

68 

81 

95 

110 

129 

151 

209 

294 

70 

75 

90 

105 

122 

143 

168 

233 

325 

75 

83 

99 

115 

133 

157 

1S4 

256 

357 

80 

91 

108 

125 

145 

171 

201 

279 

390 

85 

99 

117 

136 

157 

185 

218 

302 

423 

90 

108 

127 

147 

169 

199 

235 

325 

457 

95 

115 

136 

158 

181 

214 

252 

348 

490 

100 

123 

146 

169 

194 

229 

270 

372 

524 

110 

133 

166 

191 

219 

260 

307 

420 

592 

120 

155 

187 

213 

246 

291 

345 

470 

660 

130 

172 

208 

237 

274 

323 

384 

520 


140 

190 

230 

263 

303 

357 

424 

572 


150 

209 

252 

289 

333 

393 

466 



160 

229 

276 

315 

365 

430 

509 



170 

250 

299 

343 

399 

469 




180 

272 

323 

373 

435 

511 




190 

294 

347 

403 

472 





200 

318 

372 

435 

509 






Rem. 2. The square root of any of these areas gives in inches the side < 
a square bow of that area. The distances apart of the triangles which foi 
the web of the frame, having first been found by Rem 1 (for said Rem and tl 
table are dependent on each other), the above areas for bows 5 ft apart from c 
to cen, suffice not only to resist the pressure along the bow, but also, as squa 
beams, to sustain with a safety in no case less than about 5, the load of an 
stones resting upon them between the adjacent tops of two triangles; and w 
very trifling deflections. It is therefore unnecessary to deepen the ribs for tl 
purpose; although it may be done (preserving the same area) in ease consid 
ations of detail should render it desirable. 

As before suggested, it will generally be best, in spans exceeding 30 or 40 ft. 
give the bow a rise not exceeding about one fifth or one sixth of the span ; a 
to support the frames as at/, Fig 9. I 

The size of the chord may be the same as that of the bow ; and lik< b 
uniform from end to end; care however being taken that it be not materia |{ 
weakened by footing the bow upon its ends; or (when too long for single t jr 
bers) by the splicing necessary to prevent its being stretched or pulled apart 
































CENTERS FOR ARCHES. 


719 




ie thrust of the bow. When, however, the chord can be placed at, or a little 
elow the springs of the arch, all danger of this kind may be avoided by simply 
edging its ends well against the faces of the abutments. 

As to the size of the web members, when a bowstring truss is 
u *‘3 r loaded on top of the bow, (as is approximately the case with a center 
• its archstones,) the strains on the web members are quite insignificant, and 
rise chiefly from the weight of the center itself; but while it is being so 
)aded, they are not only greater, but are constantly changing, not only in 
nsile a ^ S0 c ^ arac ^ er —being at one period compressive, and at another 

Hence it would be very tedious.to calculate thedimensions of the web members, 
ortunately the necessity for doing so is in a great measure obviated by the fact 
iat a center being but a temporary structure, the timber composing it is not ulti- 
ately wasted il a greater quantity of it is used than is absolutely required. 

' oreover facility of workmanship is secured bv not having to employ timbers 
many different sizes. ' • 

Hence the writer will venture to suggest, entirely as a rule of thumb, to give 
»ch web member half the transverse area of the bow, 

king care to make each of them a tie-strut. 

Rem. 3. As to details of joints, we refer to the Figs on pages 611, 
3; merely suggesting here the use of long and wide iron shoes where timbers 
e subjected to great pressure sideways. 

Rein. 4. To prevent the thrust of the bow when its rise is small, from split— 
ag off the ends of the chords, the two may be united by many more bolts than 
e employed in roof trusses, &c, where only one is generally placed near each end 
the chord. But they may when required be inserted at intervals extending to 
any feet from the ends. They should have strong large washers; and may have 
out the same inclination as the shortest web member. 

iAnother way of securing the same end in smaller spans, is by completely en- 
sing the two sides of the bow and chord, to a distance of a few feet from their 
ds, in short pieces of board or plank spiked to both of them, and having about 
e same inclination as just suggested for bolts. 

Rem. 5. Build up both sides of the arch at once, in order to strain the cen- 
rs as little as possible. 

Rem. 6. When a bridge consists of more than one arch, and they are to be 
ilt one at a time, there must be at least two centers; for a center must not 
struck until the contiguous arches on both sides are finished, for fear of over- 
rning the outer unsupported pier. Therefore if there are but two arches, they 
ust be built at once, requiring two centers. 

Rem. 7. Always use supports either vertical or inclined (and pro- 
led with striking-wedges) under the frames, and intermediate of the end sup- 
rts, when possible; even if they can extend out but a few feet from the abut- 
mts, as at the left side of Fig 9. 

Rem. 8. The weight of large centers and their lagging is greater 
;* flat arches than for high ones of the same span; and also approaches nearer 
that of the supported arch. 

Rem. 9. Thickness of lagging. The following table gives thicknesses 
lich will not bend more than an eighth of an inch under the weight of any 
obable archstones adapted to the respective spans ; and generally not 
much. 

TABLE OF LAGGING.— Original. 


istance apart 
of frames, 

i the clear. 


Feet. 

6 

5 

4 

3 

2 


Span of center in feet. 


10 . 


20 . 


50. 


100 . 


150. 


200 . 


Thickness of close lagging not to bend more than % inch. 


Ins. 

4 H 

m 

i 


Ins. 

5 

3 % 


2 

iy s 


Ins. 

P 

3 

2 


With thicknesses three quarters as great as these, the bending may reach 
ull quarter inch; which may be allowed in dists apart of 3 or more ft. 

Rein. 10. Centers are framed, or put together, (like iron bridges) ou a 
m, level temporary floor or platform, ou which a full-size drawing of a frame is 





















720 CENTERS FOR ARCHES. 




"UU Fig. 13 UU 


first made. As each frame is finished, it is removed to its place on the piers or 
abuts. 

Art. 9. The Wissallickon Bridge of the Reading R R, at Philadelphia, 

has five arches of 6o ft span, 23 ft rise, 28 ft wide (archstones 3 ft deep, with dressed 
beds and joints, in cement mortar); with four cutstone piers 9.5 ft thick at toy, and 
troni So to 50 ft high. It contains about 15400 cub yds of masonry.* Ksicll center 
consisted of 7 frames or trusses of hemlock timber, of the Bowstring pattern with 
lattice (p 596) web-members; and as nearly as may be, of the same span and 
rise as the arches. They were placed 4.5 ft apart from center to center; and were 

supported near each end /, Fig 13 
(a transverse section to scale) by 
a hemlock post p, 12 ins square. 
The bow was of two thicknesses 
bb of hemlock plank, 6 ins apart 
clear, in lengths of 6 ft, with their 
upper edges cut to suit the curve 
■» of the arch. Each piece was 4 ins 
J thick, by 13.5 ins deep at its middle, 
and 12 ins at its ends. These pieces 
did not break joint; but at each 
joint were four % inch bolts, with 
nuts and washers, uniting them 
, , , , . with chocks or filling-in pieces. 

The bow, b b, footed on top ot the ends of the chords f; and the angle formed by 
their meeting (seen only in a side view) was (for about 2.5 ft horizontal and 5 5 ft 
vertical) filled up solid with vertical pieces, to afford a firmer base for resting the 
frame on n ; beyond which it extends (in a side view) about 18 ins 
The chords / were of two thicknesses of 4X12 hemlock plank, 6 ins apart 
clear, and most of them in two or three lengths; breaking joint, and with two 
inch bolts, with nuts and washers, at each joint, for bolting them together, and to 
filling-in pieces. The web members of each frame were 26 lattices, o of 
t >\,V nch hemlock , crossing each other about at right angles, at intervals of about 
3.5 tt from center to center, and passing between the two thicknesses b b of the bow 

ai Y.*u f 1 * e Cl .‘° rds - , A f ? wo{ the lat tices were in two lengths, and the joints were 
not at the crossings. The lattices were connected at each crossing by two hard wood 
treenails 9 ins long, and 2 ins diam; and one such, 18 ins long, passed through the 
intersection of each end of a lattice with a bow or chord. The first lattice foots 
about 4 ft from the end of a chord. They do not extend above the top of the bow 
All the spaces between the two thicknesses of bow or chord, where not occupied bv 
the ends of lattices, were completely filled bv chocks, well spiked 
Ea «J* frame contained about 360 cub ft of timber; and weighed about 5 
tons. They were very flexible laterally until in place, and braced together bv 4 
transverse horizontal plauks spiked to their chords; and by 5 others above them 
spiked to the lattices. ’ 

Until the keystones were placed, all the joints of the frames continued tight under 
the pressure from the arch and from the unfinished backing to the height of about 
14 ft above the springing line; but after the keystones were set, all the joints of the 
chords alone opened from .25 to .75 of an inch; and at the same time the lagging tin- 
der the haunches of the arches became slightly separated from the soffit of the masonry 

thf“?h , aJ*"r1of“c , k?ng. UtttfU " lDCh Ut ,h « tl>e P»s>uro from 

The portion of the bridge above the piers was about two thirds completed before 
the centers were struck. 

enTof e r"'r, S r T e W . e ? ,se V 32 - 5 ft lon ^ 0f 12 X 12 inch oa *) ™der each 

end ot a center. It was trimmed to form i smaller ones w, w, each 4.5 ft long and 
tapering 7 ins ; one under each end of each frame /. They played between tapered 
docks a a, of oak 2 ft long, 1 It wide, let 1 inch into the cap c, or into the piece n 

fallow when' put' Tlw Burf “ 088 ' ,tre «» lubricated with 

onlTVn^ Wer ^ 1 f T UCk with ease > at one end of a center at a time, by an 

° a * log , ' a «enng-ram 18 ft long, and nearly a ft in diam, suspended by ropes and 

the'l?; 4 , mer i- T he .y generally yielded and moved several inches ^ 

w l SSi^ ° r i 4 ft SWinj f;, Altll0l] g h each "’edge was loosened entirely 
within 2 or 3 minutes, thus lowering the centers very suddenly, yet on account of the 

* T r h ‘ a ^idge. finished without accident, in 1882, reflects much credit on the 
writer in making observations during the entire progress of the work. cord.ally assisted the 


































CENTERS FOR ARCHES. 


721 


uardJ SteTrfl? n ?’ the ell S htest crack of a mortar joint could after- 

Itl 1 * ny P ;,rt „ of the ' v °rk. After three days the average sinking of 

the keystones was only .85 of an inch ; the least was \£\ and the greatest Sof in 

shls'’about 3/of an ?n, °/ ^ P ™ tS ? com P ressed the hemlock caps c, ™d the 

had better ll nflom ! ^ ’ Sh ° W , mg th f for *™hes of this size the caps and sills 
n.ul net ter be of some harder wood, as yellow pine or oak; although probably the 

compi ession was facilitated by the large mortices, 3 by 12 ins, and 6 ins^eep. 


J 


r ( > 


t . * s' ■ • * '• 








722 


RAILROAD CONSTRUCTION 


RAILROADS. 

RAILROAD CONSTRUCTION. 


TABLE OF ACRES REQUIRED per mile, and per 100 feet 

for different widths. 


Width. 

Feet. 

Acres 

per 

Mile. 

Acres 
per 
100 Ft. 

Width. 

Feet. 

Acres 

per 

Mile. 

Acres 
per 
100 Ft. 

Width. 

Feet. 

Acres 

per 

Mile. 

Acres 
per 
100 Ft. 

Width. 

Feet. 

Acres 

per 

Mile. 

Acres 
per 
100 Ft 

1 

.121 

.002 

26 

3.15 

.060 

52 

6.30 

.119 

78 

9.45 

.179 

2 

.242 

.005 

27 

3.27 

.062 

53 

6.42 

.122 

79 

9.5S 

.1S1 

3 

.364 

.007 

28 

3.39 

.064 

54 

6.55 

.124 

80 

9.70 

.184 

4 

.485 

.009 

29 

3.52 

.067 

55 

6.67 

.126 

81 

9.82 

.186 

5 

.606 

.011 

30 

3.64 

.069 

56 

6 79 

.129 

82 

9.94 

.188 

6 

.727 

.014 

31 

3.76 

.071 

57 

6.91 

.131 


10. 

.189 

7 

.848 

.016 

32 

3.88 

.073 

% 

7. 

.133 

83 

10.1 

.190 

8 

.970 

.018 

33 

4.00 

.076 

58 

7.03 

.133 

84 

10.2 

.193 


1 . 

.019 

34 

4.12 

.078 

59 

7.15 

.135 

85 

10.3 

.195 

9 

1.09 

.021 

35 

4.24 

.080 

eo 

7.27 

.138 

86 

10.4 

.197 

10 

1.21 

.023 

36 

4.36 

.083 

61 

7.39 

.140 

87 

10.5 

.200 

11 

1.33 

.025 

37 

4.48 

.0S5 

62 

7.52 

.142 

88 

10.7 

.202 

12 

1.46 

.028 

38 

4.61 

.0S7 

63 

7.64 

.145 

89 

10.8 

.204 

13 

1.58 

.030 

39 

4.73 

.090 

64 

7.76 

.147 

90 

10.9 

.207 

14 

1.70 

.032 

40 

4.85 

.092 

65 

7.88 

.149 

% 

11. 

.209 

15 

1.82 

.034 

41 

4.97 

.094 

66 

8. 

.151 

91 

11.0 

.209 

16 

1.94 

.037 

V* 

5. 

.094 

67 

8.12 

.154 

92 

11.2 

.211 

W 

2. 

.038 

42 

5.09 

.096 

68 

8.24 

.156 

93 

11.3 

.213 

17 

2 06 

.039 

43 

5.21 

.099 

69 

8.36 

.158 

94 

11.4 

.216 

18 

2.18 

.041 

44 

5.33 

.101 

70 

8.48 

.161 

95 

11.5 

.218 

19 

2.30 

.014 

45 

5.45 

.103 

71 

8.61 

.163 

96 

11.6 

.220 

20 

2.42 

.046 

46 

5.58 

.106 

72 

8.73 

.165 

97 

11.8 

.223 

21 

2.55 

.048 

47 

5.70 

.108 

73 

8.85 

.168 

98 

11.9 

.225 

22 

2.67 

.051 

48 

5.82 

.110 

74 

8.97 

.170 

99 

12. 

.227 

23 

2.79 

.053 

49 

5.94 

.112 

% 

9. 

.170 

100 

12.1 

.230 

21 

2.91 

.055 

M 

6. 

.114 

75 

9.09 

.172 



% 

3. 

.057 

50 

606 

.115 

76 

9.21 

.174 




25 

3.03 

.057 

51 

6.18 

.117 

77 

9.33 

.177 












































RAILROAD CONSTRUCTION 


/ I 'd 


rable of grades per mile, and per 100 feet measured liori< 
zontally, and corresponding- to different angles of incli¬ 
nation. 


Deg. 

Min. 

Feet per 
mile. 

Feet per 
100 ft. 

Deg. 

Min. 

Feet per 
mile. 

Feet per 
100 ft. 

ei a 

O •— 

Q 3 

Feet pei 
mile. 

Feet per 
100 ft. 

tb a 
a> — 

Q E 

Feet per 
mile. 

Feet per 
100 ft. 

0 1 

1.536 

.0291 

0 45 

69.11 

1.3090 

1 58 

181.3 

3.4341 

3 26 

316.8 

5.9994 

2 

3.072 

.0582 

46 

70.64 

1.3381 

2 0 

184.4 

3.4924 

28 

319.8 

6.0579 

3 

4.608 

.0873 

47 

72.18 

1.3672 

2 

187.5 

3.5506 

30 

322.9 

6.1163 

4 

6.144 

.1164 

48 

73.72 

1.3963 

4 

190.6 

3.6087 

32 

326.0 

6.1747 

5 

7.680 

.1455 

49 

75.26 

1.4254 

6 

193.6 

3.6669 

34 

329.1 

6.2330 

6 

9.216 

.1746 

50 

76.80 

1.4545 

8 

196.7 

3.7250 

36 

332.2 

6 2914 

7 

10.75 

.2037 

51 

78.33 

1.4837 

10 

199.8 

3.7833 

38 

335.3 

6.3498 

8 

12.29 

.2328 

52 

79.87 

1.5128 

12 

202.8 

3.8416 

40 

338.4 

6.4083 

9 

13.82 

.2619 

53 

81.40 

1.5419 

14 

205.9 

3.8999 

42 

341.4 

6.4664 

10 

15.36 

.2909 

54 

82.94 

1.5710 

16 

208.9 

3.9581 

44 

344.5 

6.5246 

11 

16.90 

.3200 

55 

84.47 

1.6000 

18 

212.0 

4.0163 

46 

347.6 

6.5832 

12 

18.43 

.3491 

56 

86.01 

1.6291 

20 

215.1 

4.0746 

48 

350.7 

6.6418 

13 

19.96 

.3782 

57 

87.54 

1.6583 

22 

218.1 

4.1329 

50 

353.8 

6.7004 

14 

21.50 

.4073 

58 

89.08 

1.6873 

24 

221.2 

4.1911 

52 

356.8 

6.7o83 

15 

23.04 

.4364 

59 

90.62 

1.7164 

26 

224.3 

4.2494 

54 

359.9 

6.8163 

16 

24.58 

.4655 

1 

92.16 

1.7455 

28 

227.4 

4.3076 

56 

363.0 

6.8751 

17 

26.11 

.4946 

2 

95.23 

1.8038 

30 

230.5 

4.3659 

58 

366.1 

6.9339 

IS 

27.64 

.5237 

4 

98.30 

1.8620 

32 

233.5 

4.4242 

4 

369.2 

6.9926 

19 

29.17 

.5528 

6 

101.4 

1.9202 

34 

236.6 

4.4826 

5 

376.9 

V.1384 

20 

30.72 

.5818 

8 

104.5 

1.9784 

36 

239.7 

4.5409 

10 

384.6 

7.2842 

21 

32.26 

.6109 

10 

107.5 

2.0366 

38 

242.8 

4.5993 

15 

392.3 

7 4300 

22 

33.80 

.6400 

12 

110.6 

2.0948 

40 

245.9 

4.6576 

20 

400.1 

7.5767 

23 

35.33 

.6691 

14 

113.6 

2.1530 

42 

248.9 

4.7159 

25 

407.8 

7.7234 

24 

36.86 

.6982 

16 

116.7 

2.2112 

44 

252.0 

4.7742 

30 

415.5 

7.8701 

25 

38.40 

.7273 

18 

119.8 

2 2694 

46 

255.1 

4.8325 

35 

423.2 

8.0163 

26 

39.94 

.7564 

20 

122.9 

2.3277 

48 

258.2 

4.8908 

40 

431.0 

8.1625 

27 

41.47 

.7855 

22 

126.0 

2.3859 

50 

261.3 

4.9492 

45 

438.7 

8.3087 

28 

43.01 

.8146 

24 

129.1 

2.4441 

52 

264.3 

5.0075 

50 

446.5 

8.4554 

29 

44.54 

.8436 

26 

132.1 

2.5023 

54 

267.4 

5.0658 

55 

454.2 

8.6021 

30 

46.08 

.8727 

28 

135.2 

2.560t 

56 

270.5 

5.1241 

5 

461.9 

8.7489 

31 

47.62 

.9018 

30 

138.3 

2.6186 

58 

273.6 

5.1824 

5 

461). 6 

8.8951 

32 

49.16 

.9309 

32 

141.3 

2.6768 

3 

276.7 

5.2407 

10 

477.4 

9.0413 

33 

60.69 

.9600 

34 

144.4 

2.7350 

2 

279.7 

5.2990 

15 

485.1 

9.1875 

31 

52.23 

.9891 

36 

147.4 

2.7932 

4 

282.8 

5.3573 

20 

492.9 

9.3347 


53 76 

1.0182 

38 

150.5 

2.8514 

6 

285.9 

5.4158 

25 

500.6 

9.4819 

36 

55.30 

1.0472 

40 

153.6 

2.9097 

8 

289.0 

5.4742 

30 

508.4 

9.6292 

37 

56 83 

1.0763 

42 

156.6 

2.9679 

10 

292.1 

5.5326 

35 

516.1 

9.7755 

38 

58 37 

1.1054 

44 

159.7 

3.0262 

12 

295.1 

5.5909 

40 

523.9 

9.9218 

39 

59 90 

1.1345 

46 

162.8 

3.0844 

14 

298.2 

5.6493 

45 

531.6 

10.068 

40 

(>1 44 

1.1636 

48 

165.9 

3.1427 

16 

301.3 

5.7077 

50 

539.4 

10.215 

44 

62.97 

1.1927 

50 

169.0 

3.2010 

18 

304.4 

5.7660 

55 

547.2 

10.362 

42 

61.51 

1.2218 

52 

172.0 

3.2592 

20 

307.5 

5.8244 

6 

555. 

10.610 

43 

66.04 

1.2509 

54 

175.1 

3.3175 

22 

310.5 

5.8827 




44 

67.57 

1.2800 

56 

178.2 

3.3758 

24 

313.6 

5.9410 





On a turnpike road 1° 38', or about 1 in 35, or 151 feet per nule, is tli > 
i greatest slope that will allow horses to trot down rapidly with safety. In crossing 
mountains, this is often increased to 3°, or eveir to 5°. It should never exceed 2 y 2 °, 
except when absolutely necessary. 





































I 4-t 


KiULiiUAU UUJN&J-ltL L, J.J.UJ.N 


SLOPES IN FEET PER 100 FT. HORIZONTAL. 


The fractions of minutes are given only to 34 feet in 100. 

A clinometer graduated by the 3d column, and numbered by the first one, 
will give at sight the slopes in feet per 100 feet. No errors. Original. 


Rise in ft 
per 100 
ft hor. 

Length of 
slope per 
100 ft hor. 

Angle of 
slope. 

Rise in ft 
per 100 

ft hor. 

Length of 
slope per 
100 ft hor. 

Angle of 
slope. 

Rise in ft 

per 100 
ft hor. 

Length of 
slope per 
100 ft hor. 

Angle of 

8 lope. 


Feet. 

Deg. 

Min. 


Feet. 

Deg. 

Min. 


Feet. 

Deg.- 

Min. 

] 

100.005 

0 

34.4 

35 

105 948 

19 

17 

69 

121.495 

34 

36 

2 

100.020 

1 

8.7 

36 

106.283 

19 

48 

70 

122.066 

35 

0 

3 

100.045 

1 

43.1 

37 

106.626 

20 

18 

71 

122 642 

35 

23 

4 

100.080 

2 

17.5 

38 

106.977 

20 

48 

72 

123.223 

35 

45 

5 

100 125 

2 

51.8 

39 

107.336 

21 

18 

73 

123.810 

36 

8 

6 

100.180 

3 

26.0 

40 

107.703 

21 

48 

74 

124.403 

36 

30 


100.245 

4 

0.3 

41 

108.079 

22 

18 

75 

125.000 

36 

62 

8 

100.319 

4 

34.4 

42 

108.462 

22 

47 

76 

125.603 

37 

14 

9 

100.404 

5 

8.6 

43 

108.853 

23 

16 

77 

126.210 

37 

36 

10 

100.499 

5 

42.6 

44 

109.252 

23 

45 

78 

126.823 

37 

57 

11 

100.603 

6 

16.6 

45 

109.659 

24 

14 

79 

127.440 

38 

19 

12 

100 717 

6 

50.6 

46 

110.073 

24 

42 

80 

128 062 

38 

40 

13 

100.841 

7 

24.4 

47 

110.494 

25 

10 

81 

128.690 

39 

1 

14 

100.975 

7 

58.2 

48 

110.923 

25 

38 

82 

129.321 

39 

21 

15 

101.119 

8 

31.9 

49 

111.359 

26 

6 

83 

129.958 

39 

42 

16 

101.272 

9 

5.4 

50 

111 803 

26 

34 

84 

130.599 

40 

2 

17 

101.435 

9 

38.9 

51 

112.254 

27 

1 

85 

131.244 

40 

22 

18 

101.607 

10 

12.2 

52 

112.712 

27 

28 

86 

131.894 

40 

42 

19 

101.789 

10 

45.5 

53 

113.177 

27 

55 

87 

132.548 

41 

1 

20 

101.980 

11 

18.6 

54 

113.649 

28 

22 

88 

133.207 

41 

21 

21 

102.181 

11 

51.6 

55 

114.127 

28 

49 

89 

133.869 

41 

40 

22 

102.391 

12 

24.5 

56 

114.612 

29 

15 

90 

134.536 

41 

59 

23 

102.611 

12 

57.2 

57 

115.104 

29 

41 

91 

135.207 

42 

18 

24 

102.840 

13 

29.8 

58 

115.603 

30 

7 

92 

135.882 

42 

37 

25 

103.078 

14 

2.2 

59 

116.108 

30 

32 

93 

136 561 

42 


26 

103.325 

14 

34.5 

60 

116.619 

30 

58 

94 

137.244 

43 

14 

27 

103.581 

15 

6.6 

61 

117.137 

31 

23 

95 

137.931 

43 

32 

28 

103.846 

15 

38.5 

62 

117.661 

31 

48 

96 

138.622 

43 


29 

104.120 

16 

10 3 

624 

118.191 

32 

13 

97 

139 316 

44 

8 

30 

104.403 

16 

42.0 

64 

118 727 

32 

37 

98 

140.014 

44 

25 

31 

104.695 

17 

13.4 

65 

119.269 

33 

1 

99 

140.716 

44 

43 

32 

104.995 

17 

44.7 

66 

119.817 

33 

25 

100 

141.421 

45 

00 

33 

105.304 

18 

15.8 

67 

120.370 

33 

49 

101 

142.130 


17 

34 

105.622 

18 

46.7 

68 

120.930 

34 

13 

102 

142.843 

45 

34 


Any hor <lisf is = sloping dist X cosine ang of slope. 

“ sloping <list is = hor dist h- cosine “ “ “ 

“ vert height is = hor dist X tangent “ “ “ 

or — sloping dist X sine “ “ “ 


i 

ii 




































GRADES 


725 


Table of grades per mile; or per 100 feet measured hori¬ 
zontally. 


Grade 
in ft. 
per mile. 

Grade 
in ft. 

per 100 ft. 

Grade 
in ft. 
per mile. 

Grade 
in ft. 
per 100 ft. 

Grade 
in ft. 
per mile. 

Grade 
in ft. 
per 100 ft. 

Grade 
in ft. 
per mile 

Grade 
in ft. 
per 100 ft 

1 

.01894 

39 

.73-64 

77 

1.45833 

115 

2.17803 

2 

.03788 

40 

.75758 

78 

1.47727 

116 

2.19697 

3 

.05682 

41 

.77652 

79 

1.49621 

117 

2.21591 

4 

.07576 

42 

.7 9545 

80 

1.51515 

118 

2.23485 

5 

.09470 

43 

.81439 

81 

1.53409 

119 

2.25379 

6 

-11364 

44 

.83333 

82 

1.55303 

120 

2.27273 

7 

.13258 

45 

.85227 

83 

1.57197 

121 

2.29167 

8 

.15152 

46 

.87121 

84 

1.59091 

122 

2.31061 

9 

.17045 

47 

.89015 

85 

1.609S5 

123 

2.32955 

10 

.18939 

48 

.90909 

86 

1.62879 

124 

2.34848 

11 

.20333 

49 

.92803 

87 

1.64773 

125 

2.36742 

12 

.22727 

50 

.94697 

88 

1.66666 

126 

2.38636 

13 

.24621 

51 

.96591 

89 

1.6S561 

127 

2.40530 

14 

.26515 

52 

.98485 

90 

1.70455 

128 

2 4 2 424 

15 

.28409 

53 

1.00379 

91 

1.72348 

129 

2.44318 

16 

.30303 

54 

1.02 173 

92 

1.74212 

130 

2.46212 

17 

.32197 

55 

1.04167 

93 

1.76136 

131 

2.4S106 

IS 

.34091 

56 

1.06061 

94 

1.78030 

132 

2.50000 

19 

.35985 

57 

1.07955 

95 

1.79924 

133 

2.51894 

20 

.37879 

58 

1.09848 

96 

4.81818 

134 

2.53788 

21 

.39773 

59 

1.11742 

97 

1.83712 

135 

2 55682 

22 

.41667 

60 

1.13636 

98 

1.85606 

136 

2.57576 

23 

.43561 

61 

1.15530 

99 

1.87500 

137 

2.59470 

24 

.45455 

62 

1.17424 

100 

1.89394 

138 

2.61364 

25 

.47348 

63 

1.19318 

10 L 

1.91288 

139 

2.63258 

26 

.49242 

64 

1.21212 

102 

1.93182 

140 

2.65152 

27 

.51136 

65 

1.23106 

103 

1.95076 

141 

2.67045 

28 

.53030 

66 

1.25000 

104 

1.96969 

142 

2.6-939 

29 

.54924 

67 

1.26394 

105 

1.98864 

143 

2 70833 

30 

.56818 

68 

1.28788 

10 1 

2.00758 

14 4 

2 72727 

31 

.58712 

69 

1.30682 

107 

2.02652 

145 

2.74621 

32 

.69606 

70 

1.32576 

108 

2.04545 

146 

2 76515 

33 

.6250) 

71 

1.34470 

109 

2.06439 

147 

2.7S409 

34 

.64894 

72 

1.36364 

110 

2.0S333 

148 

2.80303 

35 

.68288 

73 

1.3S258 

111 

2.10227 

149 

2.82197 

36 

.68132 

74 

1.40152 

112 

2.12121 

150 

2.84091 

37 

.70076 

75 

1.42045 

113 

2.14015 

151 

2.85985 

38 

.71970 

76 

1.43939 

114 

2.15909 

152 

2.87879 




If the grade per mile should consist of feet and tenths, add to the grade per 100 
feet in the foregoing table, that corresponding to the number of tenths taken from 
the table below; thus, for a grade of 43.7 feet per mile, we have .81439 + .01320 = 

.82765 feet per 100 feet. 

« 


Ft. per Mile. 

Per 100 Feet. 

Ft. per Mile. 

Per 100 Feet. 

Ft. per Mile. 

Per 100 Feet. 

.05 

.00094 

.4 

.00758 

.7 

.01326 

.1 

.00189 

.45 

.00852 

.75 

. .01420 

.15 

.00283 

.5 

.00947 

.8 

.01515 

.2 

.00379 

.55 

.01041 

.85 

.01609 

.25 

.00473 

.6 

.01136 

.9 

.01705 

.3 

.00568 

.65 

.01230 

.95 

.01799 

.35 

.00662 













































I Ad L 

Pal 

ing. 

Defl 

o 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

2 

4 

6 

8 

10 

12 

14 

16 

18 


RAILROADS, 


dii. 


Middle Ordinates, <fec, of Carves. 


Contains no error as great as 1 in the last figure. 


Chord 100 feet. 


Defl. 
Dist. 
in ft. 

Tang. 
Dist. 
in ft. 

Mid. 

Ord. 

Ang. of 
Defl. 

.029 

.014 

.004 

O ' 

1 36 

.058 

.029 

.007 

38 

.087 

.043 

.011 

40 

.116 

.058 

.014 

42 

.145 

.072 

.018 

44 

.175 

.087 

.022 

46 

.204 

.102 

.025 

48 

.233 

.116 

.029 

50 

.262 

.131 

.033 

52 

.291 

.145 

.036 

54 

.320 

.160 

.040 

56 

.349 

.174 

.043 

58 

.378 

.189 

.047 

2 

.407 

.203 

.051 

2 

.436 

.218 

.054 

4 

.465 

.232 

.058 

6 

.494 

.247 

.062 

8 

.524 

.262 

.065 

10 

.553 

.276 

.069 

12 

.582 

.291 

.073 

14 

.611 

.305 

.076 

16 

.640 

.320 

.080 

18 

.669 

.334 

.083 

20 

.698 

.349 

.087 

22 

.727 

.363 

.091 

24 

.756 

.378 

.095 

26 

.785 

.392 

.098 

28 

.814 

.407 

.102 

30 

.844 

.422 

.105 

32 

.873 

.436 

.109 

34 

.902 

.451 

.113 

36 

.931 

.465 

.116 

38 

.960 

.480 

.120 

40 

.989 

.494 

.123 

42 

1.018 

.509 

.127 

44 

1.047 

.623 

.131 

46 

1.076 

.538 

.134 

48 

1.105 

.552 

.138 

50 

1.134 

.567 

.142 

52 

1.164 

.582 

.145 

54 

1.193 

.596 

.149 

56 

1.222 

.611 

.153 

58 

1.251 

.625 

.156 

3 

1.280 

.640 

.160 

2 

1.309 

.654 

.164 

4 

1.338 

.669 

.167 

6 

1.367 

.683 

.171 

8 

1.396 

.698 

.174 

10 

1.425 

.712 

.178 

12 

1.454 

.727 

.182 

14 

1.483 

.741 

.185 

16 

1.513 

.757 

.189 

18 

1.542 

.771 

.193 

20 

1.571 

.786 

.197 

22 

1.600 

.800 

.200 

24 

1.629 

.815 

.204 

26 

1.658 

.829 

.207 

28 

1.687 

.844 

.211 

30 

1.716 

.858 

.214 

32 

1.745 

.872 

.218 

34 

1.803 

.902 

.225 

36 

1.862 

.931 

.233 

38 

1.920 

.960 

.240 

40 

1.978 

.989 

.247 

42 

2.036 

1.018 

.255 

44 

2.094 

1.047 

.262 

46 

2.152 

1.076 

.269 

48 

2.211 

1.105 

.276 

50 

2.269 

1.134 

.284 

52 

2.327 

1.163 

.291 

54 

2.385 

1.192 

.298 

56 

2.443 

1.221 

.305 

58 

2.502 

1.251 

.313 

4 

2.560 

1.280 

.320 

5 

2.618 

1.309 

.327 

10 

2.676 

1.338 

.334 

15 

2.734 

1.367 

.342 

20 


Rad. 
in ft. 

Defl. 
Dist. 
in ft. 

Tang. 
Dist. 
in ft. 

Mid. 

Ord. 

3581 

2.793 

1.396 

.349 

3508 

2.851 

1.425 

.356 

3438 

2.909 

1.454 

.364 

3370 

2.967 

1.483 

.371 

3306 

3.025 

1.512 

.378 

3243 

3.084 

1.542 

.385 

3183 

3.142 

1.571 

.393 

3125 

3.200 

1.600 

.400 

3070 

3.257 

1.629 

.407 

3016 

3.316 

1.658 

.414 

2964 

3.374 

1.687 

.422 

2913 

3.433 

1.716 

.429 

2865 

3.490 

1.745 

.436 

2818 

3.549 

1.774 

.443 

2773 

3.606 

1.803 

.451 

2729 

3.664 

1.832 

.458 

2686 

3.723 

1.861 

.465 

2645 

3.781 

1.890 

.473 

2605 

3.839 

1.919 

.480 

2566 

3.897 

1.943 

.487 

2528 

3.956 

1.978 

.495 

2491 

4.014 

2.007 

.502 

2456 

4.072 

2.036 

.509 

2421 

4.131 

2.065 

.516 

2387 

4.189 

2.094 

.523 

2355 

4.246 

2.123 

.531 

2323 

4.305 

2.152 

.538 

2292 

4.363 

2.182 

.545 

2262 

4.421 

2.210 

.552 

2232 

4.480 

2.240 

.560 

2204 

4.537 

2.268 

.567 

2176 

4.596 

2.298 

.574 

2149 

4.654 

2.327 

.582 

2122 

4.713 

2.356 

.589 

2096 

4.771 

2.385 

.596 

2071 

4.829 

2.414 

.603 

2046 

4.888 

2.444 

.611 

2022 

4.946 

2.473 

.618 

1999 

5.003 

2.501 

.625 

1976 

5.061 

2.530 

.632 

1953 

5.120 

2.560 

.640 

1932 

5.176 

2.588 

.647 

1910 

5.235 

2.618 


1889 

5.294 

2.647 

.662 

1869 

5.350 

2.675 

.669 

1848 

5.411 

2.705 

.676 

1829 

5.467 

2.734 

.683 

1810 

5.526 

2.763 

.691 

1791 

5.583 

2.792 

.698 

1772 

5.643 

2.821 

.705 

1754 

5.707 

2.850 

.713 

1736 

5 760 

2.880 

.720 

1719 

5.817 

2.908 

.727 

1702 

5.875 

2.937 

.734 

1685 

5.935 

2.967 

.742 

1669 

5.992 

2.996 

.749 

1653 

6.050 

3.025 

.756 

1637 

6.108 

3.054 

.764 

1622 

6.166 

3.083 

.771 

1607 

6.223 

3.112 

.778 

1592 

6.281 

3.140 

.785 

1577 

6.341 

3.170 

.793 

1563 

6.398 

3.199 

.800 

1549 

6.456 

3.228 

.807 

1536 

6 515 

3.257 

.814 

1521 

6.575 

3.287 

.822 

1508 

6.631 

3.316 

.829 

1495 

6.689 

3.345 

.836 

1482 

6.748 

3.374 

.843 

1469 

6.807 

3.403 

.851 

1457 

6.863 

3.432 

.858 

1445 

6.920 

3.460 

.865 

1433 

6.980 

3.490 

.873 

1403 

7.125 

3.562 

.891 

1375 

7,271 

3.635 

.909 

1348 

7.416 

3.708 

.927 

1323 | 

7.561 

3.781 i 

.945 

















































RAILROADS 


727 


Table of Radii, Middle Ordinates, dfcc, of Carves. Chord 100 feet. 

(Continued.) 

The Tangential Angle is always one-half of the Angle of Deflection. 


Ang. of 
Defl. 

Rad. 
in ft. 

Defl. 
Dist. 
in ft. 

Tang. 
Dist. 
in ft. 

Mid.* 

Ord. 

Ang. of 
Defl. 

Rad. 
in ft. 

Defl. 
Dist. 
in ft. 

Tang. 
Dist. 
in ft. 

Mid.# 

Ord. 

O ' 

4 26 

1298 

7.707 

3.854 

.963 

O ' 

10 15 

559.7 

17.87 

8.942 

2.238 

30 

1274 

7.852 

3.927 

.982 

30 

546.4 

18.30 

9.160 

2.292 

35 

1250 

7.997 

3.999 

1.000 

45 

533.8 

18 73 

9.378 

2.347 

40 

1228 

8.143 

4.072 

1.018 

11 

521.7 

19.17 

9.596 

2.402 

45 

1207 

8.288 

4.145 

1.036 

15 

510.1 

19.60 

9.814 

2.456 

50 

1186 

8.433 

4.218 

1.054 

30 

499.1 

20.04 

10.03 

2.511 

55 

1166 

8.579 

4.290 

1.072 

45 

488.5 

20.47 

10.25 

2.566 

5 

1146 

8.724 

4.363 

1.091 

12 

478.3 

20.91 

10.47 

2.620 

5 

1128 

8.869 

4.436 

1.109 

15 

468.6 

21.34 

10.69 

2.675 

10 

1109 

9.014 

4.508 

1.127 

30 

459.3 

21.77 

10.90 

2.730 

15 

1092 

9.160 

4.581 

1.145 

45 

450.3 

22.21 

11.12 

2.785 

20 

1075 

9.305 

4.654 

1.164 

13 

441.7 

22.64 

11.34 

2 839 

25 

1058 

9.450 

4.727 

1.182 

15 

433.4 

23.07 

11.56 

2.894 

30 

1042 

9.596 

4.799 

1.200 

30 

425.4 

23.51 

11.77 

2.949 

35 

1027 

9.741 

4.872 

1.218 

45 

417.7 

23.94 

11.99 

3.003 

40 

1012 

9.886 

4.945 

1 236 

14 

410.3 

24.37 

12.21 

3.058 

45 

996.9 

10.03 

5.017 

1.255 

15 

403.1 

24.81 

12.43 

3.113 

50 

982.6 

10.18 

5.090 

1.273 

30 

396.2 

25.24 

12.65 

3 168 

55 

968.8 

10.32 

5.163 

1.291 

45 

389.5 

25.67 

12.86 

3.223 

6 

955.4 

10.47 

5.235 

1.309 

15 

383.1 

26.11 

13.08 

3.277 

5 

942.3 

10.61 

5.308 

1.327 

15 

376.8 

26.54 

13.30 

3.332 

10 

929.6 

10.76 

5.381 

1.346 

30 

370.8 

26.97 

13.52 

3.387 

15 

917.2 

10.90 

5.453 

1 364 

45 

364.9 

27.40 

13.73 

3.442 

20 

905.1 

11.05 

5.526 

1.382 

16 

359.3 

27.84 

13.95 

3.496 

25 

893.4 

11.19 

5.599 

1.400 

30 

348.5 

28.70 

14.39 

3.606 

30 

882.0 

11.34 

5.672 

1.418 

17 

338.3 

29.56 

14.82 

3.716 

35 

870.8 

11.48 

5.744 

1.437 

30 

328.7 

30.43 

15.26 

3.825 

40 

859.9 

11.63 

5.817 

1.455 

18 

319.6 

31 29 

15.69 

3.935 

45 

849.3 

11.77 

5.890 

1.473 

30 

311.1 

32.15 

16.13 

4.045 

50 

839.0 

11.92 

5.962 

1.491 

19 

302.9 

33.01 

16.56 

4.155 

55 

828.9 

12.07 

6.035 

1.510 

30 

295.3 

33.87 

17.00 

4.265 

7 

819.0 

12.21 

6.108 

1.528 

20 

287.9 

34.73 

17.43 

4.375 

5 

809.4 

12.36 

6.180 

1.546 

21 

274.4 

36.44 

18.30 

4.594 

10 

800.0 

12.50 

6 253 

1.564 

22 

262.0 

38.17 

19.17 

4.815 

15 

790.8 

12.65 

6.326 

1.582 

23 

250.8 

39.87 

20.04 

5.035 

20 

781.8 

12.79 

6.398 

1.600 

24 

240.5 

41.58 

20.91 

5.255 

25 

773.1 

12.94 

6.471 

1.619 

25 

231.0 

43.29 

21.77 

5.476 

30 

764.5 

13.08 

6.544 

1.637 

26 

222.3 

44.98 

22.64 

5.696 

35 

756.1 

13.23 

6.616 

1.655 

27 

214.2 

46.69 

23.51 

5.917 

40 

747.9 

13.37 

6.689 

1.673 

28 

206.7 

48.38 

24.37 

6.139 

45 

739.9 

13.52 

6.762 

1.691 

29 

199.7 

50.08 

25.24 

6.361 

50 

732.0 

13 66 

6.835 

1.710 

30 

193 2 

51.76 

26 11 

6.582 

55 

724.3 

13.81 

6.907 

1.728 

31 

187.1 

53.45 

26.97 

6.805 

8 

716.8 

13.95 

6.980 

1.746 

32 

181.4 

55.13 

27.83 

7.027 

15 

695.1 

14.39 

7.198 

1.801 

33 

176.0 

56.82 

28.70 

7.252 

30 

674.7 

14.82 

7.416 

1.855 

34 

171.0 

58.48 

29.56 

7.473 

45 

655.5 

15.26 

7.634 

1.910 

35 

166.3 

60.13 

30.42 

t.695 

9 

637.3 

15.69 

7.852 

1.965 

36 

161.8 

61.80 

31.29 

7.919 

15 

620.1 

16.13 

8 070 

2.019 

37 

157.6 

63.45 

32.15 

8.142 

30 

603.8 

16.56 

8.288 

2.074 

38 

153.6 

65.10 

33.01 

8.366 


588.4 

17.00 

8.506 

2.128 

39 

149.8 

66.76 

33.87 

8.591 

10 

573.7 

17.43 

8.724 

2.183 

40 

146.2 

68.40 

34.73 

8.816 


For ordinates 5 ft apart, for Chords of 100 ft, see p 730. 

To find tangential and deflection angles for any given rad and 
chord. Divide half the chord by the rad. The quot will be nat sine of the tangl 
ang. Find this tangl ang in the table of nat sines; and mult it by 2 for the def ang. 

To find the def dist for chords 100 ft long. Div 10000 by the 
rad in feet. „ . . .. 

To find the def dist for equal chords of any given length. 
Div chord by rad. Mult quot by chord. Or div sq of chord by rad. 

To find the tangl dist for equal chords of any given length. 
First tind the tangl ang as above. Divide it by 2. Find in the table of nat sines 
the nat sine of the quot. Mult this nat sine by the given chord. Mult prod by 2. 

To find the rad to any given def ang. for equal chords of any 
length. Divide the def ang by 2. Find nat sine of the quotient. Divide hall 
the°chord by this nat sine. _ 


* 'Phe middle ordinate for a rad of 600 ft or more, (chord 100 ft,) may in 
practice be taken at one-fourth of the tang dist. Even in 400 ft rad it will be too short only 6 in the 
third decimal. 







































728 


CIRCULAR CURVES. 


Radii, Arc, of Curves; in metres. Chord, 20 metres — 2 

deka metres. 


The stakes, at the ends of the 2-dekanietre chords, should he numbered 2, 4, 6, Ac; 
meaning 2, 4, 6, Ac. dekamrtres. The tangential angle in the table will then give 
the amount of deflection per unit (dekametre) of measurement. 


6 

li 

a 

03 

e 

0) 

O 

Tangential 

angle. 

^Radius. 

Metres. 

W W 

e r. 0) 

*5 z 

0) 

® 8 

Q 

Tangl dist. 

Metres. 

Mid. ord. 

Metres. 

Deft angle. 

Tangential 

angle. 

Radius. 

Metres. 

Defl dist. 

Metres. 

Tangl dist. 
Metres. 

Mi<1. ord. 

Metres. 1 

0° 10' 

0° 5' 

6875.50 

.058 

.029 

.007 

8° 

0' 

4° 0' 

143.36 

2.790 

1.396 

.3*9 

20 

10 

3437.75 

.116 

.058 

.015 


10 

5 

140.44 

2.848 

1.425 

.356 

30 

15 

2291.84 

.175 

.087 

.022 


20 

10 

137.63 

2.906 

1.454 

.364 

40 

20 

1718.88 

.233 

.116 

.029 


30 

15 

134.94 

2.964 

1.483 

.371 

50 

25 

1375.11 

.291 

.145 

.036 


40 

20 

132.35 

3.022 

1.512 

.378 

1° 0' 

30 

1145.93 

.349 

.175 

.044 


50 

25 

129.85 

3.080 

1.541 

.386 

10 

35 

982.23 

.407 

.204 

.051 

9° 

0' 

30 

127.45 

3.138 

1.570 

.393 

20 

40 

859.46 

.465 

.233 

.058 


10 

35 

125.14 

3.196 

1.599 

.400 

30 

45 

763.97 

.524 

.262 

.065 


20 

40 

122.91 

3.254 

1.629 

.407 

40 

50 

687.57 

.582 

.291 

.073 


30 

45 

120.76 

3.312 

1.658 

.415 

50 

55 

625.07 

.640 

.320 

.080 


40 

50 

118.68 

3.370 

1.687 

.422 

2° 0' 

1° 0' 

572.99 

.698 

.349 

.087 


50 

55 

1 16.68 

3.428 

1.716 

.429 

10 

5 

528.92 

.756 

.378 

.095 

10° 

0' 

5° 0' 

114.74 

3.486 

1.745 

.437 

20 

10 

491.14 

.814 

.407 

.102 


20 

10 

111.05 

3.602 

1.803 

.451 

30 

15 

458.40 

.873 

.436 

.109 


40 

20 

107.58 

3718 

1.861 

.466 

40 

20 

429.76 

.931 

.465 

.116 

11° 

0' 

30 

104.33 

3.834 

1.919 

.4so 

50 

25 

401.48 

.989 

.494 

.124 


20 

40 

101.28 

3.950 

1.977 

.495 

3° 0' 

30 

382.02 

1.047 

.524 

.131 


40 

50 

98.39 

4.065 

2.035 

.509 

10 

35 

361.91 

1.105 

.553 

.138 

12° 

0' 

6° 0' 

95.67 

4.181 

2.093 

.524 

20 

40 

343.82 

1.163 

.582 

.145 


20 

10 

93.09 

4.297 

2.152 

.559 

30 

45 

327.46 

1.222 

.611 

.153 


40 

20 

90.65 

4.413 

2.210 

.553 

40 

50 

312.58 

1.280 

.640 

.160 

13° 

O' 

30 

88.34 

4.528 

2.268 

.568 

50 

55 

298.99 

1.338 

.669 

.167 


20 

40 

86.14 

4.644 

2.326 

.582 

4° O' 

2° 0' 

286.54 

1.396 

.698 

175 


40 

50 

84.05 

4.759 

2.384 

.597 

10 

5 

275.08 

1.454 

.727 

.182 

14° 

O' 

7° 0' 

82.06 

4.875 

2.442 

.612 

20 

10 

264.51 

1.512 

.756 

.189 


20 

10 

80.16 

4.990 

2.500 

.626 

30 

15 

254.71 

1.570 

.785 

.196 


40 

20 

78.34 

5.106 

2.558 

.641 

40 

20 

245.62 

1.629 

.814 

.204 

15° 

O' 

30 

76.61 

5.221 

2.616 

.655 

50 

25 

237.16 

1.687 

.844 

.211 


20 

40 

74.96 

5.336 

2.674 

.670 

5° 0' 

30 

229.26 

1.745 

.873 

.218 


40 

50 

73.37 

5.452 

2.732 

.685 

10 

35 

221.87 

1.803 

.902 

.225 

10° 

0' 

8° 0' 

71.85 

5.567 

2.790 

.699 

20 

40 

214.94 

1.861 

.931 

233 


to 

10 

70.40 5.682 

2.848 

.714 

30 

45 

208.43 

1.919 

.960 

.240 


40 

20 

69.00 5.797 

2.906 

.729 

40 

50 

202.30 

1.977 

.989 

.247 

17° 

0' 

30 

67.65 

5.912 

2.964 

.743 

50 

55 

196.53 

2.035 

1.018 

255 


20 

40 

66.36 

6 027 

3.022 

.758 

6° 0' 

3° O' 

191.07 

2.093 

1.047 

.262 


40 

50 

65.12 

6.142 

3.080 

.772 

10 

5 

185.91 

2.152 

1.076 

.269 

18° 

O' 

9° 0' 

63.92 

6.257 

3.138 

.7 87 

20 

10 

181.03 

2.210 

1.105 

.276 


20 

10 

62.77 

6.372 

3.196 

.802 

30 

15 

176.39 

2 268 

1.131 

.284 


40 

20 

61.66 

6.487 

3.254 

.816 

40 

20 

171.98 

2.326 

1.163 

.291 

19° 

0' 

30 

60.59 

6.602 

3.312 

.831 

50 

25 

167.79 

2.384 

1.192 

.298 


to 

40 

59.55 

6.717 

3.370 

.846 

7° 0' 

30 

163.80 

2.442 

1.222 

.306 


40 

50 

58.55 

6.831 

3.428 

.860 

10 

35 

160.00 

2500 

1.251 

.313 

20° 

0' 

10° 0' 

57.59 

6.946 

3.486 

.875 

20 

40 

156.37 

2.558 

1.280 

.320 

21° 

0' 

30 

54.87 

7.289 

3.C60 

.919 

30 

45 

152.90 

2.616 

1.309 

.827 

22° 

O' 

11° 0' 

52.41 

7.632 

3.834 

.963 

40 

50 

149.58 

2.674 

1.338 

.335 

23° 

0' 

30 

50.16 

7.975 4.008 

1.007 

50 

55 

146.10 

2.732 

1.367 

.342 

24° 

O' 

12° 0' 

48.10 

8.316 4.181 

1.051 







25° 

O' 

30 

46.20 

8.65814.355 

1.095 


Radius = 


Half the chord 


Sine of tangential angle 

~ .. ... Square of chord 

Deflection dist = - 


- = Half the chord v co8ecant of tangential 

angle. 


Twice the chord X B * ne °* tangential 

angle. 


, Radius an^ie. 

Tangential dist = Twice the chord X sine of half the tangential an^le 

Middle ord = Radius X (1 — cosine of tangential angle) = Half the chord X 
tangent of half the tangential angle. 

For curves of 6J metres, or greater, radius, the ordinate at 5 metres from 
the end of the 20-metre chord, or midway between the end of the chord and the mid¬ 
dle ordinate, may be taken at three-fourths of the middle ordinate. 


t 

twi 

I» 


til 

It 


I 











































TABLE OF LONG CHORDS. 


Table of Long Chords. 


1W 


Ang. 

of 

Defl. 

1 Sta. 

2 Sta. 

3 Sta. 

4 Sta. 

Ang. 

of 

Defl. 

1 Sta. 

2 Sta. 

3 Sta. 

4 Sta. 

1° 

100 

200.0 

300.0 

400.0 

% 

100 

199.7 

298.9 

397.5 

\A 

100 

200.0 

300.0 

399.9 

6° 

100 

199.7 

298.8 

397.3 

<x 

100 

200 0 

300.0 

399.9 


100 

199.7 

298.7 

397.0 


100 

200.0 

300.0 

399.8 

u 

100 

199.7 

298.6 

396.7 

/4 

2° 

100 

200.0 

299.9 

399.7 

% 

100 

199.6 

298.5 

396.5 

_ \/L 

100 

200.0 

299.9 

399.6 

7° 

100 

199.6 

298.4 

396.2 


100 

200.0 

299.8 

399.5 


100 

199.6 

298.3 

396.0 

Q 

100 

200.0 

299.8 

399.4 


100 

199.6 

298.2 

395.7 

3° 

100 

200.0 

299.7 

399.3 

% 

100 

199.6 

298.1 

395.4 


100 

200.0 

299.7 

399.2 

8° 

100 

199.6 

298.0 

395.1 


100 

200.0 

299.6 

399.1 

Y 

100 

199.5 

297.9 

394.8 

& 

100 

200.0 

299.6 

399.0 


100 

199.5 

297.8 

394.5 

/4 

4° 

100 

199.9 

299.6 

398.9 


100 

199.4 

297.7 

394.3 

1/ 

100 

199.9 

299.5 

398.7 

9° 

100 

199.4 

297.5 

394.1 


100 

199.9 

299.4 

398.5 


100 

199.4 

297.4 

393 7 

n 

100 

199 9 

299.3 

398.3 

E/ 

100 

199.3 

297.3 

393.2 

50^ 

100 

199.9 

299.2 

398.0 

% 

100 

199.2 

297.2 

39218 


100 

199 8 

299.1 

397.8 

10° 

100 

199.2 

297.0 

392.4 

*1 

100 

199.8 

299.0 

397.6 







111 Of O II It r roll Ill UU1VCO uiwivnwuii; — — —~ ' i 

f vpl in It npr sec X gauge in ins) -s- (Rad of curve in ft X 32.2). Experience 
as shown that half an incl for each degree of def angle (100 ft chords) does very 
;efl for 4 it 8.5 ins gauge up to 40 miles per hour. At 60 miles use 1 inch per deg. 
n dangerous places this may be increased for safety against high winds. Ap- 

. _ _ 1 • 4 \— Aiitnr -roil at. rflt.ft of 1 lDCll XU GO OI 


iroaching the curve raise the outer rail at the rate of 1 inch in about CO or 80 ft 
If the ends of the curves are tapered off with very easy “ t : ™ n a 8, £°" 

»f the rise may he made on said easier curves instead of on the tangent, or ail ot 

t if they are iong enough. 












































730 


TABLE OF ORDINATES, 


Table of Ordinates 5 ft apart. Chord 100 ft. 

For Railroad Curves. 


Ordinates for angles intermediate of those in the table can at once be found by 
simple proportion. 


Distances of the Ordinates from the end of the 100 feet Chord. 


Ang. of 
Defl. 

Mid. 
50 ft. 

45 ft. 

40 ft. 

35 ft. 

j 30 ft. 

25 ft. 

1 

1 20 ft. 

1 

j 15 ft. 

O ' 

4 

.014 

.014 

.014 

.013 

I 

! .012 

.010 

| 

• .008 

— 

.008 

8 

.029 

.029 

.028 

.026 

! .024 

.022 

.018 

.015 

12 

.043 

.043 

.041 

.038 

i .037 

.033 

.028 

.022 

10 

.058 

.058 

.056 

.052 

, .049 

.044 

.037 

.030 

20 

.073 

.072 

.070 

.086 

< .061 

.055 

.017 

.037 

24 

.087 

.086 

.083 

.077 

I .074 

.066 

.056 

.015 

28 

.102 

.101 

.098 

.092 

.086 

.077 

.065 

.052 

32 

.116 

.115 

.112 

.106 

.098 

.088 

.075 

.058 

36 

.131 

.130 

.126 

.119 

.110 

.099 

.084 

.066 

40 

.115 

.114 

.140 

.133 

.123 

.110 

.093 

.074 

44 

.160 

.158 

.153 

.145 

.135 

.121 

.103 

.081 

48 

.174 

.172 

.167 

.158 

.147 

.132 

.112 

.088 

52 

.189 

.187 

.181 

.171 

.159 

.143 

.122 

.095 

56 

.204 

.202 

.195 

.185 

.171 

.154 

.131 

.103 

i 

.218 

.216 

209 

.198 

.183 

.164 

.140 

111 

4 

.233 

.231 

.223 

.211 

.196 

.175 

.150 

.118 

8 

.247 

.245 

.237 

.224 

.208 

.186 

.159 

.125 

12 

.262 

.260 

.252 

.237 

.220 

.196 

.168 

.133 

16 

.276 

.274 

.265 

.251 

.232 

.207 

.177 

.140 

20 

.291 

.288 

.279 

.264 

.244 

.218 

.187 

. 148 

24 

.306 

.303 

.293 

.277 

.256 

.229 

.197 

.155 

28 

.320 

.317 

.307 

.291 

.269 

.240 

.206 

.163 

32 

.334 

.331 

.321 

.304 

.281 

.251 

.215 

.171 

36 

.349 

.345 

.335 

.317 

.293 

.262 

.224 

.178 

40 

.364 

.360 

.349 

.330 

.305 

.273 

.233 

.185 

44 

.378 

.374 

.363 

.343 

.318 

.284 

.242 

.192 

48 

.393 

.389 

.377 

.356 

.330 

.295 

.251 

.200 

52 

.407 

.403 

.391 

.370 

.342 

.305 

.261 

208 

56 

.422 

.418 

.405 

.883 

.354 

.316 

.270 

.215 

2 

.436 

.43*2 

.419 

.397 

.366 

.327 

.280 

.222 

4 

.451 

.446 

.433 

.409 

.379 

.338 

.289 

.230 

8 

.465 

.461 

.447 

.425 

.391 

.349 

.298 

237 

12 

.480 

.475 

.461 

.437 

.403 

.360 

.308 

.245 

16 

.495 

.490 

.475 

.450 

.415 

.371 

.317 

.252 

20 

.509 

.504 

.489 

.463 

.428 

.382 

.326 

.260 

24 

.523 

.518 

.503 

.476 

.440 

.393 

.334 

.267 

28 

.538 

.533 

.517 

.489 

.452 

.404 

.346 

.275 

32 

.552 

.547 

.531 

.503 

.465 

.415 


28*2 

36 

.567 

.562 

.545 

.516 

.477 

.425 

.364 

289 

40 

.582 

.576 

.559 

.529 

.489 

.436 

.373 

297 

44 

.596 

.590 

.573 

.542 

.501 

.447 

.382 

.304 

48 

.611 

.605 

.587 

.555 

.513 

.458 

.391 

312 

52 

56 

.625 

.640 

.619 

.634 

.601 

.615 

.569 

.582 

.526 

.538 

.469 

.480 

.401 

.410 

.319 

.326 

8 

.654 

.648 

.629 

.595 

.550 

.491 

.419 

.334 

4 

.669 

.662 

.643 

.608 

.562 

.502 

.428 

.341 

8 

.683 

.677 

.657 

.621 

.574 

.512 

.438 

349 

12 

.698 

.691 

.671 

.635 

.587 

.523 

.448 

.357 

16 

.713 

.705 

.685 

.649 

.599 

.534 

.457 

364 

20 

.727 

.720 

.699 

.662 

.611 

.545 

.466 

371 

24 

.742 

.734 

.713 

.675 

.623 

.556 


378 

28 

.766 

.749 

.727 

.688 

.635 

.567 


386 

32 

.771 

•763 

.741 

.702 

.648 

.578 

.494 

394 

36 

.786 

.777 

.755 

.715 

.600 

589 

.503 

401 

40 

44 

48 

.800 

.814 

.829 

.792 

.806 

.821 

.769 

.783 

.797 

.728 

.741 

.754 

.673 

.685 

.697 

.600 

.611 

.621 

.512 

•521 

.531 

I 408 

.415 

423 

52 

56 

4 

10 

20 

30 

40 

50 

.843 

.858 

.873 

.909 

.945 

.981 

1.017 

1.054 

.835 

.850 

.864 

.900 

.936 

.972 

1.008 

1.044 

.811 

.825 

.839 

.874 

.909 

.944 

.979 

1.014 

.768 

.781 

.794 

.827 

.860 

.893 

.926 

.959 

.709 

.721 

.734 

.764 

.795 

.825 

.855 

.886 

.632 

.643 

.655 

.682 

.709 

.736 

.764 

.791 

.541 

.550 

.559 

.582 

.606 

.629 

.652 

.431 

.438 

.445 

.461 

.482 

.501 

.519 

538 

5 

1.091 

1.080 

1.048 

.993 

.917 

.818 

.699 

557 

10 

1.127 

1.116 

1.083 

1.026 

.947 

.845 

779 , 


20 

1.164 

1.152 

1.118 

1.058 

.978 

.872 

.746 

594 

30 

1.200 

1.188 

1.153 

1.092 

1.009 

.900 

.769 

.613 


10 ft. 

5 ft. 

.005 

.003 

.010 

.005 

.015 

.008 

.020 

.011 

.026 

.014 

.031 

.017 

.036 

.019 

.042 

.022 

.047 

.024 

.052 

.027 

.057 

.030 

.062 

.033 

.068 

.035 

.073 

.038 

.078 

.041 

.083 

.043 

.088 

.046 

.094 

.049 

.099 

.052 

.104 

.055 

.109 

.057 

.114 

.060 

.120 

.063 

.125 

.066 

.130 

.069 

.135 

.072 

.141 

.075 

.147 

.077 

.152 

.080 

.157 

.083 

.162 

.086 

.167 

.088 

.173 

090 

.178 

.093 

.183 

.096 

.188 

.099 

.194 

.102 

.199 

.104 

.204 

.107 

.209 

.110 

.214 

.113 

.219 

.316 

.225 

.118 

.230 

.121 

.235 

.124 

.240 

.127 

.246 

.130 

.251 

.132 

.257 

.135 

.262 

.138 

.267 

.141 

.272 

.144 

.278 

.146 

.283 

.149 

.288 

.152 

.293 

.155 

.298 

.158 

.304 

.160 

.309 

.163 

.314 

.166 

.327 

.173 

.340 

.179 

.354 

.186 

.367 

.193 

.380 

.199 

.393 

.207 

.406 

.211 

.419 

.220 

.432 

.228 












































TABLE OF OBDINATES, 


731 


Table of Ordinates 5 ft apart. — (Continued.) 


Distances of the Ordinates from the end of the 100 feet Chord. 


Ang. of 
DeU. 

Mid. 

50 ft. 

45 ft. 

40 ft. 

35 ft. 

30 ft. 

25 ft. 

20 ft. 

15 ft. 

10 ft. 

5 ft. 

o - 

5 40 

1.236 

1.224 

1.188 

1.124 

1.039 

.927 

.792 

.631 

.445 

.235 

50 

1.273 

1.260 

1.223 

1.157 

1.070 

.954 

.816 

.649 

.458 

.241 

6 

1.309 

1.296 

1.258 

1.191 

1.100 

.982 

.839 

.668 

.472 

.24b 

10 

1.345 

1.332 

1.293 

1.224 

1.130 

1.009 

.862 

.686 

.485 

.255 

20 

1.382 

1.368 

1.328 

1.256 

1.161 

1.036 

.886 

.705 

.498 

.262 

SO 

1.419 

1.404 

1.362 

1.290 

1.192 

1.064 

.909 

.724 

.511 

.269 

40 

1.455 

1-440 

1.397 

1.323 

1.222 

1.091 

.932 

.742 

.524 

.276 

50 

1.491 

1.476 

1.432 

1.355 

1.253 

1.118 

.956 

.761 

.537 

.283 

7 

1.528 

1.512 

1.467 

1.389 

1.284 

1.146 

.979 

.779 

.551 

.290 

10 

1.564 

1.548 

1.502 

1.422 

1.314 

1.173 

1.002 

.798 

.564 

.297 

20 

1.600 

1.584 

1.537 

1.454 

1.345 

1.200 

1.026 

.816 

.576 

.304 

30 

1.637 

1.620 

1.572 

1.488 

1.375 

1.228 

1.048 

.835 

.590 

.311 

40 

1.073 

1.656 

1.607 

1.521 

1.405 

1.255 

1.071 

.854 

.603 

.318 

50 

1.710 

1.692 

1.641 

1.553 

1.436 

1.282 

1.095 

.872 

.616 

.324 

8 

1.746 

1.728 

1.677 

1.587 

1.467 

1.310 

1.118 

.891 

.629 

.332 

30 

1.855 

1.836 

1.782 

1.687 

1.559 

1.392 

1.188 

.946 

.669 

.353 

9 

1.965 

1.944 

1.886 

1.787 

1.651 

1.474 

1.258 

1.002 

.708 

.373 

30 

2.074 

2.052 

1.991 

1.887 

1.742 

1.556 

1.328 

1.057 

.748 

.394 

10 

2.183 

2.161 

2.096 

1.987 

1.834 

1.637 

1.398 

1.114 

.787 

.415 

30 

2.292 

2.269 

2.201 

2.087 

1.926 

1.719 

1.468 

1.170 

.827 

.436 

11 

2.401 

2.377 

2.306 

2.186 

2.018 

1.802 

1.538 

1.226 

.866 

.457 

30 

2.511 

2.486 

2.411 

2.286 

2.110 

1.884 

1.609 

1.282 

.906 

.478 

12 

2.620 

2.594 

2.516 

2.386 

2.203 

1.967 

1.680 

1.339 

.946 

.499 

30 

2.730 

2.703 

2.621 

2.485 

2.295 

2.049 

1.750 

1.395 

.985 

.520 

13 

2.839 

2.811 

2.726 

2.585 

2.387 

2.132 

1.820 

1.451 

1.025 

.541 

30 

2.919 

2.920 

2.832 

2.685 

2.479 

2.214 

1.891 

1.507 

1.065 

.562 

14 

3 058 

3.028 

2.937 

2.785 

2.571 

2.297 

1.961 

1.564 

1.105 

.583 

30 

3.168 

3.136 

3.042 

2.884 

2 6fJ*l 

2.379 

2.031 

1.620 

1.144 

.604 

15 

3.277 

3.245 

3.147 

2 984 

2.756 

2.462 

2.102 

1.676 

1.184 

.625 

30 

3.387 

3.354 

3.252 

3.084 

2.848 

2.544 

2.172 

1.732 

1.224 

.646 

16 

3 496 

3.462 

3.358 

3.184 

2.941 

2.627 

2.243 

1.789 

1.264 

.667 

17 

3.716 

3.680 

3.569 

3.384 

3.125 

2.792 

2.384 

1.902 

] .344 

.709 

18 

3.935 

3.897 

3.779 

3.584 

3.310 

2.958 

2.525 

2.014 

1.424 

.751 

19 

4.155 

4.115 

3.990 

3.784 

3.495 

3.123 

2.666 

2.127 

1.504 

.793 

20 

4.375 

4.332 

4.201 

3.984 

3.680 

3.288 

2.808 

2.240 

1.583 

.836 

22 

4.815 

4.768 

4.624 

4.386 

4.050 

3.620 

3 093 

2 467 

1744 

922 

24 

5.255 

5.204 

5.048 

4.789 

4.423 

3.952 

3.379 

2.695 

1.905 

1 008 

26 

5.697 

5.642 

5.473 

5.192 

4.798 

4.286 

3.665 

2.924 

2.068 

1.094 

28 

6.139 

6 079 

5.898 

5.595 

5.171 

4.622 

3.952 

3.154 

2.232 

1.181 

30 

6 582 

6.517 

6.323 

5.999 

5 544 

4.958 

4.239 

3.385 

2.396 

1.268 

32 

7-027 

6.957 

6.751 

6.406 

5.922 

5.297 

4.530 

3.619 

2.565 

1.356 

34 

7.472 

7.398 

7.179 

6.813 

6.300 

5.637 

4.822 

3.854 

2.733 

1.445 

36 

7.918 

7.841 

7.609 

7.222 

6.679 

5.978 

5.115 

4.090 

2.901 

1.535 

38 

8.367 

8.286 

8.041 

7.633 

7.060 

6.320 

5.410 

4.327 

3.069 

1.626 

40 

8.816 

8.731 

8.474 

8.044 

7.442 

6.663 

5.705 

4.565 

3.238 

1718 


For middle ordinates for bending rails, see p 761 












































732 


LEVEL CUTTINGS 


To prepare a Table. T, of Level Cuttings, for every ^ofa 
foot of height. or depth. For Tables of Level Cuttings, see pp 733 to 740. 
For cost of Earthwork, p 742, &c. 





i~ Ti / 


L 


a 


Let the fig represent the cutting; or. if inverted, 
the filling; in which the horizontal lines are sup¬ 
posed to be -jL. foot apart. First calculate the 
area in square feet, of the layer ab co, adjoining 
the roadway a b. Then find how many cubic 
yards that area gives in a distance of 100 feet. 
These cubic yards we will call Y; they form the 
first amount to be put into the Table T. 

Next calculate the area in square feet of the triangle a n o. Multiply this area by 4. Find how 
many cubic yards this increased area gives in a distance of 100 feet. Or they will be found ready 
calculated below. We will call them y. This is all the preparation that is needed before 

commencing the table. 

ExAUl.-Let the roadbed a b be 18 feet, and the side-slopes 1 to 1. Then for the area of ab c o 
since the side-slopes are ljsj to 1; and s t is .1 foot; c o must be 18.3 feet; and the mean length of 
ab c o must be 16.15 feet. Consequentlv, the area is 18.15 X -l— 1.815 square feet; which, in a 

181 5 

distance of 100 feet, gives 181.5 cubic feet; which is equal to —-— =6.7222 cubic yards ; or Y. 

Next, as to the triangle a n o: its height a n being .1 foot, and its base n o .15 feet; its area 
_ .1 X .15 _.015 

- 5 - — ——.0075 square ft. This multiplied by 4, gives .03 square feet; which, in a distance of 

L L g 

100 feet, gives .03 X 100 = 3 cubic feet: which is equal to -- = .1111 cubic yard; or y. 


Having thus found Y and y, proceed to make out the table in the manner following, which is so 
plain as to require no explanation. The work should be tested about every 5 feet, by calculating the 
area of the full depth arrived at; multiply it by 100, and divide the product by 27 for the cubic yards 
The cubic yards thus found should agree with the table. 


Y. 6.7222 . Y. 6.722 .1 

y - .1111 


6.8333 6.8333 i. able. i. 

y .1111 - 

- 13.5555 .2 - 

6.9444 6.9444 „ . , . 

y .HU - Cub. Yds. 

--- 20 5000 .3 * oet - _ 

7.0555 7.0555 , fi-9 v 

y .mi - i . ,^ 2Y - 

- 27.5555 .4 '* . i^.6 

7.1666 7.1666 ,6 . . ; 

y .. .1111 r . 2k6 

- 34 72*>2 5 . 34-7 

7 ., 77 - i'Lfzf * .6 . 42.0 

7.2; 71 ^2u7 Ac . 

42.0000 .6 ___ 

The follow ing table contains y. ready calculated for different side-slopes. It plainly 
remains the same for all widths of roadbed. J \ 


Table T. 


Height. 

Feet. 

Cub. Yds. 

.1 . 

6.72 Y. 

.2 . 

13.6 

.3 . 

20 5 

.4 . 

27.6 

.5 . 

34.7 

.6 .... 

42.0 

&c. 


Side-slope. 

y 

Side-slope. 

y 

K to i . 


1% to 1 . 


34 to l . 


2 to l .. .. 

1 JQ*> 

% to 1 . 


2]X to 1 .... 


1 to 1 . 


2% to 1 .. .. 

1 q*w 

to 1 . 


3 to 1 .. 

9999 

1to 1 . 


4 t.o 1 . 





















































































RAILROADS 


733 


* Table 1. Level Cuttings.* 

Roadway 14 feet wide, side-slopes 1% to 1. 


4 1 For single-track embankment. 


p 

111 

»! 

eight 
. Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

IK 


Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

hi 

0 


5.24 

10.6 

16.1 

21.6 

27.3 

33.1 

39 0 

45.0 

51.2 


1 1 

57.4 

63.8 

70.2 

76.8 

83.5 

90.3 

97.2 

104.2 

111.3 

118.6 

it 

2 

125 9 

133.4 

141.0 

148.6 

156.4 

164.4 

172.4 

180.5 

1S8.7 

197.1 


3 

205.6 

214.1 

222.8 

231.6 

240.5 

249.5 

258.7 

267.9 

277.3 

286.7 


4 

296.3 

306.0 

315.8 

325.7 

335.7 

345.8 

356.1 

366.4 

376.9 

387.5 

0 

5 

398.1 

408.9 

419.9 

430.9 

442.0 

453.2 

464.6 

476.1 

487.6 

499 3 

0 

6 

511.1 

523.0 

535.0 

547.2 

559.4 

571.8 

584.2 

596.8 

609.5 

622.3 

ill 

7 

635 2 

648.2 

661.3 

674.6 

687.9 

701.4 

714.9 

728.6 

742.4 

756.3 


8 

770.3 

784.5 

798.7 

813.1 

827.5 

842.1 

856.8 

871.6 

886.5 

901.5 


9 

916.7 

931.9 

947.3 

962.7 

978.3 

994.0 

1010 

1026 

1042 

1058 

W 

10 

1074 

1090 

1107 

1123 

1140 

1157 

1174 

1191 

1208 

1225 

ol 

11 

1243 

1260 

1278 

1295 

1313 

1331 

1349 

1367 

1385 

1404 


12 

1422 

1411 

1459 

1478 

1497 

1516 

1535 

1554 

1574 

1593 


13 

1613 

1633 

1652 

1672 

1692 

1712 

1733 

1753 

1773 

1794 

10 

14 

1815 

1835 

1856 

1877 

1898 

1920 

1941 

1962 

1984 

2006 

tie 

15 

2028 

2050 

2072 

2094 

2116 

2138 

2161 

2183 

2206 

2229 

jj 

16 

2252 

2275 

2298 

2321 

2344 

2368 

2391 

2415 

2439 

2463 


17 

2487 

2511 

2535 

2559 

2584 

2608 

2633 

2658 

2683 

2708 


18 

2733 

2759 

2784 

2809 

2835 

2861 

2886 

2912 

2938 

2964 


19 

2991 

3017 

3044 

3070 

3097 

3124 

3151 

3178 

3205 

3232 


20 

3259 

3287 

3314 

3312 

3370 

3398 

3426 

3454 

3482 

3510 


21 

3539 

3567 

3596 

3625 

3654 

3683 

3712 

3741 

3771 

3800 


22 

3830 

3859 

3889 

3919 

3949 

3979 

4009 

4040 

4070 

4101 


23 

4132 

4162 

4193 

4224 

4255 

4287 

4318 

4349 

4381 

4413 


24 

4144 

4176 

4508 

4541 

4573 

4605 

4638 

4670 

4703 

4736 


25 

4769 

4802 

4835 

4868 

4901 

4935 

4968 

5002 

5036 

5070 


26 

5104 

5138 

5172 

5206 

5241 

5275 

5310 

5345 

5380 

5415 


27 

5450 

5485 

5521 

5556 

5592 

5627 

5663 

5699 

5735 

5771 


28 

5807 

5844 

5880 

5917 

5953 

5990 

6027 

6064 

6101 

6139 


29 

6176 

6213 

6251 

6289 

6326 

6364 

6402 

6440 

6479 

6517 


30 

6556 

6594 

6633 

6672 

6711 

6750 

6789 

6828 

6867 

6907 


31 

6916 

6986 

7026 

7066 

7106 

7146 

7186 

7226 

7267 

7307 


32 

7318 

7389 

7430 

7471 

7512 

7553 

7595 

7636 

7678 

7719 


33 

7761 

7803 

7815 

7887 

7929 

7972 

8014 

8057 

8099 

8142 


34 

8185 

8228 

8271 

8315 

8358 

8401 

8445 

8489 

8532 

8576 


35 

8620 

8661 

8709 

8753 

8798 

8842 

8887 

8932 

8976 

9022 


36 

9067 

9112 

9157 

9203 

9248 

9294 

9340 

9386 

94.32 

9478 

ij 

37 

9524 

9570 

9617 

9663 

9710 

9757 

9804 

9851 

9*98 

9915 


38 

9993 

10040 

10088 

10135 

10183 

10231 

10279 

10327 

10375 

10124 


39 

10472 

10521 

10569 

10618 

10667 

10716 

10765 

10815 

10864 

10913 


40 

10963 

11013 

11062 

11112 

11162 

11212 

11263 

11313 

11364 

11414 


41 

11465 

11516 

11567 

11618 

11669 

11720 

11771 

11*23 

11874 

11926 


42 

11978 

12029 

12031 

12134 

12186 

12238 

12291 

12343 

12396 

12449 


43 

12502 

12555 

12608 

12661 

12715 

12768 

12822 

12875 

12929 

12983 


44 

13037 

13091 

13145 

13200 

13254 

13309 

13363 

13418 

13473 

13528 


45 

13583 

13639 

13694 

13749 

13805 

13861 

13916 

13972 

14028 

14084 


46 

14141 

11197 

14254 

14310 

14367 

14424 

14480 

14537 

14595 

14652 


47 

14709 

14767 

14824 

14882 

14940 

14998 

15056 

15114 

15172 

15230 


48 

15289 

15347 

15406 

15465 

15524 

15583 

15642 

15701 

15761 

15826 


49 

15880 

15939 

15999 

16059 

16119 

16179 

16239 

16300 

16360 

16421 


50 

16481 

16542 

16603 

16664 

16725 

16787 

16848 

16909 

16971 

17033 


51 

17094 

17156 

17218 

17280 

17343 

17405 

17467 

17530 

17593 

17656 


52 

17719 

17782 

17845 

17908 

17971 

18035 

18098 

18162 

18226 

18290 


53 

18351 

18418 

18482 

18546 

18611 

18675 

18740 

18805 

18*70 

18935 


54 

19000 

19065 

19131 

19196 

19262 

19327 

19393 

19459 

19525 

19591 


55 

19657 

19724 

19790 

19857 

19923 

19990 

20057 

20124 

20191 

20259 


56 

20326 

20393 

20461 

20529 

20596 

20664 

20732 

20800 

20869 

20937 


57 

21005 

21074 

21143 

21212 

21280 

21349 

21419 

21488 

21557 

21627 


58 

21696 

21766 

21836 

21906 

21976 

22046 

22116 

22186 

22257 

22327 


59 

22398 

22469 

22540 

22611 

22682 

22753 

22825 

22896 

22968 

23039 


60 

23111 

23183 

23255 

23327 

23399 

23472 

23544 

23617 

23689 

23762 


* From the Author’s “ Measurement and Cost of Earthwork.” 




































734 RAILROADS. 

Table 2 . Level Cutting?*. 


Roadway 24 feet wide, side-slopes 1J4 to 1. 
For double-track embankment. 


Height 
in Ft. 

,0 

.1 

O 

• <CJ 

.3 

.4 

.5 

6 

.7 

.8 

.9 


Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds 

0 


8.94 

18.0 

27.2 

36.4 

45.8 

55.3 

64.9 

74.7 

84. 

1 

94.4 

104.5 

114.7 

124.9 

135.3 

145.8 

156.4 

167.2 

178.0 

188. 

2 

200.0 

211.2 

222.4 

233.8 

245.3 

256.9 

268.6 

280.5 

292.4 

304. 

3 

316.6 

328.9 

341.2 

353.7 

366.3 

379.0 

391.9 

404 8 

417.8 

431.', 

4 

444.4 

457 8 

471.3 

484.9 

498.6 

512.4 

526.4 

640.4 

5546 

568 

5 

583.3 

597.8 

612.4 

627.1 

642.0 

6569 

671.9 

687.1 

702.3 

717. 

6 

733.3 

748.9 

764.7 

780.5 

796.4 

812.5 

828.7 

8449 

8d.3 

877. 

7 

894.4 

911.2 

928.0 

944.9 

962.0 

979.2 

996.4 

1014 

1031 

1049 

8 

1067 

1085 

1102 

1121 

1139 

1157 

1175 

1194 

1212 

1231 

9 

1250 

1269 

1288 

1307 

1326 

1346 

1365 

1385 

1405 

1425 

10 

1444 

1465 

1485 

1505 

1525 

1546 

1566 

1587 

1608 

1629 

11 

1650 

1671 

1692 

1714 

1735 

1757 

1779 

1800 

1822 

1845 

12 

1867 

1889 

1911 

1934 

1956 

1979 

2002 

2025 

2048 

2071 

13 

2094 

2118 

2141 

2165 

2189 

2213 

2236 

2261 

2285 

2309 

14 

2333 

2358 

2382 

2407 

2432 

2457 

2482 

2507 

2532 

2558 

15 

2583 

2009 

2635 

2661 

2686 

2713 

2739 

2765 

2791 

2818 

16 

2844 

2871 

2898 

2925 

2952 

2979 

3006 

3034 

3061 

3089 

17 

3117 

3145 

3172 

3201 

3229 

3257 

3285 

3314 

3342 

3371 

18 

3400 

3429 

3458 

3487 

3516 

3546 

3575 

3605 

3635 

3665 

19 

3f94 

3725 

3755 

3785 

3815 

3846 

3876 

3907 

3938 

3969 

20 

4000 

40:;l 

4062 

4094 

4125 

4157 

4189 

4221 

4252 

4285 

21 

4317 

4349 

4381 

4414 

4446 

4479 

4M2 

4545 

4578 

4611 

22 

4644 

4678 

4711 

4745 

4779 

4813 

4846 

4881 

4915 

4949 

23 

4983 

5018 

5052 

5087 

5122 

5157 

5192 

5227 

5262 

5298 

24 

5333 

5369 

5405 

5441 

5476 

5513 

5549 

5585 

5621 

5658 

25 

5694 

5731 

5768 

5805 

5842 

5879 

5916 

5954 

5991 

6029 

26 

6067 

6105 

6142 

6181 

6219 

6257 

6295 

6334 

6372 

6411 

27 

6450 

6489 

6528 

6567 

6606 

6646 

6(85 

6725 

6765 

6805 

28 

6844 

6885 

6925 

6965 

7005 

7046 

7086 

7127 

7168 

7209 

29 

7250 

7291 

7332 

7374 

7415 

7457 

7499 

7541 

7682 

7 6'’5 

30 

7667 

7709 

7751 

7794 

7836 

7879 

7922 

7965 

8008 

8051 [ 

31 

8094 

8138 

8181 

8225 

8269 

8313 

8356 

8401 

8445 

8489 

32 

8533 

8578 

8622 

8667 

8712 

8757 

8802 

8847 

8892 

8938 

33 

8983 

9029 

9075 

9121 

9166 

9212 

9259 

9305 

9351 

9398 

34 

9444 

9491 

9538 

9585 

9632 

9679 

9726 

9774 

9821 

9869 

35 

9917 

9965 

10012 

10061 

10109 

10157 

10205 

10254 

10302 

10351 

36 

10400 

10449 

10498 

10547 

10596 

10646 

10695 

10745 

10795 

1084c 

37 

10894 

10945 

10995 

11045 

11095 

11146 

11196 

11247 

11298 

11349 

38 

11400 

11451 

11502 

11554 

11605 

11657 

11709 

11761 

11812 

11865 

39 

11917 

11969 

12021 

12074 

12126 

12179 

12232 

12285 

12338 

12391 

40 

12444 

12498 

12551 

12605 

12659 

12713 

12766 

12821 

12875 

12929 

41 

12983 

13038 

13092 

13147 

13202 

13257 

13312 

13367 

13422 

13478 

42 

13533 

13689 

13645 

13701 

13756 

13813 

13869 

13925 

13981 

14038 

43 

14094 

14151 

14208 

14265 

14322 

14379 

14436 

14494 

14551 

14609 

44 

14667 

14725 

14782 

14840 

14899 

14957 

15015 

15074 

15132 

15191 

45 

15250 

15309 

15368 

15427 

15486 

15546 

15605 

15665 

15725 

15785 

46 

15844 

15905 

15965 

16025 

16085 

16146 

16206 

16267 

16328 

16389 

47 

16450 

16511 

16572 

16634 

16695 

16757 

16819 

16881 

16942 

17005 

48 

17067 

17129 

17191 

17254 

17316 

17379 

17442 

17505 

17568 

17631 

49 

17694 

17758 

17821 

17885 

17949 

18013 

18076 

18141 

18205 

18269 

50 

18333 

18398 

18462 

18527 

18592 

18657 

18722 

18787 

18852 

18918 

51 

18983 

19049 

19115 

19181 

19246 

19313 

19379 

19445 

19511 

19578 

52 

19644 

19711 

19778 

19845 

19912 

19979 

20046 

20114 

20181 

20249! 

53 

20317 

20385 

20452 

20521 

20589 

20657 

20725 

20794 

20862 

20931 

54 

21000 

21069 

21138 

21207 

21276 

21346 

21415 

21485 

21555 

21625 1 

55 

21694 

21765 

21835 

21905 

21975 

22046 

22116 

22187 

22258 

22329[ 

56 

22400 

22471 

22542 

22614 

22685 

22757 

22829 

22901 

22972 

23045 I 

57 

23117 

23189 

23261 

23334 

23406 

23479 

23552 

23625 

23698 

23771 

58 

23844 

23918 

23991 

24065 

24139 

24213 

24286 

24361 

24435 

24509 

59 

24583 

24658 

24732 

24807 

24882 

24957 

25032 

25107 

25182 

2525^ 

60 

25323 

25409 

25485 

25561 

25636 

25713 

25789 

25865 

25941 

26015 
__ 1 , 


For continuation to 100 feet, see Tabi.k 7. 






































RAILROADS, 


735 


Table 3. Isevel Cutting's. 

Roadway 18 feet wide, side-slopes 1 to 1. 


For single-track excavation, 


)epth 
u Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 


Cu. Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 


6.70 

13.5 

20 3 

27.3 

34.3 

41.3 

48.5 

55.7 

63.0 

1 

70.4 

77.8 

85.3 

92 9 

100.6 

108 3 

116 1 

124.0 

132.0 

140.0 

2 

148.1 

156.3 

164.6 

172.9 

181.3 

189.8 

198.4 

207.0 

215 7 

224.5 

3 

2333 

242.3 

251.3 

260.3 

269.5 

278.7 

288.0 

297.4 

306.8 

316.3 

4 

325.9 

335.6 

345.3 

355.1 

3650 

375.0 

385.0 

395.1 

405.3 

415.6 

5 

425 9 

436.3 

446.8 

457.4 

468.0 

478.7 

489.5 

500.3 

511.3 

522.3 

6 

533.3 

544.5 

555.7 

567.0 

578.4 

589.8 

601.3 

612.9 

624.6 

636.3 

7 

648.1 

660.0 

672.0 

684.0 

696.1 

708.3 

720.6 

732.9 

745.3 

757.8 

8 

770.4 

7830 

795.7 

808.5 

821.3 

834.3 

847.3 

860.3 

873.5 

886.7 

9 

900.0 

913.4 

926.8 

9403 

953.9 

967.6 

981.3 

995.1 

1009 

1023 

10 

1037 

1051 

1065 

1080 

1094 

1108 

1123 

1137 

1152 

1167 

11 

1181 

1196 

1211 

1226 

1241 

1256 

1272 

1287 

1302 

1318 

12 

1333 

1349 

1365 

1380 

1396 

1412 

1428 

1444 

1460 

1476 

13 

1493 

1509 

1525 

1542 

1558 

1575 

1592 

1608 

1625 

1642 

14 

1659 

1676 

1693 

1711 

1728 

1745 

1763 

1780 

1798 

1816 

15 

1833 

1851 

1869 

1887 

1905 

1923 

1941 

1960 

1978 

1996 

16 

2015 

2033 

2052 

2071 

2089 

2108 

2127 

2116 

2165 

2184 

17 

2204 

2223 

2242 

2262 

2281 

2301 

2321 

2340 

2360 

2380 

18 

2400 

2420 

2440 

2460 

2481 

2501 

2521 

2542 

2562 

2583 

19 

2604 

2624 

2645 

2666 

2687 

2708 

2729 

2751 

2772 

2793 

.20 

2815 

2836 

2858 

2880 

2901 

2923 

2945 

2967 

29 S9 

3011 

21 

3033 

3056 

3078 

3100 

3123 

3145 

3168 

3191 

3213 

3236 

22 

3259 

3282 

3305 

3328 

3352 

3375 

3398 

3422 

3445 

3469 

23 

3493 

3516 

3540 

3564 

3588 

3612 

3636 

3660 

3685 

3709 

24 

3733 

3758 

3782 

3807 

3832 

3856 

3881 

3906 

3931 

3956 

25 

3981 

4007 

4032 

4057 

4083 

4108 

4134 

4160 

4185 

4211 

26 

4237 

4263 

4289 

4315 

4341 

4368 

4394 

4420 

4417 

4473 

27 

4500 

4527 

4553 

4580 

4607 

4634 

4661 

4688 

4716 

4743 

28 

4770 

4798 

4825 

4853 

4881 

4908 

4936 

4964 

4992 

5020 

29 

5048 

5076 

5105 

5133 

5161 

5190 

5218 

5247 

5276 

5304 

L 30 

5333 

5362 

5391 

5420 

5449 

5479 

5508 

5537 

5567 

5596 

31 

5626 

5656 

5685 

5715 

5745 

5775 

5805 

5835 

5865 

5S96 

32 

5926 

5956 

5987 

6017 

6048 

6079 

6109 

6140 

6171 

6292 

33 

6233 

6264 

6296 

6327 

6358 

6390 

6421 

6453 

6485 

6516 

34 

6548 

6580 

6612 

6644 

6676 

6708 

6741 

6773 

6S05 

6838 

35 

6870 

6903 

6936 

6968 

7001 

7034 

7067 

7100 

7133 

7167 

36 

7200 

7233 

7267 

7300 

7334 

7368 

7401 

7435 

7469 

7503 

37 

7537 

7571 

7605 

7640 

7674 

7708 

7743 

7777 

7812 

7847 

38 

7881 

7916 

7951 

7986 

8021 

8056 

8092 

8127 

8162 

8198 

: 39 

8233 

8269 

8305 

8340 

8376 

8412 

8448 

8484 

8520 

8556 

40 

8593 

8629 

8665 

8702 

8738 

8775 

8812 

8848 

8885 

8922 

41 

8959 

8996 

9033 

9071 

9103 

9145 

91 S3 

9220 

9258 

9296 

42 

9333 

9371 

9409 

9447 

9485 

9523 

9561 

9600 

9638 

9676 

43 

9715 

9753 

9792 

9831 

9869 

9908 

9947 

9986 

10025 

10064 

44 

10104 

10143 

10182 

10222 

10261 

10301 

10341 

10380 

10420 

10460 

45 

10500 

10540 

10580 

10620 1 

10661 

10701 

10741 

10782 

10822 

10863 

46 

10904 

10944 

10985 

11026 

11067 

11108 

11149 

11191 

11232 

11273 

47 

11315 

11356 

11398 

11440 

11481 

11523 

11565 

11607 

11649 

11691 

48 

11733 

11776 

11818 

11860 

11903 

11945 

11988 

12031 

12073 

12116 

49 

12159 

12202 

12245 

12288 

12332 

12375 

12418 

12162 

12505 

12549 

50 

12593 

12636 

126S0 

12724 

12768 

12812 

12856 

12900 

12945 

12989 

51 

13033 

13078 

13122 

13167 

13212 

13256 

13301 

13346 

13391 

13436 

52 

13481 

13527 

13572 

13617 

13663 

13708 

13754 

13*00 

13845 

13891 

53 

13937 

13983 

14029 

14075 

14121 

14168 

14214 

14260 

14307 

14353 

54 

14400 

14447 

14493 

14540 

14587 

14634 

14681 

14728 

14776 

14823 

55 

14870 

14918 

14965 

15013 

15061 

15108 

15156 

15204 

15252 

15300 

56 

15348 

15396 

15445 

15493 

15541 

15590 

15638 

15687 

15736 

15784 

57 

15833 

15882 

15931 

15980 

16029 

16079 

16128 

17177 

16227 

16276 

58 

16326 

16376 

16425 

16175 

16525 

16575 

16625 

16675 

16725 

16776 

59 

16826 

16876 

16927 

16977 

17028 

17079 

17129 

17180 

17231 

17282 

60 

17333 

17384 

17436 

17487 

17538 

17590 

17641 

17693 

17745 

17796 


For continuation to 100 feet deep, see Table 7. 

















































736 


RAILROADS 


Table 4. Level Cuttings. 

Roadway 18 feet, side-slopes 1% to 1. 


For single-track excavation, 


Depth 
iu Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 


Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Y 

0 


6.72 

13.6 

20.5 

27.6 

34.7 

42.0 

49.4 

56.9 6 

1 

72.2 

80.1 

88.0 

96.1 

104.2 

112.5 

120.9 

129.4 

13S.0 

14 

2 

155.5 

164.5 

173.5 

182.7 

191.9 

201.3 

210.8 

220.4 

230.1 

24 

3 

249.9 

260.0 

270.1 

280.4 

290 8 

301.3 

311.9 

322.6 

333.4 i 34 

4 

355.5 

366.7 

37S.0 

3^9.4 

400.9 

412.5 

424.2 

436 0 

448.0, 46 

5 

472.2 

484.5 

496.9 

509.4 

522.0 

534.7 

547.6 

560.5 

573.6 

58 | 

6 

600.0 

613.4 

626.9 

640.5 

654.2 

668.1 

682.0 

696.1 

710.2 

72 

7 

7:38.9 

753.4 

768.0 

782.7 

797.6 

812.5 

827.6 

842.7 

858.0 

87 

8 

888.9 

904.5 

920.2 

936.1 

952.0 

968.1 

984.2 

1001 

1017 

103 

9 

1050 

1067 

1084 

1101 

1118 

1135 

1152 

1169 

1187 

120 

10 

1222 

1240 

1258 

1276 

1294 

1313 

1331 

1349 

1368 

138' 

11 

1406 

1425 

1444 

1463 

1482 

1501 

1521 

1541 

1560 

158 

12 

1600 

1620 

1640 

1661 

1681 

1701 

1722 

1743 

1764 

178 

13 

1806 

l s ‘27 

1848 

1869 

1891 

1913 

1934 

1956 

1978 

200 

11 

2022 

2045 

2067 

2039 

2112 

2135 

2158 

2181 

2204 

222 

15 

2250 

2273 

2297 

2321 

2344 

2368 

2392 

2416 

2440 

246 

16 

2489 

2513 

2538 

2563 

2588 

2613 

2638 

2663 

2688 

271 

17 

2739 

2765 

‘2790 

2816 

2842 

2868 

2894 

2921 

2947 

297 

18 

3000 

3027 

3054 

3081 

3108 

3135 

3162 

3189 

3217 

324 

19 

3272 

3300 

3328 

3356 

?3S4 

3413 

3441 

3469 

3498 

352 

20 

3556 

35S5 

3614 

3643 

3672 

3701 

3731 

3761 

3790 

382 

21 

3S50 

3880 

3910 

3941 

3971 

4001 

4032 

4063 

4094 

412 

22 

4156 

4187 

4218 

4249 

4281 

4313 

4344 

4376 

4408 

444 

23 

4472 

4505 

4537 

4569 

4G02 

4635 

4668 

4701 

4734 

476 

24 

4800 

4833 

4867 

4901 

4934 

4968 

5002 

5036 

5070 

510 

25 

5139 

5173 

52C8 

5243 

5278 

5313 

5348 

5383 

5418 

545 

26 

5489 

5525 

5560 

5596 

5632 

5668 

5704 

5741 

5777 

581 | 

27 

5850 

5887 

5924 

5961 

5998 

6035 

6072 

6109 

6147 

618 

28 

6222 

6260 

6298 

6336 

6374 

6413 

6451 

6489 

6528 

656 

29 

6606 

6645 

6684 

6723 

67 62 

6801 

6841 

6881 

6920 

696 

30 

7000 

7040 

7080 

7121 

7161 

7201 

7242 

7283 

7324 

736 

31 

7406 

7447 

7488 

7529 

7571 

7613 

7654 

7696 

7738 

778 

32 

7822 

7865 

7907 

7949 

7992 

8035 

8078 

8121 

8164 

820 


8250 

8293 

8337 

8381 

8424 

8468 

8512 

8556 

8600 

864 

34 

86S9 

8733 

8778 

8823 

8S68 

8913 

8958 

9003 

9048 

909 

35 

9139 

9185 

9230 

9276 

9322 

9368 

9414 

9461 

9507 

95J 

30 

9600 

9647 

9604 

9741 

9788 

9835 

9882 

9929 

9977 

lOOl 

37 

10072 

10120 

10168 

10216 

10264 

10313 

10361 

10409 

10458 

1050 

38 

10556 

10605 

10654 

10703 

10752 

10801 

10851 

10901 

10950 

1100 

39 

11050 

11100 

11150 

11200 

11251 

11301 

11352 

11403 

11454 

1150 

40 

11556 

11607 

11658 

11709 

11761 

11813 

11864 

11916 

11968 

1202 

41 

12072 

12125 

12177 

12229 

12282 

12335 

12388 

12441 

12494 

1254 

42 

12600 

12653 

12707 

12761 

12814 

12868 

12922 

12976 

13030 

1308 

43 

13139 

13193 

13248 

13303 

13358 

13413 

13468 

13523 

13578 

1363 

44 

136S9 

13745 

13800 • 

13856 

13912 

13968 

14024 

140S1 

14137 

1419 

45 

14250 

14307 

14364 

14421 

14478 

14535 

14592 

14649 

14707 

1476 

40 

14822 

14880 

14938 

14996 

15054 

15113 

15171 

15229 

152 s 8 

1534 

47 

15406 

154G5 

15524 

15583 

15642 

15701 

15761 

15821 

158S0 

ir,94 

48 

16000 

16060 

16120 

16181 

16241 

16301 

16362 

16423 

164S4 

1654 

49 

16600 

16667 

16728 

16789 

16851 

16913 

16974 

17036 

17098 

1716 

50 

17222 1 

17285 

17347 

17409 

17472 

17535 

17598 

17661 

17724 

177?. 

61 

17850 l 

17913 

17977 

18041 

18104 

18168 

18232 

18296 

18360 

1S4S 

62 

18489 

18553 

18618 

18683 

18748 

18813 

18878 

18943 

19008 

1907 

63 

19139 

19205 

19270 

19336 

19402 

19468 

19534 

19601 

19667 

197c 

64 

19800 

19867 

19934 

20000 

20068 

20135 

20202 

20269 

20337 

204( 

65 

20472 

20540 

20608 

20676 

20744 

20S13 

20881 

20949 

21018 

210£ 

66 

21156 

21225 

21294 

21363 

21432 

21501 

21571 

21641 

21710 

217 £ : 

67 

21850 

21920 

21990 

22061 

22131 

22201 

22272 

22343 

22414 

224 ; 

58 

22550 

22627 

22698 

22709 

22841 

22913 

22984 

23056 

23128 

232( 

59 

23272 

23345 

23417 

23489 

23562 

23635 

23708 

23781 

2385 4 

230* 

60 

24000 

24073 

24147 

24221 

24294 

24.368 

24442 

24516 

24590 

246t 1 

„_. . . __ ' —--- -— K 


For continuation to 100 feet deep, see Table 7. 


























































RAILROADS 


737 


Table 5. I<evel Cuttings. 

Roadway 28 feet wide, side-slopes 1 to 1. 
For double-track excavation. 


•epth 
a b't. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 


Cu. Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 


10.4 

20.9 

31.4 

42.1 

52.8 

63.6 

74.4 

85.3 

96.3 

1 

107.4 

118.6 

129.8 

141.1 

152.4 

163.9 

175.4 

187.0 

198.7 

210.4 

2 

222.2 

234.1 

246.1 

258.1 

270.2 

282.4 

294.7 

307.0 

319.4 

331.9 

3 

344.4 

357.1 

369.8 

3S2.6 

395.4 

408.3 

421.3 

434.4 

447.6 

460.8 

4 

474.1 

487.4 

500 9 

514.4 

528.0 

541.7 

555.4 

569.2 

583.1 

597.1 

5 

611.1 

625.2 

639.4 

653.7 

668.0 

682.4 

696.9 

711.4 

726.1 

740.8 

6 

755.6 

770.1 

785.4 

800.4 

815.5 

830.6 

845.8 

861.1 

876.5 

891.9 

7 

907.5 

923.0 

938.7 

951.5 

970.3 

986.2 

1002 

1018 

1034 

1050 

8 

1067 

1083 

1099 

1116 

1132 

1149 

1166 

1182 

1199 

1216 

9 

1233 

1250 

1267 

1285 

1302 

1319 

1337 

1354 

1372 

1390 

10 

1407 

1425 

1443 

1461 

1479 

1497 

1515 

1534 

1552 

1570 

11 

1589 

1607 

1626 

1645 

1664 

1682 

1701 

1720 

1739 

1759 

12 

1778 

1797 

1816 

1836 

1855 

1875 

1895 

1914 

1934 

1954 

i 13 

1974 

1994 

2014 

2034 

2055 

2075 

2095 

2116 

2136 

2157 

14 

2178 

2199 

2219 

2240 

2261 

2282 

2304 

2325 

2346 

2367 

i 15 

2389 

2410 

2432 

2454 

2475 

2497 

2519 

2541 

2563 

2585 

16 

2607 

2630 

2652 

2674 

2697 

2719 

2742 

2765 

2788 

2810 

17 

2833 

2856 

2879 

2903 

2926 

2949 

2972 

2996 

3019 

3043 

18 

3067 

3090 

3114 

3138 

3162 

3186 

3210 

3234 

3259 

3283 

19 

3307 

3332 

3356 

3381 

3406 

3431 

3455 

3480 

3505 

3530 

20 

3556 

3581 

3606 

3631 

3657 

3682 

3708 

3734 

3759 

3785 

21 

3811 

3837 

3863 

3889 

3915 

3942 

3968 

3994 

4021 

4047 

22 

4074 

4101 

4128 

4154 

4181 

4208 

4235 

4263 

4290 

4317 

23 

4344 

4372 

4399 

4427 

4455 

4482 

4510 

4538 

4566 

4594 

24 

4622 

4650 

4679 

4707 

4735 

4764 

4792 

4821 

4850 

4879 

25 

4907 

4936 

4965 

4994 

5024 

5053 

5082 

5111 

5141 

5170 

26 

5200 

5230 

5259 

5289 

5319 

5349 

5379 

5409 

5439 

5470 

27 

5500 

5530 

5561 

5591 

5622 

5653 

5684 

5714 

5745 

5776 

28 

5807 

5839 

5870 

5901 

5932 

5964 

5995 

6027 

6059 

6090 

29 

6122 

6154 

6186 

6218 

6250 

6282 

6315 

6347 

6379 

6412 

30 

6444 

6477 

6510 

6543 

6575 

6508 

6641 

6674 

6708 

6741 

31 

6774 

6807 

6841 

6874 

6908 

6942 

6975 

7009 

7043 

7077 

32 

7111 

7145 

7179 

7214 

7248 

7282 

7317 

7351 

73S6 

7421 

33 

7456 

7490 

7525 

7560 

7595 

7631 

7666 

7701 

7736 

7772 

31 

7807 

7843 

7879 

7914 

7950 

7988 

8022 

8058 

8094 

8130 

35 

8167 

8203 

8239 

8276 

8312 

8349 

8386 

8423 

8459 

8496 

36 

8533 

8570 

8608 

8645 

8682 

8719 

8757 

8794 

8832 

8870 

37 

8907 

8945 

8983 

9021 

9059 

9097 

9135 

9171 

9212 

9250 

38 

9289 

9327 

9366 

9405 

9444 

9482 

9521 

9560 

9599 

9639 

39 

9678 

9717 

9756 

9796 

9835 

9875 

9915 

9954 

9994 

10034 

40 

10074 

10114 

10154 

10194 

10235 

10275 

10315 

10356 

10396 

10437 

41 

10178 

10519 

10559 

10600 

10641 

10682 

10724 

10765 

10806 

10847 

42 

10889 

10930 

10972 

11014 

11055 

11097 

11139 

11181 

11223 

11265 

43 

11307 

11350 

11392 

11434 

11477 

11519 

11562 

11605 

11648 

11690 

41 

11733 

11776 

11819 

11863 

11906 

11949 

11992 

12036 

12079 

12123 

45 

12167 

12210 

12254 

12298 

12342 

12386 

12430 

12474 

12519 

12563 

46 

12607 

12652 

12696 

12741 

12788 

12831 

12875 

12920 

12965 

18010 

47 

13056 

13101 

13146 

13191 

13237 

13282 

13328 

13374 

13419 

13465 

48 

13511 

13587 

13603 

13649 

13695 

13742 

13788 

13834 

13881 

13927 

49 

13974 

14021 

14068 

14114 

14161 

14208 

14255 

14303 

14350 

14397 

50 

14444 

14492 

14539 

14587 

14635 

14682 

14730 

14778 

14826 

14874 

51 

14922 

11970 

15019 

15067 

15115 

15164 

15212 

15261 

15810 

15359 

52 

15407 

15450 

15505 

15554 

15604 

15653 

15702 

15751 

1580! 

15850 

63 

15900 

15950 

15999 

16049 

16099 

16149 

16199 

16249 

16299 

16350 

51 

16400 

16450 

16501 

16551 

16602 

16653 

16704 

16754 

16805 

1*6856 

55 

18907 

16959 

17010 

17061 

17112 

17164 

17215 

17267 

17819 

17370 

56 

17122 

17474 

17526 

17578 

17630 

17682 

17736 

17787 

17839 

17892 

57 

17911 

17997 

18050 

18103 

18155 

18208 

18261 

18314 

18368 

18421 

58 

18174 

18527 

18581 

18634 

IS'688 

18742 

18795 

18849 

18903 

18957 

59 

19011 

19065 

19119 

19174 

19228 

19282 

19337 

19391 

19146 

19501 

60 

19556 

19610 

19665 

19790 

19775 

19831 

19886 

19941 

19996 

20052 


For continuation to 100 feet, see Table 7. 































738 


RAILROADS 


Table 6. lievel C'uttlnffS. 

p 

Roadway 28 ft wide, side-slopes \)^ to 1. 

For double-track excavation. 

- ---;---:——b 


in Ft 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 ]i 


Hu. Yds. 

Cu.Yds. 

Du. Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds.! 

Cu.Yds. 

Cu.Yds. 

Cu.Ydsf 

0 


10.4 

21.0 

31.6 

42.4 

53.2 

64.2 

75 3 

86.5 


1 

109.3 

120.8 

132.5 

144.3 

156.1 

168.1 

180.2 

192.4 

204.8 

217.1 

2 

229.6 

242.3 

255.0 

267.9 

280.9 

294.0 

307.2 

320.5 

334.0 

347. 

3 

361.2 

374.9 

388.8 

402.8 

416.9 

431.1 

445.4 

459.9 

474.4 

489.; 

4 

503.7 

518.6 

533.6 

548 6 

563.9 

579.3 

594.7 

610.2 

625.8 

641.1 

5 

657.5 

673.4 

689.5 

705.7 

722.1 

738.5 

755 0 

771 7 

788.4 

805. 

6 

822.2 

839.3 

856.5 

873.8 

891.2 

908.8 1 

926.4 

944 2 

962.0 

980. 

7 

998.1 

1016 

1035 

1053 

1072 

1090 

1109 

1128 

1147 

1166 

8 

1185 

1204 

1224 

1243 

1263 

1283 

1303 

1322 

1343 

1363 

9 

1383 

1403 

1424 

1445 

1465 

1486 

1507 

1528 

1549 

1571 

10 

1592 

1614 

1635 

1657 

1679 

1701 

1723 

1745 

1767 

1790 

11 

1812 

1835 

1858 

1881 

1904 

1927 

1950 

1973 

1997 

2020 

12 

2044 

2068 

2092 

2116 

2140 

2164 

2189 

2213 

2238 

2262 

13 

2287 

2312 

2337 

2362 

2387 

2413 

2438 

2464 

2489 

2515 

14 

2541 

2567 

2593 

2619 

2645 

2672 

2698 

2725 

2752 

2779 

15 

2806 

2833 

2860 

2887 

2915 

2942 

2970 

2997 

3025 

3053 

16 

3081 

3109 

3138 

3166 

3195 

3223 

3252 

3281 

3310 

3339 

17 

3368 

3397 

3427 

3456 

3486 

3516 

3546 

3576 

3606 

3636 

18 

3667 

3697 

3728 

3758 

3789 

3820 

3851 

3882 

3913 

3944 

19 

3976 

4007 

4039 

4070 

4102 

4134 

4166 

4198 

4231 

4363 

20 

4296 

4328 

4361 

4394 

4427 

4460 

4493 

4527 

4560 

4594 

21 

4627 

4661 

4695 

4729 

4763 

4797 

4832 

4S66 

4900 

4935 1 

22 

4970 

5005 

5040 

6075 

5111 

5146 

5181 

6217 

5253 

5288 

23 

5324 

5360 

5396 

5432 

5469 

5505 

5542 

5578 

5615 

5652 i 

24 

5689 

5726 

5763 

6800 

5838 

5875 

5913 

5951 

5989 

6027 

25 

6065 

6103 

6141 

6179 

6218 

6257 

6295 

6334 

6373 

6412 M 

26 

6451 

6491 

6530 

6570 

66 < 9 

6649 

6689 

6729 

6769 

6809 1, 

27 

6S50 

6890 

6931 

6971 

7012 

7053 

7094 

7135 

7176 

7217 

28 

7259 

7300 

7342 

7384 

7426 

7468 

7510 

7552 

7594 

7637L 

29 

7680 

7722 

7765 

7808 

7851 

7894 

7937 

7981 

8024 

8067! 

30 

8111 

8155 

8199 

8243 

8287 

8331 

8375 

8420 

8464 

8509 i. 

31 

8554 

8598 

8643 

8688 

8734 

8779 

8824 

8870 

8915 

8961 1 

32 

9007 

9053 

9099 

9145 

9191 

9238 

9284 

9331 

9378 

9425], 

33 

9472 

9519 

9;' 66 

9613 

9061 

9708 

9756 

9804 

9851 

990<i 1 

34 

9948 

9997 

10045 

10093 

10142 

10190 

10239 

10188 

10337 

10386:, 

35 

10435 

10484 

10534 

10583 

10633 

10683 

10732 

10782 

10832 

1088- 1 

36 

10933 

10983 

11034 

11084 

11135 

11186 

11237 

11288 

11339 

11897 it 

37 

11443 

11494 

11546 

11598 

11649 

11701 

11753 

11806 

11858 

11910 ’ 

38 

11963 

12016 

12068 

12121 

12174 

12227 

12281 

12334 

12387 

12441 

39 

12494 

12548 

12602 

12656 

12710 

12764 

12819 

12873 

12928 

12982 

40 

13037 

13092 

13147 

13202 

13257 

13312 

13368 

13423 

13479 

13535 

41 

13591 

13647 

13703 

13759 

13S15 

13872 

13928 

13985 

14042 

14099 

42 

14156 

14213 

14270 

14327 

14385 

14442 

14500 

14558 

14615 

14672i- 

43 

14731 

14790 

14848 

14906 

14965 

15024 

15082 

15141 

15200 

15259' 

44 

15318 

15378 

15437 

15497 

15556 

15616 

15676 

15736 

15796 

15856 

45 

15917 

15977 

16038 

16098 

16159 

16220 

16281 

16342 

16403 

16465- 

46 

16526 

165S7 

16649 

16711 

16773 

16835 

16897 

16959 

17021 

17084!' 

47 

17146 

17209 

17272 

17335 

17398 

17461 

17524 

17*>8< 

17651 

17714 ' 

48 

17778 

17842 

17905 

17969 

18033 

18098 

18162 

18226 

18291 

18356| s 

49 

18420 

18485 

18550 

18615 

18680 

18746 

18811 

18877 

18942 

19008 i & 

50 

19074 

19140 

19206 

19272 

19339 

19405 

19472 

19538 

19605 

196)71 * 

51 

19739 

19806 

19873 

19940 

20008 

20075 

20143 

20211 

20279 

20347 1 

52 

20415 

20483 

20551 

20620 

20688 

20757 

20826 

20894 

20963 

21031 - 

53 

21102 

21171 

21211 

21310 

213*0 

21450 

21519 

21589 

21659 

21736 ! 

54 

21800 

21870 

21941 

22012 

22082 

22153 

22224 

22295 

22366 

22438 

55 

22509 

22581 

22662 

22724 

22796 

22868 

22940 

23012 

23085 

23157 - 

56 

23230 

23302 

23375 

23448 

23521 

23594 

23667 

23741 

23814 

2388! 

57 

23961 

24035 

24109 

24183 

24257 

24331 

24405 

24480 

24554 

24621 

58 

24704 

24779 

24854 

24929 

25004 

25079 

25155 

25230 

25306 

2638 

59 

25457 

25533 

25609 

25686 

25762 

25838 

25915 

25992 

26068 

2614! 

60 

26222 

26299 

26376 

26454 

26531 

26609 

26686 

26764 

26842 

26926 


For continuation to 100 feet, see Table 7. 




























































RAILROADS. 


730 


Table 7. Level Cutting’s. 


Continuation of the six foregoing Tables of Cubic Contents, to 100 feet of height or depth. 


ft 

^eight 
Depth 
t Feet. 

Table 

1 

Table 

o 

A 

Table 

3 

Table 

4 

Table 

3 

Table 

6 

\ 


Cu. Yds. 

Cu. Yds. 

Cu. Yds. 

Cu. Yds. 

Cu. Yds. 

Cu. Yds. 

7 

! 61 

23835 

26094 

17818 

24739 

20107 

26998 

1. 

.5 

21201 

26479 

18103 

25113 

203S6 

27390 


62 

21570 

26 367 

1S370 

25489 

20667 

27785 

■1 

*£) 

21912 

27257 

18C34 

25SG8 

20949 

' 28183 


63 

25317 

27650 

18900 

26250 

21231 

1 28583 

■i'i 

.5 

25691 

28016 

19168 

26635 

21519 

! 2S986 

,6 

61 

26071 

2S441 

19437 

27022 

21807 

29393 

8 

.5 

26157 

28816 

19708 

27413 

22097 

29S01 

'I 

65 

26813 

29250 

199S1 

27806 

22389 

30213 

10 

.5 

27231 

29637 

20256 

28201 

22682 

30627 

;o 

66 

27622 

30067 

20533 

2S600 

22978 

31041 

12 

.5 

28016 

30479 

20S12 

29001 

23275 

31161 

j 

67 

28113 

30894 

2 1 093 

29406 

23571 

31887 

9 

.5 

28S12 

31313 

21375 

29813 

23875 

32312 

)3 

68 

29215 

S1733 

21659 

30222 

21178 

32741 

!9 

.5 

29620 

32157 

21915 

30035 

21182 

33172 

ifi 

63 

30028 

325S3 

22233 

31050 

21789 

33605 

H 

•5 

30133 

33013 

22523 

31468 

25097 

34042 

53 

70 

30852 

33444 

22511 

31889 

25 07 

31181 

4 

.5 

31268 

33879 

23103 

33313 

25719 

31924 

)5 

71 

31687 

34317 

23401 

32739 

26031 

35369 

^8 

.5 

32108 

34757 

23701 

33168 

26 149 

35816 


72 

32533 

35200 

21000 

33600 

26607 

36267 

>7 

.5 

32960 

35646 

21301 

34035 

26986 

36720 

’2 

73 

33390 

36094 

21601 

34472 

27307 

37176 

0 

.5 

33823 

36516 

21907 

34913 

27631 

37635 

7 

71 

31259 

37000 

25214 

35356 

27956 

33096 

!' 

.5 

31697 

37457 

25522 

35501 

28282 

38561 


75 

35139 

37917 

25832 

36250 

28611 

39028 

■9 

.5 

35582 

38379 

26144 

36701 

28942 

39493 

,] 

i76 

36029 

38314 

26458 

37156 

29171 

39970 

' 

.5 

36.79 

39313 

26774 

37613 

29608 

40446 

Hi 

77 

36931 

39783 

27092 

38072 

29911 

40921 

5 


373S6 

40257 

27411 

38535 

30282 

41405 

" 

78 

37311 

40733 

27733 

39000 

30622 

41889 

if 5 

33305 

41213 

28056 

39168 

30964 

42375 

(i 

79 

38768 

41694 

28381 

39939 

31307 

42865 

| 

.5 

392 55 

42179 

28708 

40113 

31053 

43357 


80 

39704 

426G7 

29037 

40889 

32000 

43852 

i 

81 

40650 

43650 

29700 

41859 

32700 

41850 

; 

82 

41607 

44641 

30370 

42822 

33107 

45859 


83 

42576 

45650 

31048 

43S06 

34122 

46880 

5 

84 

43555 

46667 

31733 

41800 

34S14 

47911 

f 

85 

44546 

47694 

32126 

45S06 

35571 

■ 48954 


86 

45518 

48733 

33126 

46822 

36311 

50098 

1 

87 

46561 

49783 

33833 

47850 

37056 

51072 

I 

88 

47585 

60841 

34548 

48889 

37807 

52148 

5 

89 

48620 

51917 

35270 

49939 

38567 

63235 

9 

30 

49667 

53000 

36000 

51000 

39533 

54333 

) 

31 

50724 

54094 

36737 

52072 

40107 

55413 


32 

51793 

55200 

37481 

53156 

40889 

56563 

) 

33 

52872 

56317 

38233 

51250 

41678 

57694 

) 

31 

53963 

57411 

38993 

55356 

42474 

58837 


35 

55065 

58583 

39759 

56172 

43278 

59990 


36 

56178 

59733 

40533 

57600 

44089 

61155 


37 

57302 

60891 

41315 

58739 

41907 

62331 


38 

58437 

62067 

42101 

59889 

45733 

63518 


39 

59583 

63250 

42900 

61050 

46567 

64716 


)0 

60741 

61144 

43704 

62222 | 

47407 

65926 






51 












































740 


RAILROADS. 


Table 8, 

Of Cubic Yards in a 100-foot station of level cutting or filling, to be added to, or sul 
tracted from, the quantities in the preceding seven tables, in case the excav) 
tions or embankments should be increased or diminished 2 feet in width. 


Cubic Yards in a length of 100 feet; breadth 2 feet; and of different depths. 


Height or 
Depth 
in Feet. 

Cubic 

Yards. 

Height or 
Depth 
in Feet. 

Cubic 

Yards. 

Height or 
Depth 
in Feet. 

Cubic 

Yards. 

Height or 
Depth 
in Feet. 

Cubic 

Yards. 

Height or 
Depth 
in Feet. 

Cubic 

Yards 

A 

3.70 

.5 

152 

.5 

300 

.5 

448 

.5 

596 

1 

7.41 

21 

156 

41 

304 

61 

452 

81 

600 

A 

11.1 

.5 

159 

.5 

307 

.5 

456 

.5 

604 

2 

14.8 

22 

163 

42 

311 

62 

459 

82 

607 

A 

18.5 

.5 

167 

.5 

315 

.5 

463 

.5 

611 

3 

22.2 

23 

170 

43 

319 

63 

467 

83 

615 

A 

25.9 

.5 

174 

.5 

322 

.5 

470 

.5 

619 

4 

29.6 

24 

178 

44 

326 

64 

474 

84 

622 

A 

33.3 

.5 

181 

.5 

330 

.5 

478 

.5 

626 

5 

37.0 

25 

185 

45 

333 

65 

481 

85 

630 

.5 

40.7 

.5 

189 

.5 

337 

.5 

485 

.5 

633 

6 

44.4 

26 

193 

46 

341 

66 

489 

86 

637 

A 

48.1 

.5 

196 

.5 

344 

.5 

493 

.5 

641 

7 

51.9 

27 

200 

47 

348 

67 

496 

87 

644 

A 

55.6 

.5 

204 

.5 

352 

.5 

500 

.5 

648 

8 

59.3 

28 

207 

48 

356 

68 

504 

88 

652 

A 

63.0 

.5 

211 

.5 

359 

.5 

507 

.5 

656 

9 

66.7 

29 

215 

49 

363 

69 

511 

89 

659 

A 

70.4 

.5 

219 

.5 

367 

.5 

515 

.5 

663 

10 

74 l 

30 

222 

50 

370 

70 

519 

90 

667 

.5 

77.8 

.5 

226 

.5 

374 

.5 

522 

.5 

67 C 

11 

81.5 

31 

230 

51 

378 

71 

526 

91 

674 

A 

85 2 

.5 

233 

.5 

3S1 

.5 

530 

.5 

67 £ 

12 

88.9 

32 

237 

52 

385 

72 

533 

92 

681, 

A 

92.6 

.5 

241 

.5 

389 

.5 

537 

.5 

685 

13 

96.3 

33 

244 

63 

393 

73 

541 

93 

6SS 

A 

100 

.5 

248 

.5 

896 

.5 

544 

.5 

69» 

14 

104 

34 

252 

54 

400 

74 

548 

94 

69* 

A 

107 

.5 

256 

.5 

404 

.5 

552 

.5 

70( 

15 

111 

35 

259 

55 

407 

75 

556 

95 

70- 

Jd 

115 

.5 

263 

.5 

411 

A 

559 

.5 

70' 1 

16 

119 

36 

267 

56 

415 

76 

563 

96 

711 

A 

122 

.5 

270 

.5 

419 

.5 

567 

.5 

711 

17 

126 

87 

274 

57 

422 

77 

570 

97 

71( 

.5 

130 

.5 

278 

.5 

426 

.5 

574 

.5 

72' 

18 

133 

38 

281 

58 

430 

78 

578 

98 

72 i 

.5 

137 

.5 

285 

.5 

433 

.5 

581 

.5 

73- 

19 

141 

39 

289 

59 

437 

79 

585 

99 

73: 

.5 

144 

.5 

293 

.5 

441 

.5 

589 

.5 

73' 

20 

148 

40 

296 

60 

444 

80 

593 

100 

74 


Remark. The foregoing tables of level cuttings may also 
used for widths of roadway greater than those at the liea 
of the tables. Thus, suppose we wish to use Table 1, for a roadbed m n , l 1 
wide, instead of c b, which is only 14 ft. and for which the table was calculated, 
is only necessary first to find the vert dist s a, between these two roadbeds ; anr 
add it mentally to each height t s, of the given embkt, when taking out from 







































RAILROADS. 


741 


W.m£n f w yd8 , C ° rreSpon,lin " t0 the hei S hts - By this means we obtain 
th eml ’ kt c ! > °P’ t° r an y required dist. Next, from these contents 

,i! nufi C 0 l ' res h° ndlu g to the height s a, for the same dist. The remainder will 
uamly be the embkt m nop. 

KwStJn cn ' 1 su ,ffi c ! e ntly correct to take 5 n to the nearest tenth of a 
oot, winch will save trouble in adding it mentally to the heights in the tables. 

I f the roadbed is narrower than the table, as, for instance if mn be 
,he width m the table but we wish to find the contents for the width cb then 
irst find sa, and calculate the cubic yards in 100 feet length of cbmn. Then 
n aking out the cubic yards from the table, first subtract sa mentally from’ 
,ach height; and to the cubic yards taken out for each 100 feet, opposite this 
educed height, add the cubic yards in 100 feet of cbmn. 1 

To avoid trouble with contractors about the measurement of rock 
u ®, stipulate m the contract, either that it shall conform with the theoretical 
ross section; or that an extra allowance of say about 2 feet of width of cut 
vill be made, to cover the unavoidable irregularities of the sides. 

■ Shrinkage of Embankment. Although earth, when first dug and 
Hoosely thrown out, .swells about ^ part, so that a cubic yard in place averages 
•u?" 1 or 1 - 2 . c uhic yards when dug: or 1 cubic yard dug is equal to f, or to 
’ U 3 cub 10 y ard in place; yet when made into embankment it gradually 
41 ubsides settles, or shrinks, into a less bulk than it occupied before beinq duq. 

« lhe following are approximate averages of the shrinkage; or, in other words 
he earth measured in place in acut, will, when made into embankment.occupy 
. bulk less than before by about the following proportions: 


Gravel or sand.about 8 per ct; or 1 in 12% less. 


Clay .. 

Loam. “ 

Loose vegetable surface soil. “ 

Puddled clay. “ 


10 per ct; or 1 in 10 “ less. 
12 per ct; or 1 in 8% less. 
15 per ct; or 1 in less. 
25 per ct; or 1 in 4 less. 


The writer thinks, from some trials of his own, that 1 cubic yard of any hard 
in place, will make from 1% to 1% cubic yards of embankment; say on an 
tge 1.7 cubic yards. Or that 1 cubic yard of rock embankment requires 


iock 

verage ±., muuim j»iua. v^i mai. 1 tuuiu yarn oi rocK emoanxinent requires 
>882 of a cubic yard in place. He found that a solid cubic yard when broken 



Cubic 

Of which there were 

O’ | 

yards. 

Solid 

Voids 

1 n loose heap. 


52.6 per cent. 

47.4 per cent. 

] Carelessly piled. 


57 “ 

43 “ 

1 Carefully piled. 


63 “ 

37 “ 

. tubble, very carelessly scabbled. 

.. 1.5 

67 “ 

33 “ 

. tubble, somewhat carefully scabbled... 
1 

.. 1.25 

80 “ 

20 “ 


For trestles, see p 755. 

For culverts and stone bridges, see pp 693, Ac. 























742 


COST OF EARTHWORK 


COST OF EARTHWORK. 


Art. 1. It is advisable to pay for this kind of work by the cubic yard of excavation only ; in 
stead of allowing separate prices for excavation and embankment. By this means we get rid of tn 
difficulty of measurements, as well as the controversies and lawsuits which otten atteud the detei 
initiation of the allowance to be made for the settlement or subsidence of the embankments. 

It is, moreover, our opinion that justice to the contractor should lead to the L.llgl ISI1 B>ra<* 
tice of uayins the laborers by the cubic yard, instead of by the da> 
Experience fully proves that when laborers are scarce and wages high, men cun scarcely be depende 
upon to do three-fourths of the work which they readily accomplish when wages are low and whe 
fresh hands are waiting to be hired in case any are discharged. The contractor is thus placed at tl 
mercy of his men. The writer has known the most satisfactory results to attend a system or tasl j 
w ork, accompanied by liberal premiums for all overwork. By this means the interests of the , j ib erei 
are identilied with that or the contractor; and every man takes care that the others shall do the j 

fair share of the task. , , . ... I 

Ell wood Morris. C E, of Philadelphia, was, we believe, the firsUperson who properly investigate , 
the elements of cost of earthwork, and reduced them to such a form as to enable us to calculate tr 
total with a considerable degree of accuracy. He published his results in the Journal of the rrankli 
Institute in 1811. His paper forms the basis on which, with some variations, we shall consider tl 
matter; and on which we shall extend it to wheelbarrows, as well as to carts. Throughout this pap< 
we speak of a cubic yard considered only as solid in its place, or before it is loosened for removal, 
is scarcely necessary to add that the various items can of course only be regarded as tolerably clo 
approximations, or averages. As before stated, the men do less work when wages are high ; and mo 
when they are low. A great deal besides depends on the skill, observation, and energy of the coi 
tractor aud his superintendents. It is no unusual thing to see two contractors working at the san 
prices, in precisely similar material, where one is making money, and the other losing it, front a w a 
of tact in the proper distribution of his forces, keeping his roads in order, having his carts and ba 
rows well filled, Ac, Ac. Uncommonly long spells of wet weather may seriously affect the cost of ex 
cuting earthwork, by making it more difficult to loosen, load, or empty ; besides keeping the roads 
bad order for hauliug. 

The aggregate cost of excavating and removing earth is made up by the following items, namely 

1st. Loosening the earth ready for the shovellers. 

2d. Loading it by shovels into the carts or barrows 

3d. Hauling, or wheeling it away, including emptying and reluming. 

4th. Spreading it out into successive layers on the embankment. 

5th. Keeping the hauling-road for carts, or the plank gangways for barrows, in good order. 

6th. Wear, sharpening, depreciation, and interest on cost of tools. 

7th. Superintendence, and water-carriers. 

8th. Profit to the contractor. 

We will consider these items a little in detail, basing our calculations on the assumption that cot 
mon labor costs $1 per day. of 10 working hours. The results in our tables must therefore be 
creased or diminished in about the same proportion as common labor costs more or less than this. 

Arf. 2. liOosoniiiK tlie oarth ready for the shovellers. This 

generally done cither by ploughs or by picks ; more cheaply by the first. A plough with two hors 
and two men to manage them, at $1 per day for labor. 75 cents per day for each horse and 37 cet 
per day for plough, including harness, wear, repairs. Ac. or a total of $3 87, will loosen, of stro 
heavy soils, from 200 to 300 cubic yards a day, at from 1.93 to 1.29 cents per yard ; or of ordiur 
loam, from 400 to 600 cubic vards a day, at from .97 to .64 of a cent per yard. Therefore, as an o» 
nary average, we may assume the actual cost to the contractor for loosening by the plough, as • 
lows : strong heavy soils. 1.6 cents ; common loam, .8 cei t; light sandy soils, .4 cent. Very stiff pt 
slay, or obstinate cemented gravel, may be set down at 2 5 cents ; they require three or four horse 

Ry the pick, a fair day's work is about 14 yards of stiff pure clay, or or cemented gravel; 25 vat 
Of strong heavy soils; 40 yards of common loam; 60 yards of light sandy soils —all measured j 
place; which, at $1 per dav for labor, gives, for stiff clay, 7 cents; heavy soils, 4 cents: loam, { 
cents; light sandy soil, 1.666 cents. Pure sand requires but very little labor for loosening; .5 o , 
cent will cover it. I 

Art.S. Shovelling- the loosened earth into earts. The amm 

shovelled per day depends partly upon the weight of the material, but more upon so proportion! 
the number of pickers and of carts to that of shovellers, as not to keep the latter waiting for eitl 
material or carts. In fairlv regulated gangs, the shovellers into carts are not actually engaged i 
shovelling for more than six-tenths of their time, thus being unoccupied but four-tenths of it; wh ( 
under bad management, they lose considerably more than one-half of it. A shoveller can read i 
load into a cart one-third of a cubic yard measured in place (and which is an average working c; 
ioad), of sandy soil, in five minutes ;'of loam, in six minutes ; and of any of the heavy soils, in se' 
minutes. This would give, for a day of 10 working hours, 120 loads, or 40 cubic yards of light sat 
toil; 100 loads, or 33t£ cubic yards of loam ; or .86 loads, or 28.7 yards of the heavy soils. But fr ( 
these amounts we must deduct four-tenths for time necessarily lo'st: thus reducing the actual wr i 
ing quantities to 24 yards of light sandy soil, 20 yards of loam, 17.2 yards of the heavy soils. W 
the shovellers do less than this, there is some mismanagement. 

Assuming these as fair quantities, then, at $1 per day for labor, the actual cost to the contrat 
for shovelling per cubic yard measured in place, will be, for sandy soils, 4.167 cents; loam, 5 cet 
heavy soils, clays, Ac, 5.81 cents. 

In practice, the carts are not usually loaded to any less extent with the heavier soils than with 
lighter ones. Nor. indeed, is there any necessity for so doing, inasmuch as the difference of wei 
of a cart and one third of a cubic yard of the various soils is too slight to need any attention ; e 
cially when the cart-road is kept in good order, as it will be by any contractor who understands 












COST OF EARTHWORK. 


743 


own interest, 
may occur in 


5t ^necessary to modify the load on account of any slight inclinations which 
the grading of roads. An earth-cart weighs by itself about ^ a ton w tucn 

„ Art * ^ Hfii'lins away the earth; dumping, or einotvin"' a 

pe"hour e or“)0?LVpef mh lu ^ wMc)!*is IJuaT^ W way *'—* ^ 

as the Uistauce to whioh the. e.nrth _ «. I a, . / * 



lengths contained in the distance to which the earth has to be removed; that is, 

The number (600) of m inutes in a working day _ the number of trips, or loads 
4 the number of 100-feet lengths in the lead removed per day, per cart. 

:i . cu . b ‘ c .yard measured before being loosened, makes an average cart load, the num- 
thP l^r 33 ’ l , ! dud i , ;J ?' w ' 1 . 1 glve lhe number of cubic yards removed per day by each cart- and 
hauling 0 >ard3 dlvlded lnt0 the total ex P e use of a cart per day, will give the cost per cubic yard fur 

Remark. When removing loose rock, which requires more time for loading, say, 

No. o^minutes (600) in a working day = No _ of loads removed< 
it 6 -j- A r °- of 100-/eef lengths of lead per day, per cart. 

U In leads of ordinary length one driver can attend to 4 carts ; which, at $1 per day, is 25 cents per 
-1 c;irt - . When labor is $1 per day, the expense of a horse is usually about 75 cents; aud that of the 

* cart, including harness, tar, repairs, &c, 25 cents, making the total daily cost per cart $1.25. The 
expense of the horse is the same on Sundays and on rainy days, as when at work ; and this consid- 

* cation is included in the 75 cents. Some contractors employ a greater number of drivers who also 
if help to load the carts, so that the expense is about the same in either case. 

Example. How many cubic yards of loam, measured in the cut. can be hauled bv a horse and cart 
in a day of 10 working hours. (600 minutes,) the lead, or length of haul of earth being 1000 feet (or 
10 lengths of 100 feet.) and what will be the expense to the contractor for hauling per cubic yard 
assuming the total cost of cart, horse, and driver, at $1.25? J ’ 

TT 600 minutes 600 

Sere, I" . ' ™ . = — = 43 loads. 


4 10 lengths of 100 feet, 14 

. . 125 cents 

And 


43 loads 

And —-— — 14.3 cubic yards. 

O 


= 8.74 cents per cubic yard. 


14.3 cub yds 

In this manner the 2d and 3d columns of the following tables have been calculated. 

Art. 5. Spreading-, or levelling: off the earth into regular 
thin layers on the embankment, a bankman win spread from so to loocubic 
yards of either commou loam, or any of the heavier soils, clays, &c, depending on their dryness 
This, at $1 per day, is 1 to 2 cents per cubic yard; and we may assume cents as a fair averaee 
for such soils; while 1 cent will suffice for light sandy soils. b 

This expense for spreading is saved when the earth is either dumped over the end of the embank¬ 
ment, or is wasted; still, about % cent per yard should be allowed in either case for keenine the 
lumping-places clear and in order. * ° 

Art. 6. Keeping the cart-road in good order for hauling. 

, Vo ruts or puddles should be allowed to remain uufilled; rain should at once be led off by shallow 
litches ; and the road be carefully kept in good order; otherwise the labor of the horses, and the wear 
)f carts, will be very greatly increased. It is usual to allow so much per cubic yard for road repairs; 
but we suggest so much per cubic yard, per 100 feet of lead ; say yL 0 f a cent. 

Art. 7. Wear, sharpening, and depreciation of picks and 

shovels. Experience shows that about % of a cent per cubic yard will cover this item. 

Superintendence and water-carriers. These expenses win vary with 
ocal circumstances ; but we agree with Mr. Morris, that 1J^ cents per cubic yard will, under ordinary 
lircumstances. cover both of them. An allowance of about y A cent mav in justice be added for extra 
rouble in digging the side-ditches; levelling off the bottom of the cut to grade; and general trimming 
jp. In very light cuttings this may be increased to }4 cent per cubic yard. 

At }4 cent, all the items in this article amount to 2 cents per cubic yard of cut. 

Art. S. Profit to the cont ractor. This may generally be set down at from 6 to 
15 per cent, according to the magnitude of the work, the risks incurred, and various incidental cir- 
mmstances. Out of this item the contractor generally has to pay clerks, storekeepers, and other 
igents, as well as the expenses of shanties, &c ; although these are in most cases repaid by the profits 
i if the stores; and by the rates of boarding and lodging paid to the contractors by the laborers. 

Art. 9. A knowledge of the foregoing items enables ns to 
calculate with tolerable accuracy the cost of removing earth. 

f-’or example, let it be required to ascertain the cost per cubic yard of excavating cowmon loam, meas- 
-ired in place; and of removing it into embankment, with an average haul or lead of 1000 reel; the 
wages of laborers being $1 per day of 10 working hours; a horse 75 cts a day ; and a cart 25 cts. One 
driver to four carts. 


* When an entire cut is made into an embankment, the mean haul is the dist between centers 
>f gravity of the cut and embkt. 
















744 


COST OF EARTHWORK 


Cents. 

Here we have cost of loosening, sag l>y pick, Art 2, per cubic yard, say, 2.50 
Loading into carls, Art. 3, •• •* 5.00 

Hauling 1000 feet, as calculated previously in example, Art. 4, “ 874 

Spreading into layers. Art. b, “ 1.50 

Keeping cart-road in repair, Art. (i, 10 lengths of 100 ft, 1.00 

Various items in Art. 7, 2.00' 

_ 

Total cost to contractor, 20.74 

Add contractor’s profit, say 10 per cent, 2.074 


Total cost per cubic yard to the company, 22.814 
It is easy to construct a table like the following, of costs per cubic yard, for different leugths of leac 1 
Columns 2 and 3 are first obtained by the Kule in Article 4; then to each amount in coluniu 3 is addt 

the variable quautity of of a cent for every 100 feet length of lead, for keeping the road iu ordei 
aud the constant quantity (for any given kind of soil) composed of the prices per cubic yard, f< 
loosening, loading, spreading, or wasting, Ac, either taken from the preceding articles; or modifie 
to suit particular circumstances. In this manner the tables have been prepared. 


By Carts. Labor $1 per day, of 10 working: hours. 


A 

Z, & 

C O 
<•— 

<d a 

o JZ 
o _ 

cS ^3 

13 

© 


Cl 
C 0 
XZ o> 
xz 

tL «-* 

a 

0 


Feet. 


•2j3 
o 
or. CS 
'O O) 

a 

.2 « 

3 ^ 

3 Us 

O v 
Q. 
c 


a- 

3 


Cu.Yds. 


p 

to 

flfl 

"2 g. 
£8 
I'S 

•3 03 

§ 

a 

u. 

-•a 

CO 

o 

o 


Cts. 


25 

47.0 

50 

44.4 

75 

42.1 

100 

40.0 

150 

36.4 

200 

33.3 

300 

28.6 

400 

25 0 

500 

22.2 

600 

20.0 

700 

18.2 

800 

16.7 

900 

15.4 

1000 

14.3 

1100 

13.3 

1200 

12.5 

1300 

11.8 

1400 

11.1 

1500 

10.5 

1600 

10.0 

1700 

9.52 

1800 

9.09 

1900 

8.70 

2000 

8.33 

2250 

7 54 

2500 

6 90 

*4 mile 

6.58 

3000 

5.88 

8250 

5.48 

3500 

5.13 

3750 

4.82 

4000 

4.54 

4250 

4.30 

4500 

4.08 

4750 

3.88 

5000 

3.70 

1 mile 

3 52 

1% m. 

2.86 

1M m. 

2.40 

1 Vs m. 

2.07 

2 ID. 

1.82 


L 


2.66 
2.81 
2.97 
3.12 
3 43 
3.75 
4.37 
5.00 
5.63 
6.25 
6.87 
7.48 
812 
8.74 
9.40 
10.0 
10.6 

11.2 

11.9 

12.5 

13.1 

13.7 

14.4 
15.0 

16.6 

18.1 
19.0 

21.2 

22.8 

24.3 
25 9 

27.5 

29.1 

30.6 

32.2 

33.8 
85.5 

43.8 
52.1 

60.4 

68.7 


Common Loam. 


TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 


Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

n 

O) T3 
xz - 

a c i 

O 

Oh ** 

Cts. 

Cts. 

Cts. 

Cts. 

13.69 

12 44 

11.99 

10.74 

13.86 

12.61 

12.16 

10.91 

14.05 

12.80 

12.35 

11.10 

14.22 

12.97 

12.52 

11.27 

14.58 

13.83 

12.88 

11.63 

14.95 

13.70 

13.25 

12.00 

15.67 

14.42 

13.97 

12.72 

16.40 

15.15 

14.70 

13.45 

17.13 

15 88 

15.43 

14.18 

17.85 

16.60 

16.15 

14.90 

18.57 

17.32 

16.87 

15.62 

19.28 

18.03 

17.58 

16.33 

19.92 

18.67 

18.22 

16.97 

20.74 

19.49 

19.04 

17.79 

21.50 

20.25 

19.80 

18.55 

22.20 

20.95 

20.50 

19 25 

22.90 

21.65 

21.20 

19.95 

23.60 

22.35 

21.90 

20.65 

24.40 

23.15 

23.70 

21.45 

25.10 

23.85 

23.40 

22.15 

25.80 

24.55 

24.10 

22.85 

26.50 

25.25 

24.80 

23.55 

27.30 

26 05 

25.60 

24.35 

28.00 

26.75 

26.30 

25.05 

29.85 

28.60 

28.15 

26.90 

31.60 

30.35 

29 90 

28.65 

32.64 

31.39 

30.94 

29.69 

35.20 

33.95 

33.50 

32.25 

37.05 

35.80 

35.35 

34.10 

38.80 

37.55 

37.10 

35.85 

40.65 

39.40 

38.95 

37 70 

42.50 

41.25 

40.80 

39.55 

44.35 

43.10 

42.65 

41.40 

46.10 

44.85 

44.40 

43.15 

47.95 

46.70 

46.25 

45.00 

49.80 

48.55 

48.10 

46.85 

51.78 

50.53 

50.08 

48.83 

61.40 

60.15 

59.70 

58.45 

71.02 

69.77 

69.32 

68.07 

80.64 „ 

79.39 

78.94 

77.69 

90.26 " 

89.01 

88.56 

87.31 


Strong Heavy Soils. 


TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 


Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

Ploughed 

and 

Wasted. 

Cts. 

Cts. 

Cts. 

Cts. 

16.00 

14.75 

13.50 

12.25 

16.17 

14.92 

13.67 

12.42 

16.36 

15.11 

13.86 

12.61 

16.53 

15.28 

14.03 

12.78 

16.89 

15 64 

14.39 

13.14 

17.26 

16.01 

14.76 

13.51 

17.98 

16.73 

15.48 

14.23 

18.71 

17.46 

16.21 

14.96 

19.44 

18.19 

16.94 

1569 

20.16 

18.91 

17.66 

16.41 

20.88 

19.63 

18.38 

17.13 

21.59 

20.34 

19.09 

17.84 

22.23 

20.98 

19.73 

1H. 4 >- 

23.05 

21 80 

20.55 

19.30 

23.81 

22.56 

21.31 

20.06 

24.51 

23.26 

22.01 

20.76 

25.21 

23.96 

22.71 

21.46 

25.91 

24.66 

23.41 

22.16 

26.71 

25.46 

24.21 

22.96 

27.41 

26.16 

24 91 

23.66 

28.11 

26 86 

25 61 

24.36 

28.81 

27.56 

26.31 

25.06 

29.61 

28.36 

27.11 

25.86 

30.31 

29.06 

27.81 

26.56 

32.16 

30.91 

29.66 

28.41 

33.91 

32.66 

31.41 

30.16 

34.95 

33.70 

32.45 

31.20 

37.51 

36.26 

35.01 

33.76 

39.36 

38.11 

36.86 

35.61 

41.11 

39.86 

38.61 

37.36 

42.96 

41.71 

40.46 

39.21 

44.81 

43.oft 

42.31 

41.06 

46.66 

45.41 

44.16 

42.91 

48.41 

47.16 

45.91 

44.66 

50.26 

49.01 

47.76 

46.51 

52.11 

50 86 

49.61 

48.36 

54.09 

52.84 

51.59 

50.34 

63.71 

62.46 

61.21 

59.96 

73.33 

72.08 

70.83 

69.51 

82.95 

81.70 

80.45 

79.21 

92.57 

91.32 

90.07 

88.8! 














































COST OF EARTHWORK 


745 


By Carts. Labor $1 per day, of 10 working 1 hoars. 


lit 

In 

fo 

5tc 

’ 

Length of Lead, or distance to which 
the earth is hauled, iu feet. 

Number of cubic yards in place, 
hauled per day by eueh cart. 

| Cost per cubic yard in place, for 
hauling aud emptying only. 

Pure 

stiff Clay, or cementec 
Gravel. 

Light Sandy Soils. 

TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 

TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 

Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

aud 

Spread. 

Ploughed 

and 

Wasted. 

Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

Ploughed 

and 

Wasted. 


Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 


25 

47.0 

2.66 

19.00 

17.75 

14.50 

13.25 

11.52 

10.77 

10.25 

9 50 


50 

44.4 

2.81 

19.17 

17.92 

14.67 

13.42 

11.69 

10.94 

10.42 

9 67 


75 

42.1 

2.97 

19.36 

18.11 

14.86 

13.61 

11.88 

11.13 

10 61 

9 86 


100 

40.0 

3.12 

19.53 

18.28 

15.03 

13.78 

12.05 

11.30 

10.78 

10 03 

— 

150 

36.4 

3.43 

19.89 

18.64 

15.39 

14.14 

12.41 

11.66 

11.14 

10.39 


200 

33.3 

3.75 

20.26 

19.01 

15.76 

14.51 

12.78 

12.03 

11.51 

10 76 


300 

28.6 

4.37 

20.98 

19.73 

15.48 

15.23 

13.50 

12.75 

12.23 

11.48 


400 

25.0 

5.00 

21.71 

20.46 

17.21 

15.96 

14.23 

13.48 

12.46 


F 

500 

22.2 

5.63 

22.44 

21.19 

17.94 

16.69 

14.96 

14.21 

13.69 

12.94 


600 

20.0 

6.25 

23.16 

21.91 

18.66 

17.41 

15.68 

14.93 

14.41 


It* 

700 

18.2 

6.87 

23.88 

22.63 

19.38 

18.13 

16.40 


15.13 

14.38 


800 

16.7 

7.48 

24.59 

23.34 

20.09 

18.84 

17.11 

16.36 

15.84 

15 09 

— 

900 

15.4 

8.12 

25.23 

23.98 

20.73 

19.48 

17.75 

17.00 

16.48 



1000 

14.3 

8.74 

26.05 

24.80 

21.55 

20.30 

18.57 

17.82 

17.30 



1100 

13.3 

9.40 

26.81 

25.56 

22.31 

21.06 

19.33 

18.58 

18.06 

17.31 

2 

1200 

12.5 

10.0 

27.51 

26.26 

23.01 

21.76 

20.03 

19.28 

18.76 

18.01 


1300 

11.8 

10.6 

28.21 

26.96 

23.71 

22.46 

20.73 

19.98 

19.46 

18.71 


1400 

11.1 

11.2 

28.91 

27.66 

24.41 

23.16 

21.43 

20.68 

20.16 

19.41 


1500 

10.5 

11.9 

29.71 

28.46 

25.21 

23.96 

22.23 

21.48 

20.96 

20.21 

— 

1600 

10.0 

12.5 

30.41 

29.16 

25.91 

24.66 

22.93 

22.18 

21.66 

20.91 

. 

1700 

9.52 

13.1 

31.11 

29.86 

26.61 

25.36 

23.63 

22.88 

22.36 

21.61 

5 

1800 

9.09 

13.7 

31.81 

30.56 

27.31 

26.06 

24.33 

23.58 

23.06 

22.31 

2 

1900 

8.70 

14.4 

32.61 

31.36 

28.11 

26.86 

25.13 

24.38 

23.86 

23.11 

SI 

2000 

8.33 

15.0 

33.31 

32.06 

28.81 

27.56 

25.83 

25.08 

24.56 

23.81 

s 

2250 

7.54 

16.6 

35.16 

33.91 

30.66 

29.41 

27.68 

26.93 

26.41 


l 

2500 

6.90 

18.1 

36.91 

35.66 

32.41 

31.16 

29.43 

28.68 

28.16 

27.41 

1 

H mile 

6.58 

19.0 

37.95 

36.70 

33.45 

32.20 

30.47 

29.72 

29.20 

28.45 

s 

3000 

5.88 

21.2 

40.51 ' 

39.26 

36.01 

34.76 

33.03 

32.28 

31.76 

31.01 

% 

3250 

5.48 

22.8 

42.36 

41.11 

37.86 

36.61 

34.88 

34.13 

33.61 

32.86 

9 

3500 

5.13 

24.3 

44.11 

42.86 

39.61 

38.36 

36.63 

35.88 

35.36 

34.61 

1 

3750 

4.82 

25.9 

45.96 

44.71 

41.46 

40.21 

38.48 

37.73 

37.21 

36.46 

1 

4000 

4.54 

27.5 

47.8L 

46.56 

43.31 

42.06 

40.33 

39.58 

39.06 

38.31 

\ 

4250 

4.30 

29.1 

49.66 

48.41 

45.16 

43.91 

42.18 

41.45 

40.93 

40 18 

IS 

4500 

4.08 

30.6 

51.41 

50.16 

46.91 

45.66 

43.93 

43.18 

42.66 

41.91 

HI 1 ) 

4750 

3.88 

32.2 

53.26 

52.01 

48.76 

47.51 

45.78 

45.03 

44 51 

43 76 

:i 

5000 

3.70 

33.8 

55.11 

53.86 

50.61 

49.36 

47.63 

46.88 

46.36 

45.61 

4 

1 mile 

3.52 

35.5 

57.09 

55.84 

52.59 

51.34 

49.61 

48.86 

48 34 

47.59 

e 1 

Ya m. 

2.86 

43.8 

66.91 

65.46 

62.21 

60.90 

59.23 

58.48 

57.96 

57.21 

> m. 

2.4CT 

52.1 

76.33 

75.08 

71.83 

70.58 

68.85 

68.10 

67.58 

66.83 

■ IH m. 

2.07 

60.4 

85.95 

84.70 

81.45 

80.20 

78.47 

77.72 

77.20 ! 

76.45 

S 

s 

1 ID. 

1.82 j 

68.7 

95.57 

94.32 

91.07 

89.82 

88.09 

87.34 

86.82 

86.07 


, Art. 10. By wheelbarrows. The cost by barrows may be estimated in the same 
Jj inner by carts. See Articles 1, &c. Men in wheeling move at about the same average rate as 
rses do in hauling, that is, 2% miles an hour, or 200 feet per minute, or 1 minute per every 100-feel 
igth of lead. The time occupied in loading, emptying, &c (when, as is usual, the wheeler loads his 
; n barrow,) is about 1.25 minutes, without regard to length of lead; besides which, the time lost in 
casional short rests, in adjusting the wheeling plank, and in other incidental causes, amounts to 
ou * Yjy part of his whole time; so that we must in practice consider him as actually working hut 
aours out of his 10 working ones. Therefore 

The nu mber of minute .s in a working day X .9 <ft, number of trips or of loads 
1.25 -f- the number of 100 feet lengths of l-ead removed per day per harrow. 

See Remark, next page. 

The number of loads divided by 14 will give the number of cub yards, since a cub yard, measnred 
place, averages about 14 loads. And the cost of a wheeler and'barrow per day, (sav $1 per man. 
d 5 cents per barrow,) divided by the number of cub yards, vrill give the cost per yard for loading 
leeling, and emptying. 





















































746 


COST OF EARTHWORK. 


Ex. How many cubic yards of common loam, measured in place, will one man load, wbee 
and empty, per day of 10 working hours, (or 600 minutes;) the lead, or distance to which the earth 
removed being 1000 feet, (or 10lengths of 100 feet;) and what will be the expense per yard, supposit 
the laborer and barrow to cost $1.05 per day 1 


Here, 
48 


600 minutes X -9 540 , . 

-——;—rr-i-r- — r: — . — 48 trips, or loads per day. 

1.25 + 10 lengths 11.25 


And 3.43 cub yds per day. 


And 


105 cents 


~ 30.6 cents 


3.43 cub yds 

per cub yard for loading, wheeling away, emptying, and returning. This would be increased almo 
Inappreciably by the cost of the shovel, which, in the following tables, however, is included in tl 
cost of tools. 

Rem. for rock, which requires more time for loading, say 

No of minutes i n a working day X -9 _ No of loads removed 
1.6 -f- No of 100 feet lengths of lead ~ per day, per barroie. 

Art. II. The following tables are calculated as in the case of carts, by first finding columns 
aud 3 by means of the Rule in Art 10.and then adding to each sum in column 3, the variable quauti 
of .1 of a cent per cubic yard per 100 feet of lead for keeping the wheeling-planks in order ; and t 
prices of loosening, spreading, superintendence, water-carrying, &c, per cubic yard, as given iu t 
preceding Articles 2 to 1. 


By Wheelbarrows. Labor $1 per day, of 10 working lionr 


Length of Lend, or distance to which 
the earth is wheeled, in feet. 

Number of cubic yards In place, 
loaded, ond wheeled per day; 
each barrow. 

Cost per cubio yard In place, for 
loading, wheeling, and emptying, 

1 

Common Loam. 

Strong, Heavy Soils. 

TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OP 

PROPIT TO CONTRACTOR, 

TOTAL COST PER CUBIC 

YARD, EXCLUSIVE OF 

PROFIT TO CONTRACTOR. 

Picked 

and 

Spread. 

Picked 

and 

Wasted, 

Ploughed 

and 

Spread. 

Ploughed 

and 

Wasted. 

Picked 

and 

Spread. 

Q3 O 

J* •o 

« £ 3 

Ploughed 

and 

Spread. 

% *5 

5 C d 

Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

25.7 

4.09 

10.12 

8.87 

8.42 

7.17 

11.62 

10.37 

9.12 

7.87 

50 

22.1 

4.75 

10.80 

9.55 

9.10 

7.85 

12.30 

11.05 

9.80 

8.55 

75 

19.3 

5.44 

11.52 

10.27 

9.82 

8.57, 

13.02 

11 77 

10.52 

9.27 

100 

17.1 

6.14 

12.24 

10.99 

10.54 

9.29 

13.74 

12.49 

11.24 

9.99 

150 

14.0 

7.50 

13.65 

12.40 

11.95 

10.70 

15.15 

13.90 

12.65 

1 1.40 

200 

11.9 

8.82 

15.02 

13.77 

13.32 

12.07 

16.52 

15.27 

14 02 

12.77 

250 

10.3 

10.2 

16.45 

15.20 

14.75 

13.50 

17.95 

16.70 

15 45 

14.20 

300 

9.07 

11.6 

17.90 

16.65 

16.20 

14.95 

19.40 

18.15 

16.90 

15 6J 

350 

8.14 

12.9 

19.25 

18.00 

17.55 

16.30 

20.75 

19.50 

18 25 

17.01 

400 

7.36 

14.3 

20.70 

19.45 

19.00 

17.75 

22.20 

20.95 

19.70 

18.45 

450 

6.71 

15.6 

22.05 

20.80 

20.35 

19.10 

23.55 

22.30 

21.05 

19.80 

500 

6.17 

17.0 

23.50 

22.25 

21.80 

20.55 

25.00 

23.75 

22.50 

21.25 

600 

5.32 

19.7 

26.30 

25.05 

24.60 

23.35 

27.80 

26.55 

25.30 

24.05 

100 

4.67 

22.5 

29.20 

27.95 

27.50 

26.25 

30.70 

29.45 

28.20 

26.95 

800 

4.17 

25.2 

32.00 

30.75 

30.30 

29.05 

33.50 

32 25 

31.00 

29.75 

900 

3.76 

27.9 

34.80 

33.55 

33.10 

31.85 

36.30 

35.05 

33.80 

32.55 

1000 

3.43 

30.6 

37.60 

36.35 

35.90 

34.65 

39.10 

37.85 

36.60 

35.35 

1200 

2.91 

36.1 

43.30 

42.05 

41.60 

40.35 

44.80 

43.55 

42.30 

41.05 

1400 

2.53 

41.5 

48.90 

47.65 

47.20 

45.95 

50.40 

49.15 

47.90 

46.65 

1600 

2.24 

46.9 

54.50 

53.45 

52.80 

51.55 

56.00 

54.75 

53.50 


1800 

2.00 

52.5 

60.30 

59.05 

58.60 

57.35 

61.80 

60.55 

59.30 

58 05 

2000 

1.81 

58.0 

66 00 

64.75 

64.30 

63.05 

67.50 

60.25 

65.00 

63.75 

2200 

1.66 

63.3 

71.50 

70.25 

69.80 

68.55 

73.00 

71.75 

70.50 

25 

2400 

1.53 

68 6 

77.00 

75.75 

75.30 

74.05 

78.50 

77.25 

76.00 

74.75 

$$ mile. 

1.39 

75.5 

84.14 

82.89 

82.44 

81.19 

85.64 

84.39 

83.14 

81.89 

; 

—- 1 


* 









































COST OF EARTHWORK 


747 


By Wheelbarrows. Labor $1 per day, of 10 working- hoars. 


u . 
o 73 
£ V 

is 

■o s 


•c 

a 


8>> 

£4 

a 


a a) 

— a. 

« —r 

73 ^3 
u 

53 4) 
-3 

.2 * 
.O 


O ^ 
«- £ 

<V ?C* 

8 s- 

a ^3 

Z * 

*D 

t-* Wfi 

83 5f 

o U 


Pure Stiff Clay, or Ce¬ 
mented Gravel. 


TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 


Length of I. 
which the 

Number of c 
loaded, an 
each barroi 

o ^ 

u % 
a 

- o3 

CO o 
o — 

o 

Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

Ploughed 

and 

Wasted. 

Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

25.7 

4.09 

14.62 

13.37 

10.12 

8.87 

8.79 

8.04 

50 

22.1 

4.75 

15.30 

14.05 

10.80 

9.55 

9.47 

8.72 

75 

19.3 

5 44 

16.02 

14.77 

11.52 

10.27 

10.19 

9.44 

100 

17.1 

6.14 

16.74 

15.49 

12.24 

10.99 

10.91 

10.16 

150 

14.0 

7.50 

18.15 

16.90 

13.65 

12.40 

12.32 

11.57 

200 

11.9 

8.82 

79.52 

18.27 

15.02 

13.77 

13.69 

12.94 

250 

10.3 

10.2 

20.95 

19.70 

16.45 

15.20 

15.12 

14.37 

300 

9.07 

11.6 

22.40 

21.15 

17.90 

16.65 

16.57 

15.82 

350 

8.14 

12.9 

23.75 

22.50 

19.25 

18.00 

17.92 

17.17 

400 

7.36 

14.3 

25.20 

23.95 

•>0 70 

19.45 

19.37 

18.62 

450 

6.71 

15.6 

26.55 

25.30 

22.05 

20.80 

20.72 

19.97 

500 

6.17 

17.0 

28.00 

26.75 

23.50 

22.25 

22.17 

21.42 

000 

5.32 

19.7 

30.80 

29.55 

26.30 

25.05 

24.97 

24.22 

700 

4.67 

22.5 

33.70 

32.45 

29.20 

27.95 

27.87 

27.12 

800 

4.17 

25.2 

36.50 

35.25 

32.00 

30.75 

30.67 

29.92 

yoo 

3.76 

27.9 

39.30 

38.05 

34.80 

33.55 

33.47 

32.72 

]000 

3.43 

30.6 

42.10 

40 85 

37.60 

36*45 

36.27 

35.52 

1200 

2.91 

36.1 

47.80 

46.55 

43.30 

42 05 

41.97 

41.22 

1400 

2.53 

41.5 

53.40 

52.15 

48.90 

47.65 

47 57 

46.82 

1000 

2.24 

46.9 

59.00 

57.75 

54.50 

53.25 

53.17 

52.42 

IK >0 

2.00 

52.5 

64 80 

63.55 

60.30 

59.05 

58.97 

58.22 

•>000 

1.81 

58.0 

70.50 

69.26 

66.00 

64.75 

64.67 

63.92 

2 >00 

1.66 

63.3 

76.00 

74.75 

71.50 

70.25 

70 17 

69.42 

>400 


68.6 

81.50 

80.25 

77.00 

75.75 

75.67 

74.92 

mile. 

1.39 

75.5 

88.64 

87.39 

84.14 

82.89 

82.81 

82.06 


Light Sandy Soils. 


TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 


Ploughed 

aud 

Spread. 

Ploughed 

and 

Wasted. 

Cts. 

Cts. 

7.52 

6.77 

8.20 

7.45 

8.92 

8.17 

9.64 

8.89 

11.05 

10.30 

12.42 

11.67 

13.85 

13.10 

15.30 

14.55 

16.65 

15.90 

18.10 

17.35 

19.45 

18.70 

20.90 

20.15 

23.70 

22.95 

26.60 

25.85 

29.40 

28.65 

32.20 

31.45 

35.00 

34.25 

40.70 

39.90 

46 30 

45.55 

51.90 

51.15 

57.70 

66.95 

63.40 

62.65 

68.90 

68.15 

74.40 

73.65 

81.54 

80.79 


Art. 12. By wheeled scrapers and drag scrapers. The body 
of the wheeled scraper is a box of smooth sheet-steel about 3% ft square by lo ins 
fleen containing about 14 cubic yard of earth when ‘even full. lhe box is open 
in /font (in some machines it is closed by an “ end gate ” when full), and can be raised 
land lowered, and revolved on a horizontal axis. To fill the box it is lowered into 
and held down in, the earth, while the team draws the machine forward. When full, 
it is raised to about a foot above ground ; and, on reaching the dump, is unloaded by 
being overturned on its axis. All the movements of the box are made by means ot 
levers and without stopping the team, which thus travels constantly. The wheels 

have broad tires, to prevent them from cutting into the ground. . 

In the drag scraper the box, owing to the greater resistance to traction, is made 
much smaller. It contains about .15 to .25 cubic yard in place and is always open in 
front The operation of the drag scraper is similar to that of the wheeled scraper, 
except that the box, when filled, rests upon the ground and is dragged over it by the 

te< Each scraper (“wheeled” or “drag”) requires the constant use of a team of two 
horses with a driver. Besides, a number of men, depending on the shortness of the 
lead and the number of scrapers, are required in the pit and at the dump to load the 
scraners (by holding the box down into the earth) and unload them (by tipping the 
box). Except in sand, or in very soft soil, it is economical to use a plow before 

8C Theseverest work for the team is the filling of the box ; and this occurs oftenest 
where the lead is shortest. Hence smaller scrapers are used on short than on long 
hauls. We base our calculations on the following loads: 

For drag scrapers (used only on short hauls).2 cubic yard 

For wheeled scrapers „ 

lead less than 100 feet.y* u 

“ 100 to 300 feet .* „ 

“ 400 to 500 feet.. „ 

“ over 500 feet.. b 














































748 


COST OF EARTHWORK 


The daily expense per scraper, for driver’s wages and the use of a 2-horse team, is 
about $3.50. For leads of 41X) feet and over, we add 50 cts per day for use of “ snatch 
team” to help load the larger scrapers then used. One snatch team generally serves 
a number of scrapers. 

Owing to the fact that the teams are constantly in motion without rest, they travel 
somewhat more slowly than with carts. We take 150 ft per minute (or 75 ft of lead 
per minute) as .an average. 

In loading and unloading, the teams not only go out of their way in order to turn 
around, but travel more slowly than when simply hauling. To cover this we make 
an addition of 25 ft to each length of lead, whether long or short, for wheeled scrap¬ 
ers ; and 15 feet for drag scrapers. 

We add 1 cent per cubic yard for the cost of loading and dumping the scrapers; and 
estimate the approximate cost of the other items as follows: 


Repairs of cart-road fa ct per cub yd in place for each 100 ft of lead 

Light Soils Heavy Soils 

Loosening cts per cub yd in place cts per cub yd in place 

by pick. * .. 5. 

by shovel. * . 2 . 

Spreading. 1. 1.5 

Superintendence, wear and tear etc. 1. 1. 

We repeat that our figures are to be regarded merely as tolerable approximations, 
and subject to great variations according to skill of contractor and superintendent, 
strength of teams, character of material moved, state of weather etc etc. 

No. of trips per day No. (6C0)of mins in a working day 
per wheeled scraper Q f 75 f t l en gtlis in (lead -f 25 ft) 

No. of trips per day _ No. (COO) of mins in a working day 
per drag scraper No. of 75 ft lengths in (lead + 15 ft) 

No. of cub yds in place moved __ No. of trips per .. No. of cub yds in place, 
per day by each scraper day per scraper * per scraper per trip 

C forEng, y to»lin 1 r e ’ = ° f + 1 ct for losing 

dumping and returning No - of cub y ds in P lace > moved and dumping 

per day by each scraper 


i, 


Total cost per Cost per cub yd .1 ct per cub yd 

cubic yard in in place, for in place for each 

place exclusive — loading, haul- -f 100 ft of lead, -f 
of contractor’s ing, dumping, for repairs of 

profit and returning road 


Cost, per cub yd in place, , 
of loosening, spreading i 
or wasting, and super¬ 
intendence <fec. 


By Wheeled Scrapers. Labor $1 per day of 10 working hours. 


Length of lead, or dist 
to which earth is 
hauled 

Quantity in place, 
hauled per day by 
each scraper 

Cost per cub yd in 
place, for loading, 
hauling, dumping, 
and returning. 

Total cost, per cubic yard in place, exclusive of contractor's profit. 

Light Soils 

Heavy Soils 

Spread 

Wasted 

Picke 

Spread 

d and 

Wasted 

Plowt 

Spread 

d and 

Wasted 

Feet. 

cub vds 

cts 

cts 

cts 

Cts 

cts 

cts 

cts 

50 

200 

2.8 

4.9 

3.9 

10 

8.5 

7.3 

5.8 

100 

140 

3.4 

5.5 

4.5 

11 

9.5 

8 

6.5 

150 

105 

4.3 

6.5 

5.5 

12 

11 

9 

7.5 

200 

80 

5.4 

7.6 

6.6 

13 

12 

10 

8.5 

300 

56 

7.3 

9.6 

8.6 

15 

14 

12 

11 

400 

50 

8.5 

11 

10 

16 

15 

13 

12 

000 

43 

10 

13 

12 

18 

17 

15 

14 

800 

33 

13 

16 

15 

21 

20 

18 

17 

1000 

27 

16 

19 

18 

25 

24 

22 

21 


* Light soils can generally be advantageously loosened by the scrapers themselves in the act of 
loading. 







































COST OF EARTHWORK. 


749 


By Brag: Scrapers. Labor $1 per day of 10 working hours. 



Total cost, per cubic yard in place, exclusive of contractor's profit. 


2 

o r tD 







Si —' T3 c 

s .o B 

Light Soils 


Heavy Soils 


S . C o 

<V ~ 

• Ss-P 



Picked and 

Plowed and 

O o-c 5 

Q 

Spread 

Wasted 

Spread 

■Wasted 

Spread 

Wasted 

cts 

cts 

cts 

cts 

cts 

cts 

cts 

2.6 

4.6 

3.6 

10 

8.5 

7 

5.5 

3.5 

5.5 

4.5 

11 

9.5 

8 

6.5 

4.5 

6.6 

5.6 

12 

11 

9 

8 

5.4 

7.5 

6.5 

13 

12 

10 

9 

7.5 

9.6 

8.6 

15 

14 

12 

11 

9.3 

12. 

11 

17 

16 

14 

13 


Both wheeled and drag scrapers are made by Western Wheel Scraper Co, Mount 
Peasant,Iowa; by Kilboume & Jacobs Mfg Co, Columbus, Ohio; by Fay Manufactur¬ 
er Co, Elyria, Ohio, and others. A medium-sized wheeled scraper, weighing 450 lbs, 
nd carrying .4 cubic yard, costs about from $50 to $70. A drag scraper weighs about 
00 lbs, and costs about $14. 


Art. 13. By cars and locomotive, on level track. We have based our 
limitations upon the following assumptions: Trains of 10 cars, each car containing \]/ 2 
ubic yards of earth measured in place. Average speed of trains, including starting 
nd stopping, but not standing, 10 miles per hour, =--- 5 miles of lead per hour. Labor 
1 per day of 10 working hours. Loosening, loading (by shovelers), spreading, wear 
c of tools, superintendence, &c, the same as with carts, Arts 2, 3, 5, and 7. Loss of 
me in each trip for loading, unloading, &c, 9 minutes, = .15 hour. Therefore 
Number of trips per j = The ™ mber (1 0) of hours in a working day_ 


day, per train 


15 + the number of 5-mile lengths in the lead 
Number of Number (10) Number (1.5) of cubic 


umber of cubic') - -- - v , . 

)trds in place, per > = trips per day X of cars in a X yards in place in each 

ty per train J per train train car 

list per cubic yard, in place, l 
r hauling, dumping, and V 


One day’s train expenses + 1 day’s cost of track 


. , Number of cubic yards in place per day per tram 

l ni liiii^ j 

lie day’s train expenses: 

Cost of 10 cars @ $100. $1°°0 

“ locomotive. 3000 

-$4000 

One day’s interest at 6 per cent, on cost of train. $0.G7 

Wages of engine driver (who fires his own engine). 2.00 

foreman at dump. 2.00 

3 men at dump at $1... f .yO 

Fuel. 2.00 

Water.". 

Repairs of locomotive and cars. 2.33 

Total daily expense of one train.$13.00 

Tlie daily expense of track, for interest and repairs, may be taken at 
for each mile, or fraction of a mile, of lead. 

Therefore, 


il 

U 













































750 


COST OF EARTHWORK. 


Cost per cubic yard in place, 
for hauling, dumping, and 
returning 


Total cost per cubic') 
yard in place, ex- ! 
elusive of contrac- j 
tor’s profit 


$13 -f ($3 for each mile of lead) 

Number of Number (10) Number (1.5) o 
trips per day X of cars in a X cubic yards in 
per train train each car 

Cost per cubic yard, in place, foi 
, loading, spreading oi 


Cost per cub yd in Cost per cubic yard 
place lor hauling, , loosening, loading, spreading t 
dumping, and re- wasting, and superintendence, & 
turning (Arts 2, 3, 5, and 7.) 

By Cars and Locomotive. Labor $1 per day of 10 working hours. 


Length of lead, or dis¬ 
tance to which the 
earth is hauled. 

Number of cubic yards, 
in place, hauled per 
day by each train. 

Cost per cubic yard, in 
place, for hauling, 
dumping, aud re¬ 
turning. 

Miles. 

Cu vds. 

Cts. 

1 

4350 

.4 

2 

2700 

.7 

3 

1950 

1.1 

4 

1500 

1.7 

5 

1200 

2.3 

6 

1050 

3. 

7 

900 

3.8 

8 

750 

4.9 

10 

600 

7.2 


Total cost per cubic yard, in place, exclusive of contractor's profit. 


Light Sandy Soils. 


T3 

S-i 


« CO 


Cts. 

9.7 

10 . 

10.4 
11 . 
116 
12.3 

13.1 

14.2 

16.5 


J Picked and 

Wasted. 

1 Ploughed and 

Spread. 

Ploughed and 

Wasted. 

Picked and 
Spread. 

Cts. 

Cts. 

Cts. 

Ct 8 . 

8.4 

8.4 

7.2 

13.7 

8.8 

8.8 

7.5 

14. 

9.2 

9.2 

7.9 

14.5 

9.7 

9.7 

8.5 

15. 

10.4 

10.4 

9.1 

15.6 

11. 

11. 

9.8 

16.3 

11.8 

11.8 

10.6 

17.1 

13. 

13. 

11.7 

18.2 

15.2 

15.2 

14. 

20.5 


Strong Heavy Soils. 


Picked and 

Wasted. 

Ploughed and 
Spread. 

•a 

a 

*2 
u 1 

CL, 

Cts. 

Cts. 

Cts. 

12.4 

11.3 

10 . 

12.8 

11.6 

10.4 

13.3 

12.1 

10.9 

13.7 

12.6 

11.3 

14.4 

13.2 

12 . 

15. 

13.9 

12.6 

15.8 

14.7 

13.4 

17. 

15.8 

14 6 

19.2 

18. 

16.8 


..wo amount, wi «uik ate to oe uone, tne STcam excavator, land 
arcane or steam shovel generally economizes time and money. Where the 
ceptii of cutting is less than 10 ft, so much time is lost in moving from place to place 
that the excavators do not work to advantage. In still soils, cuttings may be made 
about from 17 to 20 It deep without changing the level of the machine. For greater 
depths in such soils the work is done in two levels, since the bucket or dipper cannot 

singlelevef 1 ’ BUt *** SlUld and louse gravel > mucl1 deeper cuts may be made from a 

The excavator resembles a dredging machine in its appearance and operation A 
large plate-steel bucket, like a dredging bucket, with a Hat hinged bottom, and pro- 
v i< ed with steel cutting teeth, is forced into and dragged through the earth bv 
steam power It dumps its load, by means of the hinged bottom, either into cars 
for transportation, or upon the waste bank, as desired. 

Each machine is mounted on a car of standard gauge, which can be coupled in an 
oid nary freight train, lhe car is made of wood or iron, as desired, and is provided 
i a locomotive attachment, by which it can he moved from point to point as the 
work proceeds lhe machines can be used as wrecking or derrick ears. 

Hsi!oile? Chine ia8 a Water taUk ’ 1,olding from ao ° to 550 gallons, for the supply of 

l T g i m \ ing , t .° excavate » 1,10 en<1 of the car nearest the work is lifted from 
the tiack by hydraulic or screw jacks, upon which it rests while working. 

In stiff soils the excavator leaves the sides of the cut nearly vertical • and the de¬ 
sired slope is afterwards given by pick and shovel. When the soil is hard or much 
frozen, it may be loosened by blasting in advance of the excavator 

Steam excavators are made by Osgood Dredge Co, Albany NY; by John 
Souther & Co, (the Otis excavator) Boston Mass; by Yu lean Iron Works, Toledo 
port’•^ I Y dnstnal Works > Ba y City Mich; and by Pound Manufacturing Co,, Lock- 

The Osgood is made in two sizes In No 1 the car is 34 ft X 10 ft, and its floor is 4 
It above the rails. It has a four-wheeled truck near each end. The dinner holds 2 
cubic yards, struck measure. The machine weighs, complete about 4') tons nn<t 
cos s about $7500 on track at works (Albany. N Y) 8 In tSol machin^ the car is 
2. ft X 10 ft, floor 5 ft above the rails. It has two pairs of wheels 16 ft anart from 
■center to center of axles. The dipper holds V/, cubic yard "«n.ck meLi o T e 
machine weighs, completo, about 28 tons, and costs about $G0W. I1,eabUie - 










































COST OF EARTHWORK. 


751 


The excavator lias to he moved forward (as the work advances) abt 8 ft at a time. 
\s regularly made, it can dig at a distance of 17 ft, horizontally, from the center of 
,'he car in any direction, and can dump 12 ft above the track. In sand or gravel it 
-akes out, while actually digging, 3 dipperfuls (= 4)^' to 6 cub yards in the dipper, 
= 3.75 to 5 cubic yards in place) per minute; in stiff clay, 2 dipperfuls per minute 
= 3 to 4 cub yards in the dipper, = 2.5 to 3.33 cubic yards in place). An average 
lay's work (10 hours) for a “No 1” machine, including time lost in moving the ma¬ 
chine, Ac, is about 500 cubic yards in “hard-pan,” and from 1200 to 1500 in sand and 
'ravel. This allows tor the usual aud generally unavoidable delays in having cars 
eady for the excavator. 

The excavators carry about 80 to 90 lbs of steam. They burn from 100 to 150 lbs 
>f good hard or soft coal per hour; and require one engineer, one fireman, one 
:ranesman, and 5 to 10 pitmen, including a boss. The pitmen are laborers, who 
ittend to the jacks, lay track for the excavator and for the dump cars, assist in 
noving the latter, bring or pump water, &c, &c. 

, After reaching the site of the work, about 30 minutes are required for getting the 
sxcavator into working condition; and an equal length of time, after completion 
>f the work, in getting it ready for transportation. 

The following figures are taken from the records of work done by a No 1 machine, 
rom May to Nov, 1883. The material was hard clay with pockets of sand. The 
ixpeuses per day of 12 working hours, at $1.50 per such day for labor, were 


Water (a very high allowance). $ 5.00 

Coal, 1% tons bituminous. 10.00 

Wages of engineer. 4.00 

“ “ fireman . 1.50 

“ “ cranesman or dipper-tender. 2.50 

“ “ pit boss . 3.00 

“ “ 8 pitmen at $1.50. 12.00 

Oil, waste, repairs, &c (estimated).. 5.00 

Interest on cost ($7500) of machine... 1.25 

- $44.25 


Reduced to our standard of $1 for labor per day of 10 working hours, this would 
>e say $30.00 per day. Reduced to the same standard, and allowing for the greater 
iroportional loss of time in stopping at evening and starting in the morning: the 
verage daily quantity excavated, measured in place, was, in shallow cutting, 530 
ubic yards; in deep cutting, 1200 cubic yards; average of whole operation, 800 
ubic yards. This would make the cost, per cubic yard measured in place, for 
oosening and loading into cars, 5.67 cts, 2.5 cts, and 3.75 cts respectively; while the 
ost by ploughing and shoveling, in strong heavy soils, by Arts 2 and 3, is 7.4 cts; and 
y picking and shoveling, say 10 cts. 

In sand, the No. 1 machine has dug and loaded as high as 100,000 cubic yards in 
2 working days of ten hours each; average 1390 cubic yards per day; at 2 cents 
>er yard when reduced to our basis of $1 per day for labor. 

. 

















752 


COST OF EARTHWORK. 


Art. 14. Removing: roek excavation by wheelbarrows. 

A cubic yard of hntd rock, in place , or before being blasted, will weigh about l.K tons, if sandstone 
or conglomerate. (150 tbs per cubic foot:) or 2 tons if good compact granite, gneiss, limestone, or 
marble, (168 lbs per cubic foot.) So that, near enough for practice in the case before us, we mav as- 
sume the weight of auy of them to be about 1.9 tons, or 4250 lbs per cubic yard, in place ; or 158 Ibt 
per cubic foot. 

Now, a solid cubic yard, when broken up by blasting for removal by wheelbarrows 
or carts, will occupy a space of about 1.8, or 1 i cubic yards : whereas average earth, when loosened 
swells to but about 1.2. or li of its original bulk in place; although, after being made into embank, 
rnent, it eventually shrinks into less than its original bulk. In estimating for earth, it is assumed 


that 


1 


yx cubic yard, in place, is a fair load for a wheelbarrow. Such a cubic yard will weigh on an 

2430 

average 2430 lbs, or 1.09 tons ; therefore, ——— — 174 lbs, is the weight of a barrow-load, of 2.31 cubic 


14 


feet of loose earth. Assuming that a barrow of loose rock should weigh about the same as one of 
earth, we may take it at Jy of a cubic yard; which gives — 177 lbs per load of loose rock, 


24 


occupying 2 cubic feet of space. 

In the following table, columns 2 and 3 are prepared on the same principle as for earth, as directed 
in Article 10. Column 4 is made up by adding to each amount in column 3, .2 of a cent for each 100 
feet length of lead, for keeping the wheeling-planks in order; and 45 cents per cubic yard, in place, 
as the actual cost for loosening, including tools, drilling, powder, Ac: as well as moderate drainage, 
and every ordinary contingency not embraced in column 3. Contractor's profits, of course, are not 
here included. 

Ample experience shows that when labor is at $1 per day. the foregoing 45 cents per cubic yard, in 
place, is a sufficiently liberal allowance for loosening hard rock under all ordinary circumstances. 
In practice it will generally range between 30 and 00 cents ; depending on the position of the strata, 
hardness, toughness, water, and other considerations. Soft shales, and other allied rocks, may fre¬ 
quently be loosened by pick and plough, as low as 15 to 20 cents; while, on the other hand, shallow 
cuttings of very tough rock, with an unfavorable position of strata, especially in the bottoms of ex¬ 
cavations, may cost $1, or even considerably more. These, however, are exceptional cases, of com¬ 
paratively rare occurrence. The quarrying of average hard rock requires about H r -° % tb of powder 
ner cubic yard, in place; but the nature of the rock, the position of the strata, Ac. may increase il 
lo *4 tt, or more. Soft rock frequently requires more powder than hard. A good chnrn-driller will 
drill 8 to 10 feet in depth, of holes about 2^ feet deep, and 2 inches diameter, per day, in average 
hard rock, at from 12 to 18 cents per toot. Drillers receive higher wages than common laborers. 


Hard Roek, by Wheelbarrows. 

Labor $1 per day, of 10 working hours. 


Length of 
Lead, or dis¬ 
tance to 
which the 
rock is 
wheeled. 

Number of 
cubic yards, 
in place, 
wheeled per 
day by each 
barrow. 

Cost per 
cubic yard, 
in place, 
for loading, 
wheeling, 
and 

emptying. 

Total cost 
per cubic 
yard, in 
place, ex¬ 
clusive of 
profit to 
contractor. 

Length of 
Lead, or dis¬ 
tance to 
which the 
rock is 
wheeled. 

Number of 
cubic yards, 
in place, 
wheeled per 
day by each 
harrow. 

Cost per 
cubic yard, 
in place, 
for loading, 
wheeling, 
and 

emptying. 

Total cost 
per cubic 
yard, in 
place, ex¬ 
clusive of 
profit to 
contractor 

Feet. 

Cubic Yds. 

Cents. 

Cents. 

Feet. 

Cubic Yds. 

Cents. 

Cents. 

25 

12.2 

8.61 

53.7 

600 

2.96 

35.5 

81.7 

50 

10.7 

9.81 

54.9 

700 

2.62 

40.1 

86.5 

75 

9.58 

11.0 

56.2 

800 

2.34 

44.8 

91 4 

100 

8.6(5 

12.1 

57.3 

900 

2.12 

49.5 

96.3 

150 

7.26 

14.5 

59.8 

1000 

1.94 

51.1 

101.1 

200 

6.25 

16.8 

62.2 

1200 

1.65 

63.6 

115.0 

250 

5.49 

19.1 

64.6 

1400 

] .44 

72 9 

120 7 

300 

4.89 

21.5 

67.1 

1600 

1.28 

82.2 

130.4 

350 

4.41 

23.8 

69.5 

1800 

1.15 

91.5 

140.1 

400 

4.02 

26.1 

71.9 

2000 

1.04 

100.8 

149.8 

450 

3.69 

28.5 

714 

2200 

.953 

110.2 

159.6 

500 

3.41 

30.8 

76.8 

2100 

.879 

119.5 

169.3 


Art. 15. Removing roek exeavation by carts. A cart-load of 

rock may be taken at ^ of a cubic yard, in place. This will weigh, on an average, 851 lbs; or but 41 
lbs more than a cart-load of average soil. Since the cart itself will weigh abouta ton, the total 
loads are very nearly equal in both cases. Columns 2 and 3 of the following table are prepared on the 
same principle as for earth, as directed in Art. 4. Column 4 is made up by adding to each amount in 
column 3. the following items: For blasting, (and for everything except those in column 3; loading, 
and repairs of cart-road,) 45 cents per cubic yard, in place; for loading, 8 cents, per cubic yard, in 
place: and for repairs of road, .2, or of a cent for each 100-feet length of lead. Contractor's profit 
not included. ° 





























COST OF EARTHWORK 


753 


Hard Rock, by Carts. 

Labor $1 per day, of 10 working hours. 


length of 
ead, or dis¬ 
tance to 
vhich the 
rock is 
hauled. 

Number of 
cubic yards, 
in place, 
hauled per 
day, by each 
cart. 

Cost per 
cubic yard, 
in place, 
for hauling, 
and 

emptying. 

Total cost 
per cubic 
yard, in 
place, ex¬ 
clusive of 
profit to 
contractor. 

Length of 
Lead, or dis¬ 
tance to 
which the 
rock is 
hauled. 

Number of 
cubic yards, 
in place, 
hauled per 
day, by each 
cart. 

Cost per 
cubic yard, 
in place, for 
hauling, 
and 

emptying. 

Total coRt 
per cubic 
yard,in 
place, ex¬ 
clusive of 
profit to 
contractor 

Feet. 

Cubic Yds. 

Cents. 

Cents. 

Feet. 

Cubic Yds. 

Cents. 

Cents. 

25 

19.2 

6.51 

59.6 

1800 

5.00 

25 0 

81.6 

50 

18.5 

6.77 

59.9 

1900 

4.80 

26.0 

82.8 

75 

17.8 

7.03 

60.2 

2000 

4.62 

27.1 

84.1 

100 

17.1 

7.29 

60.5 

2250 

4.21 

29.7 

87.2 

150 

16.0 

7.81 

61.1 

2500 

3.87 

32.3 

90.3 

200 

15.0 

8.33 

61.7 

mile 

3.70 

33.7 

92.0 

300 

13.3 

9.37 

63.0 

3000 

3.33 

37.5 

96.5 

400 

12.0 

10.4 

64.2 

3250 

3 12 

40.1 

99.6 

500 

10.9 

11.5 

65.5 

3500 

2.92 

42.8 

102 8 

eoo 

10 0 

12.5 

66.7 

3750 

2.76 

45.3 

105.8 

700 

9.23 

13.6 

68.0 

4000 

2.61 

47.9 

108.9 

800 

8.57 

14.6 

69.2 

4250 

2.47 

50.6 

112.1 

900 

8.00 

15.6 

70.4 

4500 

2.35 

53.2 

115.2 

1000 

7.50 

16.7 

71.7 

4750 

2.24 

55.8 

118.3 

1100 

7.06 

17.7 

72.9 

5000 

2.14 

58.4 

121.4 

1200 

6.67 

18.7 

74.1 

1 mile 

2.04 

61.2 

124.8 

1300 

6.32 

19.8 

75.4 

154 “ 

1.67 

75.0 

141.2 

j 1400 

6.00 

20.8 

76.6 

W 2 “ 

1.41 

88.8 

157.6 

1500 

5.71 

21.9 

77.9 

1 H “ 

1.22 

102.5 

174.0 

1600 

5.45 

22.9 

79.1 

2 “ 

1.08 

116.3 

190.4 

1700 

5.22 

24.0 

80.4 

2 M “ 

.962 

130.0 

206.8 


“ Loose rock ” will cost about 30 cts per yd less; and even solid rock will 

eraye about 10 cts less than the tables. 

Art. 16. Removing' rock excavation by cars and locomo- 

ve, on level track. Our calculations are based upon the following assumptions: 
ains of in cars, each car containing 1 cubic yard of rock measured in place. Aver- 
e speed of trains, including starting and stopping, but not standing, 10 miles per 
ur = 5 miles of lead per hour. Labor $1 per day of 10 working hours. Loosening, 
cts per cubic yard in place. Loading, 8 cts per cubic yard in place. Cost of track, 
r interest and repairs, $3 per day per mile of lead. The calculations are the same, in 
inciple, as those in Art. 13. 

Hard Rock, by Cars and Locomotive. 

Labor $1 per day of 10 working hours. 


ength of lead, or distance to which the rock 

is hauled.miles 

[umber of cubic yards, in place, hauled per 

day by each train. . 

ost, per cubic yard in place, for hauling, 

dumping, and returning.cents 

otal cost, per cubic yard in place, exclusive 
of contractor’s profit.cents 


1 

3 

5 

7 

10 

2900 

1300 

800 

600 

400 

.6 

1.7 

3.5 

5.7 

10.8 

53.6 

54.7 

56.5 

58.7 

63.8 



































754 


TUNNELS. 


TUNNELS. 

Tunnels for railroads should, if possible, be straight, espe¬ 
cially when t here is but a single track ; inasmuch as collisions or other accidents 
in a tunnel would be peculiarly disastrous. A tunuel will rarelv be expedient 
before the depth of cutting exceeds 60 feet. Firm rock of moderate hardness 
and of a durable nature, is the most favorable material for a tunnel; 
especially if free from springs, and lying in horizontal strata. In soft rock or 
in shales (even if hard and firm at first), or in earth, a lining of hard brick or 
masonry in cement, is necessary. A tunnel should have a grade or incli¬ 
nation in one direction, for ease of future drainage and ventilation. No 
special arrangement is essential for ventilation either during construction 
or after, if the length does not exceed about 1000 feet; but beyond that gen¬ 
erally during construction either shafts are resorted to, or means provided for 
forcing air into the tunnel through pipes from its ends. But after the work is 
finished except under peculiar circumstances, not hing of the kind is necessary, 
fehatts often draw air downwards; and frequently, even when aided by a steep 
uniform grade, do not secure ventilation. The MontCenis tunnel under the 
Alps, completed in 1871, is 7)4 miles long, and has no shafts, although it grades 
u.P Irom each end, which is the most unfavorable of all conditions for ventila¬ 
tion without shafts. It was made so for facilitating drainage. Its ventilation 
is maintained by air forced in from the ends. The Hoosac tunnel Mass 4!4 
miles long, has shafts : one of them 1030 feet deep; but they were for expediting 
the work Shafts generally cost from 1V 2 to 3 times as much per cubic 
yard as the main tunnel, owing to the greater difficulty of excavating and re¬ 
moving the material, and getting rid of the water, all of which must be done 
by hoisting. When through earth, they must be lined as well as the tunnel- 
and the lming must, usually be au under-pinning process. Or the lining mav 
first be built over the intended shaft, and then sunk bv undermining it grad- 
uahy; see page 650. Their sectional area commonly varies from aSut % io 
100 square feet. They have the great advantage of expediting the work bv in¬ 
creasing the number of points at which it can be carried on ; but if placed too 
close together, their cost more than compensates for this. The air in some 
tunnels, while being constructed, is much more foul than in others- so that 
after the work h» teen corameuced, shafts wi,h forced air may beliniXt 
where they were not anticipated. In excavating the tunnel itself a heading 

» and 3 t0 12 is ^iven an’d a ma1nt d a?n n I 

a short distance (10 to 100 feet, or more, according to the firmness of the ma¬ 
terial) in advance of the main work. In rock, the heading is just below the 
top of the tunnel, so that, the men can conveniently drill holes''in its floor for 
blasting, but in earth, the heading is driven along the bottom of the tunnel 
that being the most convenient for enlarging the aperture to the full tunnel 
size by undermining the earth, and letting it fall. In earth the ton and 
of the heading, as well as of the tunnel, must be czreZuv ^prevented from 

eni' fn” 'U ’. ef<> K re 1 ,e lin,n g >s built; and t his is done by mean’s of rows of verti- 
cal rough timber props, and horizontal caps or overhead pieces between whieh 
and the earth rough boards are placed to form teniporarv smpporting^ sSel and 

®re 1 • eXCa - Vat J° n - The props and caps are placed firsf; and the boards 

are then driven in between them and the earthen sides of the excavation 
These are gradually removed as the lining is carried forward. The liniit»- 
when of brick, is usually from 2 to 3 bricks thick <17 to ‘>6 inrhPQUfi^o 1 ^ 

about half again as thick. It is important that the bricks or stone should he 
of excellent hard quality, and laid in good cement. The bricks should be 
moulded to he shape of the arch. As the lining is finished i”short lengths 
and before the centers are removed, any cavities or voi<l« s i 

tl,e earth should becarefully and - ” finS*,*' 

fissuml.or if not of durable character, as common shale. linln e 7s nece^arv 
The Cross-section of a single-track railroad tunnel, in the clear of everT 
thing, and for ears of li feet extreme width, should not be less than about 15 
eet vide, by 18 feet, high ; nor a double-track one, less than 27 feet wide bv 24 
feet high ; unless in the last case the material is firm rock in which n i 

nf "pessary for lining. The roof may then be much flatter so that tfheieht 
of 20 feet may answer. With cars of 10 feet extreme width the width off he 
tunnel may be reduced to 25 feet; or with 9 feet car* to WfJJ tu i 1 e 
been made 22 feet. The Mont Ceni’s is 26 

daily progress from each face of a tunnel varies from 18 ♦ V 

le„ s ,h per 21 hours, with three relays of workmen.*OnThe Mo« Ceuta the 




TRESTLES. 


755 


tremes were about 4 to 9 feet daily for a whole year, from each face. Drills 
worked by compressed air were employed in the headings, which were 12 feet 
wide by 8 feet high. Ordinarily, from 1)4 to 3 feet may be taken as averages. 
I he difference of rate of progress between a single and a double track tunnel' 
is not so great as might be supposed ; inasmuch as a larger force can be em¬ 
ployed on the wider one. If the tunnel is in earth, the construction of the 
lining about makes up for the slower excavation of one in rock. In rock, with 
labor at §1 per day, tlie cost will usually vary with the character of the’rock, 
Irom $2 to Sp per cubic yard for the main tunnel; and from $3 to $10 for the 
heading; while shafts will average about 50 per cent, more than heading. The 
cost of a single-track tunnel, when common labor is$l per day, will generally 
range between $30 and $75 per foot of length. Tunnel work, however, is liable 
to serious contingencies which cannot, be foreseen. Since the sides and roof are 
rough as blasted, the width and height should each be estimated to the con- 
tractor as about 18 inches or 2 feet greater than the established clear ones. At 
any rate, the mode of measurement should be clearly stated in the specifications 
for the work. When a tunnel is made with a uniform grade, the work gen¬ 
erally progresses in a more satisfactory manner from the lower end, because 
the descent tavors the drainage of the spring water that is usually met with; 
whereas, at the upper end, it must be removed by pumps or by bailing. The 
upper end has, however, the advantage of sooner getting rid of the smoke in 
blasting. Before commencing a tunnel, or even deciding upon one, trial 
shafts should be sunk to ascertain the nature of the material. In long ones, 
the greatest care and accuracy are necessary for preserving the line of direc¬ 
tion, so that the work from both ends shall meet properly at the center 

In the heading of the Vosburg (Pa) tunnel of the Lehigh Valley R R, built 
1884, cross-section 7% feet X 26 feet, the average progress per working 
lav of 24 hours with two shifts of 12 hours each, was as follows; by hand 
frilling 2.8 feet and 2.4 feet respectively from each end; by machine drills 
[two rival drills in competition) 5.6 feet and 7.8 feet. The material was hard 
gray sandstone. For the whole tunnel the rate was about 2 feet per day. 

For further information respecting tunnels, the reader is referred to Mr. 
H. S. Drinker’s very full treatise on the subject, published by the Messrs Wiley. 

For Stone bridges and culverts, see pp 693, &c. 

For Trusses, see pp 547, <fcc. 


TEESTLES. 



x 


Fins 1, 2, 3, 5. 6, 7, are elevations of trestles; taken across the track or 
oadway. We may consider Fig I as adapted to a height of about 10 to 20 ft; Figs 2 

52 

































































756 


TRESTLES. 


and 3. to heights from 20 to 30 ft; Fig 5, from 30 to 40 ft; Fig 6. from 40 to 60 ft, ns 
rough approximations merely. A single framework, such as that shown in each ot 
these six figures, is called a “bent.” These bents of course admit of many modifi¬ 
cations. They are usually supported by bases of masonry, as in the figures. Ihese 
preserve the lower timbers from contact with the earth, which would hasten their 
decay It is advisable to make these bases high enough to prevent injury from cattle, 
or passing vehicles, &c. Up to heights of about 40 or 50 ft, a single row of posts or up¬ 
rights, a,a, a, Figs 1 to 9,as shown at ee under Figs 1 and 6, will answer. Hut as the 
height becomes greater, more posts should he introduced, as shown at x x under rig 



posts of 12 X 12, at its top. The other timbers were 6X 12; many of them were in 
pairs embracing the posts. This single-track viaduct was begun July 1, 1851, and 
completed Aug. 14, 1852. It contained 1,602,000 ft (B M) of timber, and 10S,862 lbs 
of iron. In the foundations were 9200 cub yds of masonry. The entire cost was 
about $140,000. It was burned down in 1875, and was replaced, in less than 3 mos, 
with a single-track viaduct of wrought-iroil trestles*, (described below) 
containing, in all, 1,340,000 lbs of iron, and 130,600 ft (B M) of timber; and costing, 
complete, above the masonry, about $95,000. Frequently the posts of trestles are in 
pairs; and the other timbers pass between ; all bolted together. 

In Fig 4, the posts a, a, a, are end views of three trestles or bents, such as Fig 3; 
audit are diag braces extending from trestle to trestle; the two outer ones inclining 
in one direction; and the central one crossing them. These may he placed either 
intermediate of the posts, as in Fig 3; with the heads of the two outer ones confined 
to tlie cap c c of one trestle; and their feet to the sill y y of the next one ; or they 
may all be spiked or bolted to the posts themselves, as in Fig 4. The last is the best, 
as it serves also directly to stiffen the posts: as do also the braces o o, n n, Fig 2. 
Such bracing is too frequently omitted. During the passage of trains, the backward 
pressure of the steam, exerted through the driving wheels against the track, pro¬ 
duces a serious strain lengthwise of the road, and tending to upset the trestles; and 
the sudden application of brakes to a moving train, produces a similar strain in the 
opposite direction. These strains become more dangerous as the ht increases. Hence 
the need for such braces. Usually the outer posts may lean 1.5 to 2.5 ins to a ft. 

The posts should not he less than about 12 ins square,except in quite low trestles; 
and even then not less than about 10 X 10. The diag bracing may generally be about 
as wide as the posts; and half as thick. The dist apart of the bents, when the road¬ 
way is supported by simple longitudinal beams, should not exceed 10 or 12 ft, for 
railroads. But if these beams receive support from braces beneath, like ss, Fig 8 ; or 
from iron truss rods, as at Fig 52, page 514, the dist may be extended to 15 or 20 or 
more ft. But when the trestles become very high, and contain a great deal of tim¬ 
ber, it becomes cheaper to place them farther apart, say 30 to 60 tt; and to carry 
the railway upon regular framed trusses, as at u u, Figs 7 and 8; as in a bridge with 
stone piers. In the Genesee viaduct, the trestles were 50 ft apart, center to center. 

When such a trestle as Fig 8 becomes very narrow in proportion to its height, we 
may add to its stability by introducing beams w, extending from trestle to trestle; 
ami still further by inserting diag braces v v, as in the old Genesee viaduct. 

E-'i«s 62, p 613, “Trusses,” will show how the timbers may be joined. In de¬ 
signing trestles, (as in wooden bridges.) it is advisable, as far as practicable, to arrange the pieces 
so that any one may be removed if it becomes decayed ; and another put in its place. On curvks, 
additional strength should be given on the convex side ; as suggested by the dotted lines in Fig 5. 
On very high trestles especially, (as well as on bridges.) wheel-guards, g g, Fig 10, either inside or 
outside of the rails, should never be omitted, as is commonly done. 

In marshy ground, piles nitty be driven to support the trestles; or may be left so 
far abovo ground, as themselves to constitute the posts. Such trestles may often he 
used advantageously, even when to he afterward filled in by embkt. They then sus¬ 
tain the rails at their proper level until the embkt has reached its final settlement. 

They are generally used to avoid the expense of embkt; especially when earth can 
only be obtained from a great dist. Even when eartli and timber are equally con¬ 
venient, they will rarely much exceed about half the cost of embkt; even when hut 
about 30 ft high; hut owing to their liability to decay, they should be resorted to 
only in caso of necessity ; or as a temporary expedient. 

Iron trestle**. At the Crtinilin double-track iron viaduct,in England,1500 
ft long, (spans 150 ft.) they are about 180 ft high ; 60 by 27 ft at base; 30 by 18 at top; 
each composed of 14 cast-iron posts, arranged as along hexagon; each post being 
formed of 17-ft lengths of iron pipes, 1 ft outer diam, by 1 inch thick. At each 17-ft 
length, the pipes are firmly connected by hor iron pieces; and between these diff 
stages is diag bracing of 4 X 4 X Yi Inch rolled T iron, arranged as in Fig 7. 

The viaduct contains about 3,000,000 lbs of wrought-irou and nearly as much cast-iron. 



TRESTLES. 


757 


The new Poring 1 ® Viaduct, referred to .above, consists of iron Pratt 
trusses, resting upon six towers. Fig 11 is a lougitudiual view, and lig 12 a tians* 
verse view, of one of the tallest towers, ‘203 It 8 
ins high from top of masonry to track rail, and 
weighing 286,000 lbs. Each tower consists of 2 
bents B B, Fig 11. Each bent has two wrought- 
iron columns CC, Fig 12, inclining toward each 
lother with a batter of 1 in 8. They are 20 ft 
♦ apart at top, and, iu the tallest tower, 69 ft 8 ins 
a between centers, at base. The two bents of a 
tower are 50 ft apart, and are connected with 
each other by hor longitudinal struts S, Fig 11, 
and diag tie-rods K. The spans between 
>, towers vary from 50 to 118 ft. 

The columns are put together in lengths 
of 25 ft. Figs 13, 14, and 15 show the upper end 
>of one of these lengths. Fig 13 is an elevation, 
seen from between the feet of a bent, Fig 14 a 
across section, and Fig 15 a side view. Each col 
is composed of three plates, P P P, and 4 angle- 
bars 4 X 4 X inch, as shown. In the tallest 
towers, the 2 opposite side plates are 15 ins wide, 
and from % to % inch thick;.their thickness 
increasing with their dist from the top oi the 
tower. The third, or back , plate is 17 ins X Va 
inch throughout. The fourth side, L, has only 



Fiff.12 


moo unuugiiuui,. .... -„ a zig-zag lacing, Z Z Z, Fig 13, of 

flat bars, so that the interior of the col is accessible for painting. 

At the (inner end U of each 25-ft length, are riveted two small iron plates 
pp forming a tenon. The foot of the next length fits over this 
tenon and is confined to it by a turned iron pin, ins diam, 
passing through carefully bored holes. To this pin, the longitudinal 
diai* rods, R, Fig 11,1*4 ius diam, are attached. The loii^itud- 
inal hor struts, S, Fig 11, are light latticed girders of uniform 
width (1 ft) and depth (2 ft). They abut against the sides of the 
columns, and are bolted to lugs of angle iron, riveted to them. 
They are connected with the corresponding transverse struts, S, I lg 
12 . by hor diag angle-bars fastened toeach strut 10 It from its end. 



3 

V’ 

s 

Hr ■cH 

'])< 

r 

3> 

p 

d 

' r. ex, 

V 

—o- 1 


P 

Fig 1 .14 


The 'transverse struts are of different de¬ 
signs, depending upon their lengths. At their ends 
they are held by pins passing through holeso, Fig 
15, in the side plates of the columns. These pins 
also hold the 1% inch diag rods, R, Fig 12. Each 
of the lowest three transverse struts is in two 
lengths, and is supported by an intermediate 
vert post, I, Fig 12. shown in cross section by 
Fig 16; and each of the lowest two is connected 
with the correspouding strut in the other bent of 
the same tower by a longitudinal strut and hor 

diag rods. _ _ 

The cols rest upon east-iron pedestals. 

Each pedestal is tenoned into the foot of its col. 
The pedestals on the north side of the bridge are 
doweled tocast-iron plates built into the masonry. 
Those on the south side are on rollers, rolling at 
rbdit angles to the axis of the bridge. The two 
pedestals of each bent are connected with each 
other, and held in position, by e^e-bars, which 
prevent them from spreading; and by struts, 
which prevent them from coming together when 
the diags are screwed up. V V, Fig 12, are an¬ 
chor rods, bolted to the masonry, and attached to 
the columns - at such a lit as not to interfere with 

the hor expansion of the tower. . 

cans are tenoned and bolted into the tops of the cols 

. * .. .tiMIQChC DC kIiOWTI 111 t 12! II* 



4 ins 


a 


I 


Fig.10 


On each cap 


IBBMilpa s i 

S;“^tolhSto the caps at Ml, cad., but those belvxen the tower, are arranged 






































































758 


TRESTLES. 


like the long spans. Where the ends of a long, and of a short, span rest upon the 
same cap, they are placed respectively 3 and 6 ins from its center, so that their cen¬ 
ter of pressure coincides with the center of cross section of the col. 

The towers are made strong enough for a double-track road ; and the trusses are 
so arranged that they can be placed closer together, and additional trusses placed 
alongside of them ; so that the track can be doubled at any time. 

The bridge is so designed that the greatest compressive strain per sq 
inch that can come upon a col, under a double-track bridge and load, is 6000 lbs; 
and the greatest tensile Strain per sq inch on a diag, 15.000 lbs. The 
greatest weight on the foot of any one col will be 367,500 lbs, or 155 lbs per sq inch 
of the 4 ft sq pedestal-stone. 

The lowest section of a tower was first erected by means of a wooden framework 
on a flopring resting on the stone piers. A gin-pole 55 ft high was then lashed to 
each col, and by means of these poles the flooring and framework were raised to the 
tops of the columns, and used in placing the second section in position on top of the 
first. This process was repeated until the tower attained its full height. One of the 
tallest towers was erected in 11 days. 

^lie Kinzua Viaduct, on the Bradford branch of the New York, Lake Erie 
& 'Western R. R. in McKean Oounty, Penna., is sinele track, 2052 feet lonar. and its 
great' St height is 285 feet from the top of the masonry piers to the track-rails. 

There are 21 spans of 61 ft each, placed between 20 towers; and 20 spans of 38% 
ft each, over the towers. The towers are similar, in general arrangement, to those 
of the Portage viaduct, above described. Tlie bents are 10 ft wide at top, and 103 ft 
at the feet of the tallest towers, which are 278 ft high. The 2 bents of a tower are 
38% ft apart. The legs are of Phoenix Iron Co’s 4-segment columns, pattern C, p 
449,7^ ins inner diam, and from i to -j-|- in thick. They are made up of lengths of 
about 33 ft, which are held in place, one on top of another, by plain cylindrical 
wrought-iron tubes about 14 ins long, and of such diam that they just fit inside of 
the cols. The abutting ends of two column-lengths slide over such a tube, and meet 
at the middle of its length. The tube is held in place by 4 bolts, which pass through 
the column from side to side, two of them being at right angles to the. other two. 
These holts hold in place the longitudinal and transverse hor braces, most of which 
are lattice-girders, spaced about 30 ft apart vertically. 

The cols rest upon smooth iron platos, which allow a sliding movement of 1 inch 
transversely, and .38 inch lengthwise, of the roadway. Each plate is bolted, by two 
1%-iuch bolts, from 9 to 12 ft long, to a square, pyramidal masonry pier, from 10 to 
13 ft deep, 8 ft square at bottom and 4 ft square at top. 

Tiie cols have cast-iron caps, to which the ends of the lower chords of the 38% ft 
spans are firmly bolted, and on which those of the 61-ft spans rest. The latter are 
bolted to the shorter spans through oval bolt-holes, which allow .17 inch play, longi¬ 
tudinally, for expansion and contraction. 

The greatest compression on a col is 8000 lbs per sq in. The lilt load, 
by'TJ S Govt experiments at Watertown Arsenal, is 35,000 lbs per sq in. 
The greatest tension on the diags is 15,000 lbs per sq in. 

The heaviest column weighs about 5000 lbs. The entire structure, ' 
3,600,000 lbs. It cost $275,000, and was built by a gang averaging 125 | 
men, aided by 2 steam-hoisters and a traveling crane. Clarke, Reeves q 
& Co, Phila, builders; now (1886) Phoenix Bridge Co. 

Mode of creel ion. At each of the four corners of the site for a j 
tower a mast 60 ft long was set up. These were guyed with ropes, and I 
by means of them the 4 columns about 33 ft long, forming the lowest | 
story of a tower, were erected and braced together. The masts were then 
raised about 30 ft, and clamped one to each of the columns, and used for j 
raising the second story into position on top of the first, and so on,except j 
for the top story, which was bolted together on the ground, in two pieces, ! 
and hoisted into position by the travelling crane. 

Verrugas Viaduct, near Lima, Peru, 575 ft long, 252 ft high, car- 1 
rying the Oroya 11 R (single track) over the Agua de Verrugas. Built by | 
the late Baltimore Bridge Co, Clias. II. Latrobe, Engr, in less than four 
Llg. 17 months in 1872. It contains 1,325,000 lbs of iron, and cost $165,000. j 
There are four Fink truss spans, three of 100 ft, and one of 125 ft. They 
rest on 3 towers. Each tower consists of three parallel bents, 25 ft apart, and each 
bent has four columns, arranged as in Fig 17. The bents are 15 ft wide at top and 
the outer columns of each bent batter 1 in 12. The columns are all of the Phoenix 
segment pattern. The two outer ones of each bent have six segments each, diam 
11% ins, area of cross section, 20 sq ins. The inner ones have four segments each, 
diam 8 ins, area 13 sq ins. The columns are put together in lengths of 23% ft, 
which are joined by cast-iron couplings. The hor braces extending from col to col 
are about 25 ft apart vert. 








RAILROAD CONSTRUCTION. 


759 


BALLAST. 

Table of cubic yards of ballast per mile of road. 

Side-slope of the ballast 1 to 1. Width in clear between 2 tracks 6 ft. The ties 
and rails may be laid first, for carrying the ballast along the line; then raised a 
few ft of length at a time, and the ballast placed under them. Deduct for ties, 
as below. 


Depth 

in 

las. 


Top width, 

Single Track. 

Top width. 

Double Track. 

10 Ft. 

11 Ft. 

12 Ft. 

21 Ft. 

22 Ft. 

23 Ft. 


Cub. Y. 

Cub. Y. 

Cub. Y. 

Cub. Y. 

Cub. Y. 

Cub. Y. 

12 

2152 

2317 

2513 

1303 

4199 

4695 

18 

3371 

3667 

3960 

6000 

6894 

7188 

21 

1031 

5085 

5171 

8996 

9388 

9780 

30 

6111 

6000 

7087 

11190 

11980 

12470 


A man can break 3 to 4 cubic yards per day, of hard quarried stone to a size 
suitable for ballast; say averaging cubes of 3 inches on an edge. Where other 
ballast cannot be had, hard-burnt clay is a good substitute. The slag from iron 
furnaces is excellent. The ties decay more rapidly when gravel or sand is used 
instead of broken stone, because these do not drain off the rain, but keep the ties 
damp longer. For stone crushers, see p. 680. 


TIES. 

In the United States the life of a tie is about as follows: 


Average, 

Years. 

7 

9 

6 


Years. 
6 to 12 
4 to 7 


Average, 

Years. 

7 

5 


Chestnut, 6 to 12 7 White Oak, 

Cedar, 6 to 15 9 Spruce Pine, 

Hemlock, 

As shown by table, Art 3,p 815, the annual expense for renewal of ties, 
in the U. S. alone, is about ten millions of dollars. 

The Penna R R, in 1883, used 575,000 ties in construction on its main line 
and branches (= 2552 miles of single track) and 779,000 ties in repairs. 

It will often, especially in the caso of the softer and more perishable woods, be 
true economy to preserve ties by the injection of creosote. See p 425. Creosote 
preserves the spikes. 

The writer believes that most of the fault usually ascribed to cross-ties, as well as 
to rail-joints, is in reality due to imperfect drainage of the roadbed. Hence, he does 
not agree with those who advocate very long ties; but considers that with good 
ballast, on a well-drained roadbed, 8^ ft is as good as more; and that 8% ft, by 9 
ins, by 7 ins; and 2 \4> ft apart from center to center, is sufficient for the heaviest 
traffic. On many important roads they are but 8 ft; and on some only 7% ft Iol *s; 
track 4 ft 8^. On narrow-gauge roads the ties are generally from 6 to 7 ft long. 
The actual cost of cutting down the trees, lopping off the branches, and hewing 
the ties ready for hauling away to be laid, is about 6 to 9 cts per tie, at $1.75 per 

day per hewer. ...... 

The narrow bases of rails resting immediately on the cross-ties, without chairs, 
frequently produce in time such an amount of crushing in the ties as to injure them 
materially even before decay begins. Burnetised ties rust the spikes away rapidly. 
Creosoted ones preserve them. 

Cross-ties of 8]4 feet, by 9 inches, by 7 inches, contain 3.719 cubic feet each ; 
and if placed 234 feet apart from center to center, there will be 2112 of them per 
mile amounting to 291 cubic yards. Therefore, if they are completely embedded 
in the ballast, they will diminish its quantity by that amount. At 2 Let apart there 
will be 2640 of them, occupying 364 cubic yards; and at 3 feet apart, 1760 of them ; 
243 cubic yards. 

























760 


BAILS. 


Cubic feet contained In cross-ties of dliferent sizes. 


Dimensions. 


Dimensions. 


Ft. 

Ins. 

Ins. 

Cub. Ft. 

Ft. 

Ins. 

Ins. 

Cub. Ft. 

8 

by 8 

by 6 

2.667 

8 H 

by 10 

by 7 

4.132 

8 

9 

6 

3.000 

s y 2 

10 

8 

4.722 

8 

9 

7 

3.500 

s y 2 

12 

8 

5.667 

8 

10 

6 

3.333 

9 

8 

6 

3.000 

8 

10 

7 

3.889 

9 

9 

6 

3.375 

8 

10 

8 

4.444 

9 

9 

7 

3.938 

8 

12 

8 

5333 

9 

10 

6 

3.750 

8'A 

8 

6 

2.833 

9 

10 

7 

4.375 

8 14 

9 

6 

3.188 

9 

10 

8 

5.000 

814 

9 

7 

3 719 

9 

12 

8 

6.000 

z'A 

10 

6 

3.542 






Bessemer steel ties, furnished by International Railway Tie Co , office 210 
Washington Street, Boston, Mass., and laid in the Boston & Maine Railroad, near 
Boston, in July, 1885, under very heavy traffic, have so far given entire satisfaction. 


RAILS. 

Tons (2240 lbs.) of rail _ n y weight of rail ( exac n 
per mile of single track in lbs. per yard. ' ’ 

Every sq inch of sectional area of rail, corresponds to 10 lbs per yard of a single 
rail; or to 15.7143 tons per mile of single-track road. Consequently, 

Wt in tons per mile 

Wt in lbs per yd of rail, of single-track Area of rail 

— or 15 . 7 U 3 = in sq ins. 

Thus, a rail of 100 tons per mile of single track, will have a section of 6.364 sq 
ins ; and will weigh 63.64 lbs per yd of single rail. Add for turnouts, sidings, road- 
crossings, and a trifle for waste in cutting. When the ties are in place, and the rails 
distributed in piles at short intervals, a gang of 6 men can lay x /i a mile of rails per 
day, of single track; or after the ballast is in place, a gang of 15 men will lay about 
one mile of complete single-track superstructure per week. 

Steel rails last from 9 to 25 years; average 15 years. 

In the U. S. steel rails weig;li, on 4 ft 8 y 2 in and 5 ft gauges, usually from 
55 to 70 lbs per yard. 56 and 60 are common. Sometimes as light as 50, and as 
heavy as 76 to 80. 3 ft gauge, 30 to 40, sometimes 50. 2 ft, 25. The usual length 
of rails is 30 ft. They have been made much longer, even up to 60 ft, but such lengths 
have not come into use to any extent. They of course have fewer joints, but the 
great space necessary between the rail-ends at the joints, to allow for expansion in 
hot weather, is a serious objection. This might be obviated by beveled joints, p 763. 
For sections of rails, to scale, see pp 764 and 765. 

Annual production of rails in the United States, in tons of 2240 lbs. 

1872 1885 

83,901 959,470 

. 1,250 

808,866 13,118 

892,857 973,838 

Steel rails usually contain from .3 to .5 of one per cent of carbon. 

The Penna K K used, in 1883, on its main line and branches (— 2552 miles 
of single track), 6600 tons of steel rails in construction of new lines, and 14,300 tons 
in repairs. 

Rrice of Steel rails, at mill, in 1888, about $35 per ton of 2240 lbs. 


Bessemer steel. 

Open-hearth steel 
Iron. 



















RAILROADS. 


TuT 


Table of middle Ordinates, to be used for the bending of rails of different 
lengths, so as to form portions of curves of different radii. Ordinates for lengths 
or radii intermediate of those in the table, may be found by simple proportion. 


Def . 

Radius . 




LENGTHS 

OF KAILS . 





Aug . 

30 

2S 

26 

21 

22 

20 

18 

16 

14 

12 

10 

8 

6 

Deg . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet . 

Feet 

.5 

11460 . 

.010 

.008 

.006 

.005 

.004 

.004 

.003 

.002 

.002 

.001 

.001 

.000 

.000 

i . 

5730 . 

.020 

.016 

.013 

.011 

.009 

.008 

.006 

.005 

.004 

.003 

.002 

.001 

.001 

1.5 

3820 . 

.029 

.026 

.021 

.018 

.016 

.013 

.010 

.008 

.006 

.004 

.003 

.002 

.001 

2 . 

2865 . 

.038 

.034 

.029 

.025 

.021 

.017 

.014 

.011 

.008 

.006 

.004 

.003 

.001 

2.5 

2292 . 

.049 

.043 

.037 

.031 

.027 

.022 

.018 

.014 

.010 

.007 

.005 

.003 

.002 

3 . 

1910 . 

.058 

.051 

.044 

.037 

.031 

.026 

.022 

.017 

.012 

.009 

.006 

.004 

.002 

3.5 

1637 . 

.070 

.061 

.052 

.043 

.037 

.031 

.025 

.020 

.015 

.011 

.008 

.005 

.003 

4 . 

1433 . 

.079 

.069 

.060 

.050 

.042 

.035 

.029 

.023 

.018 

.013 

.009 

.006 

.003 

4.5 

1274 . 

.088 

.077 

.067 

.056 

.047 

.039 

.032 

026 

.020 

.015 

.010 

.007 

.004 

5 . 

1146 . 

.099 

.086 

.074 

.063 

.053 

.044 

.035 

.029 

.022 

.016 

.011 

.007 

.004 

5.5 

1042 . 

.108 

.094 

.082 

.070 

.059 

.048 

.039 

.032 

.024 

.018 

.012 

.008 

.004 

6 . 

955.4 

.117 

.102 

.088 

.076 

.064 

.052 

.042 

.034 

.026 

.019 

.013 

.008 

.005 

6.5 

882 . 

.128 

.112 

.097 

.082 

.069 

.057 

.046 

.037 

.028 

.021 

.014 

.009 

.005 

7 . 

819 . 

.137 

.120 

.104 

.088 

.074 

.061 

.049 

.039 

.030 

.022 

.015 

.010 

.005 

7.5 

764.5 

.146 

.127 

• 111 

.094 

.079 

.065 

.053 

.042 

.032 

.024 

.016 

.010 

.006 

8 . 

716.8 

.158 

.137 

.119 

.100 

.085 

.070 

.056 

.045 

.034 

.025 

.017 

.011 

.006 

8.5 

674.6 

.166 

.145 

.126 

.106 

.090 

.074 

.060 

.048 

.036 

.027 

.018 

.012 

.007 

9 . 

637.3 

.175 

.153 

.133 

.112 

.095 

.078 

.063 

.050 

.038 

.029 

.019 

.012 

.007 

9.5 

603.8 

.187 

.163 

.141 

.119 

.101 

.083 

.067 

.054 

.042 

.031 

.021 

.013 

.008 

10 

573.7 

.196 

.171 

.148 

.125 

.106 

.087 

.071 

.057 

.045 

032 

.022 

.014 

.008 

11 

521.7 

.216 

.188 

.163 

.139 

.117 

.096 

.078 

.063 

.049 

.036 

.024 

.016 

.009 

12 

478.3 

.236 

.206 

.179 

.151 

.128 

.106 

.085 

.069 

.053 

.039 

.026 

.017 

.010 

13 

441.7 

.254 

.222 

.192 

.163 

.138 

.113 

.092 

.075 

.057 

.042 

.028 

.019 

.010 

14 

410.3 

.275 

.239 

.207 

.175 

.148 

.122 

.099 

.080 

.061 

.045 

.030 

.020 

.011 

15 

383.1 

.295 

.257 

.223 

.188 

.159 

.131 

.106 

.085 

.065 

.049 

.033 

.021 

.012 

16 

359.3 

.313 

.273 

.236 

.200 

.170 

.139 

.113 

.091 

.070 

.052 

.035 

.023 

.013 

17 

338.3 

.333 

.290 

.252 

.213 

.180 

.148 

.120 

.096 

.074 

.055 

.037 

.024 

.014 

18 

319.6 

.351 

.306 

.265 

.225 

.190 

.156 

.127 

.102 

.078 

.058 

.039 

.025 

.014 

19 

302.9 

.371 

.324 

.280 

.238 

.201 

.165 

.134 

.108 

.082 

.061 

.041 

.027 

.015 

20 

287.9 

.392 

.341 

.296 

.250 

.212 

.174 

.141 

.114 

.087 

.066 

.044 

.028 

.016 

21 

274.4 

.410 

.357 

.309 

.262 

.222 

.182 

.148 

.120 

.091 

.069 

.046 

.030 

.017 

22 

262 . 

.430 

.375 

.325 

.275 

.233 

.191 

.155 

.126 

.096 

.072 

.048 

.031 

.018 

23 

250.8 

.450 

.390 

.338 

.287 

.243 

.199 

.162 

.131 

.100 

.075 

.050 

.033 

.019 

24 

240.5 

.469 

.408 

.354 

.299 

.253 

.208 

.169 

.137 

.104 

.078 

.052 

.034 

.019 


231 . 

.486 

.424 

.367 

.311 

.263 

.216 

.176 

.142 

.108 

.081 

.054 

.035 

.020 

26 

222.3 


.441 

.382 

.323 

.274 

.225 

.183 

.148 

.112 

.084 

.056 

.037 

.021 

27 

214.2 

.524 

.457 

.396 

.335 

.284 

.233 

.190 

.153 

.116 

.087 

.058 

.038 

.022 

28 

206 . ( 

.545 

.475 

.411 

.348 

.294 

.242 

.197 

.158 

.120 

.090 

.060 

.039 

.022 

29 

199.7 

.564 

.491 

.424 

.361 

.303 

.250 

.203 

.163 

.124 

.093 

.062 

.041 

.023 


For ordinates for center line of road, see pp 726 to 731 














































RAILROAD SPIKES. 


/OZ 


WEIGHT OF RAILROAD SPIKES. 


Tile liook-hea<le<l spikes t. commonly used for confining rails to 
the cross-ties, vary within the limits of the following table; the lightest ones 
for light rails on short local branches; and the heaviest ones for heavy rails 
on first-class roads. The spikes are sold in kegs usually of 150 lbs. For the 
weight of spikes ol larger dimensions, we may near enough take that of a 
square bar of the same length. What is saved at the point, suffices for the 
addition at the head. Price, Philadelphia, If 8*, from 2% cte. per lb. for % : 


inch thick; to 3 cts. per 3b. for % inch thick, 
turer. Canal St. and Germantown Ave., Philu 
dealers, 945 Ridge Ave., Phila. 


Corydon Winch, manufac- 
C.W.4H. W. Middleton, 


Size in 
Length, i 

ins. 

Side. 

No. per keg 
of 150 lbs. 

No. per !b. 

Size in 
Length, j 

ins. 

Side. 

No. per keg 
of 150 lbs. 

No. per lb. 


X 

TB 

526 

3.5 

5V* 

X 

Vz 

350 

2.33 


X 

k 

400 

2.66 

b]% 

X 

A 

289 

1.S3 

5 

X 

% 

705 

4.7 


X 

% 

218 

1.46 

5 

X 

A 

488 

3.25 

6 

X 

y 2 

310 

2.07 

5 

X 

k 

390 

2.6 

6 

X 

A 

262 

1.75 

5 

X 

* 

295 

1.97 

6 

X 

k 

196 

1.30 

5 

X 

k 

257 

1.71 






A mile of single-track road, with 2640 crcss-ties, 2 feet apart from center 
to center; and with rails of the ordinary length of 30 feet, or 15 ties to a rail; will 
have 352 rail-joints per mile ; and, with 4 spikes to each tie, will require 10560 spikes, 
or nearly 37 kegs (5500 lbs.) of 5% X iV a size in very common nse, which weighs 
a trifle more than % ft>. per spike. 

But an allowance must be made for rail-guards at road-crossings, which we may 
assume to be 30 feet wide, or the length of a rail. A guard will usually consist of 4 
extra rails for protecting the track-rails, and spiked to the 15 ties by which said track- 
rails are sustained. Consequently, such a crossing requries 15 X 8 = 120 spikes. 
For turnouts, sidings, loss, <fcc., we may roughly average 700* spikes more per mile ; 
thus making in all (if we assume one road-crossing per mile) 10560 + 120 + 700-= 
11380 spikes per mile; or say 6000 lbs. or 40 kegs of 150 lbs. 

Adhesion of spikes. Professor W. R. Johnson found that a plain spike 
.375, or % inch square, driven 3% ins into seasoned Jersey yellow pine, or unseasoned 
chestnut, required about 2000 lbs force to extract it; from seasoned white oak, about 
4000; and from well-seasoned locust, about 6000 lbs. Bevan found that a 6-penny 
nail, driven one inch, required the following forces to extract it: Seasoued beech. 
667 lbs; oak. 507 ; elm. 327 ; pine, 187. 

Very careful experiments in Hanover, Germany, by Engineer Funk 
give from 2465 to 3940 lbs. (mean of many experiments, about 3000 lbs.) 
as the force necessary to extract a plain inch square iron spike, 6 inches 
long, wedge pointed for one inch (twice the thickness of the spike), and 
driven inches into white or yellow pine. When driven 5 inches, the 
force required was about T * n part greater. Similar spikes, ^ inch square, 

7 inches long, driven 6 inches deep, required from 3700 to 6745 lbs. to 
extract them from pine; the mean of the results beiDg 4873 lbs. In all 
cases about twice as much force was required to extract them, from oak. The spikes 
driven across the grain of the wood. Experience shows that when driven 
with the grain, spikes or nails do not hold with much more than half as much force 

Jagged spikes, or twisted ones (like an auger), or those which were either swelled 

wv. hed *vl ea i r th *J ni 'l d * 1 u ° f t l lcir a11 P roved inferior to plain, square 

ones. When the length of the wedge point was increased to 4 times the thickness 

in Glossary resi8tance to drawin g out was a trifle less. But see “ Jag-spike ” 

When the length of the spike is fixed, there is probably no better shape than the 
plain, sqnaro cross section, with a wedge-point twice as long as the width of the 
spike, as per this fig. & 



15*miles of^road^ tUrn ° UtS and sidiuga amount to ab «ut 1 mile of extra track on 

























RAIL-JOINTS. 


763 


RAIL-JOINTS. 

Art. 1. A track, being weakest at tbe joints between the rails, where they 
u ire deprived of their vertical strength, has of course a greater tendency to bend at 
s those points ; and this bending produces an irregularity in the movement of the 
!i train, which is detrimental to both rolling-stock and track. Moreover, that end of a 
it rail upon which a loaded wheel is moving, bends more than the adjacent unloaded 
a ! ,ni l of the next rail; so that when the wheel arrives at said second rail, it imparts to 
t ts end a severe blow, which injures it. Thus, the ends of the rails are exposed to 
^ ar more injury than its other portions. Numerous devices have been resorted to for 
jfr strengthening the joints of the rails, with a view of preventing this bending entirely; 
a, >r, at least, of causing the two adjacent rail-ends to bend equally, and together; so 
as to avoid the blows alluded to. None of these joint-fastenings, known as chairs, 
-sf ish-plates, wooden blocks, &c, have proved entirely satisfactory. 

Much of the deficiency ascribed to the fastenings, is, however, really due to want 
of stability in the cross-ties at the joints, and more attention must be directed to 
this latter consideration, before an efficient fastening can be obtained. Observation 
shows that when the joint ties are very firmly bedded, almost any of the ordinary 
[fastenings will (if the joint is placed between two ties, iustead of resting upon a lie;,* 
(answer very well; whereas, when the cross-ties are so insecurely bedded as to play 
up and down for half an inch or more uuder the driving-wheels of the engines, the 
strongest and most effective fastenings soon become comparatively inoperative. All 
the parts of the best of them will in that case become gradually loosened, warped, 
bent, or broken. 

Experience has established the superiority of suspended joints over supported 
..piles, Long fastenings, perhaps, possess but little superiority over short ones, where 
the track is not kept in good repair; for the great bearing of the former, although 
if imparting increased firmness on a good track, becomes converted into a powerful 
ill leverage, by which it accelerates its own destruction, in a bad one. An element in 
!, the iftjory of joints, is the omission of proper fastenings at the center of the rails, 
i Each rail should be so firmly attached to the cross ties at and near its center, as to 
compel the contraction and expansion to take place equally from that point, toward 
jr each end. It would probably be somewhat difficult to accomplish this perfectly. 
4 The attempts hitherto made have failed. 

Under the extremes of temperature in the United States, bar iron expands 
i, or contracts about 1 part in 916; or 1 inch in 76^ feet; consequently, a rail 
; iO ft long will vary inch : and one 20 ft long fully y± inch. 

Beside tliis. the rails are very liable to move or creep 
bodily in the direction of the heaviest trade, especially when the 
grade descends in the same direction; and by this process also the joint-fastenings 
| are exposed to additional strain and derangement.* 

All rails appear to become elongated very slightly at their ends by use; and this 
‘ renders a full allowance for contraction and expansion the more necessary. 

Art. 2. Even joints and broken Joints. If, in the two lines of 
'> rails forming a track, the joints are placed opposite to each other, they are called 
“even joints;” while “staggered” or “broken” joints are those where each joint 
in one of the lines of rails is opposite to the middle of a rail in the other line. In 
the latter case, the jar of passing from rail to rail is less severe, but of course more 
frequent, than where both wheels make that passage at the same time. 

Art. 3. Beveled, or mitred Joints. To lessen this jar, Mr. Sayre sug¬ 
gests cutting the rails so that the vertical plane forming the rail end shall make an 
angle of 45° to 60° with the longitudinal vert plane of the web of the rail, instead of 
the usual right angle. This would permit the use of longer rails than are now laid, 
as the great space (*/£ inch or more) between the ends of such long rails in cold 
l weather, would not be so serious an objection when the ends were thus cut obliquely. 
] This method of cutting the rails has been tried, with good results, but has not yet 
come into general use. It is claimed that a comparatively inexpensive change in the 
i arrangement of the saws at the rolling mill, would permit the rails to be cut with 
i ends at any angle, as readily as with square ends, and without further increase in 
! the cost of sawing. 

*Iu the first case the joint is called a suspended one; in the last a supported 
one. 


i 









764 


RAIL-JOINTS. 


Art. 4. Fish-plates, Fig 1, and Angle-plates, Figs 2, 3, 4, and 5, hav< 

nearly supplanted all other forms of joint on the principal 
railroads of the U. S. Although the rails are almost uni¬ 
versally of Bessemer, or similar, steel, the joint-plates ai'e 
generally made of iron. They are rolled in long bars, and 
cut off in any desired length, generally about 2 it; and 
are bolted together, and to the rails, by*4 bolts, 2 in each 
rail-end. 

Art. 5. The fish-plate Joint was one of the 

earliest suggested. It was introduced upon the Newcastle 
and Frenchtown II R, in Delaware, by Robt. II. Barr, in 
1813. The weight of a complete fish-plate joint, includ¬ 
ing bolts and nuts, is about 20 lbs. 
lig 1, one-fifth of real size, represents a fish-plate joint 



Fig.l. 


- --» * - J' ^ V.v mo rv ItMIH 

"\ a £ e ^ ^ a, V^ r,a l**on Co,office 218 S 4th St. Phila, with a steel rail weigh int 
oO lbs per vard, by the same Co. The thickness of plate at /‘is yb inch. The plate.- 
weigh 414 lbs per lineal foot of a single plate. 

Art. 6. The principal advantage of the angle-plates is that the spikes 
which are driven through slots in their flanges to confine them to the cross-ties 
*iT e 1 ° *° co,interact the tendency of the rails to “creep.”* (See Art 1.) When 

» / a 8 at 'e used, tiie fianges of the rails have to be slotted for this purpose. 

A further advantage of the angle plate is that it transfers at least a part of th< 
load directly to the ties. It thus supports the rail better than the fish plate can do 
and may continue to give some support even if the bolts should become somewha 
loose, provided the spikes hold firm. Moreover, the spreading base of the anirl< 
joint adds greatly to the lateral strength of the rail at the joint. 



rr 


TTT 

INCHES 

Fi g,2. 


T - 1 



Fig. 2 -A. 




Splice. 

For rails. 

Height. 

Thickness on 
center line of 
bolt. 

Weight of 
one bar. 

Approximate cost 
of complete joint. 

No. 4, 

“ 6, 

60 lbs & 70 ft>s 
85 lbs 

3 % ins. 

Q 9 11 

^T5 

% inch 
§§ “ 

26»4 lbs 

3034 “ 

8 1 50 

1 70 


,:°«u 4 t- x ™ e .nM is r lm,h 

inch thick. ^ lD1CK > W “H nut 1 % inches square X % 


. * 0n St. Louis bridge (steel arches) and its eastern annroacli fnlnf« 
iron columns) the rails creep, in the direction of the traffic. about a foot da 
both up and down a grade of 80 feet per mile, and with such force ths?XT 1 

various fastenings were used, in order to prevent the creeping none proved effi^ 
The track is now adjusted daily to accommodate the creeps ^ For ? a verv^ 
ing account of this case, see a paper by Prof T R TnimcAn°{n r er ^ 1 _ n ^ ere8 ^ 
Associa*ion of Engineering SocfetiS Vd. IV No ^ 0^884 anVaw" 81 ^ 
editorial discussion, in Railroad Gazette, Jan. 2 d, 1885.’ Prof’johnson attributesXh 
creeping to a wave motion of the rail caused by the passage of traX attnbutes th 


























































RAIL-JOINTS. 


765 


The wheel-tread and 76 lb steel rail, shown (one-fifth of real size) in 
Fig 3, are those designed by Robt. If. Sayre, 
C. E., and used by the Eeliigh Valley R R, 
under very heavy traffic. The angle-|>l«ite 
joint was designed by Mr. John Fritz, 
Supt Bethlehem Iron Co, Bethlehem, Pa, and Mr. 
Sayre, and has been in use on the Lehigh 
Valley R R for the past 12 years. 

These forms of wheel-tread, rail, and splice, are 
the result of careful study, and each detail has 
been modified from time to time as experience 
dictated, until now they are probably the most per¬ 
fect in this country. Mr. Sayre places the stems 
of the two plates much farther apart than usual, 
thus giving the joint greater lateral strength; at 
the same time adding to its vertical strength by 
the support given to the lower side of the rail¬ 
head by the upper enlargement c; while the lower 
one a secures a full bearing on the flange of the 
rail. The joint, for 76 lb rail, complete, 2 ft long, 
with 4 bolts % inch diam, weighs 40 to 48 lbs, 
depending upon the thickness of the angle plate. 
The drilled bolt-holes in the stem of the rail, arel 
inch diam, to allow the rails to contract and expand. 
Figs 4 and 5 (one-fifth of actual size) show an ftngle-plfttc joint 

made by Cambria Iron Fo, Johns¬ 
town, Pa, office 218 S 4tli St, Pliila, and fur¬ 
nished with their patent nut-lock, 
which consists of a small piece, or “key,’’ 
p, of Bessemer steel, semi-circular in cross- 
section at one end, and tapered to a hori¬ 
zontal edge at the other. After the nut has 
been screwed to its place, the key is driven 
‘ close up to it, and then the pointed end of 
the key is bent up (as shown in Fig 5) by a 
special tool with a lever attached. The key 
is prevented from falling out sideways by 
the edge of the longitudinal groove, Fig 4, in 
the angle-plate, into which it fits. This nut- 
lock resembles that used with the old form 
)f Fisher-joint, see B Figs 15. See nut-lock washers, p 408. 

Art. 10. Both fish- and angle-plates are apt to crack vertically about the mid- 
lle of *their length, or opposite to the joint in the rail. To obviate this, the 
.6 Jamison "bar 99 (which is made either of fish or of angle foim) is rolled about 
half inch thicker at the middle than at its ends. The thickened portion is about 8 
lon ir extending say 4 ins each way from the joint, but the uppei edge of the bar, 
n which the head of the rail bears, and in fish-burs the lower edge also, are made 


Art. 



I r riv 11 rp 

0 1 


T 

Fig.4. 


3 4 

INCHES 


Fig.5. 


ins 
upon 


6 


«L 

^ of'this increased thickness throughout their length. These joints are (1886) largely 
used on the Western railroads of the U. S. Their cost is about the same as that of 
“ ordinary fish- and angle-joints. They are made by Morris Sellers & Co, office Iso 

Ashland Block, Chicago, III. , . , , 

Art. 11. Fish- and angle-plates, of all the patterns shown, and others, are 
-rolled to suit different sizes and shapes of rails, lhe holt 
heads are usually round, and the shoulders of the bolts, immediately under 
the heads are therefore made of oval cross-section, fitting into corresponding oval 
holes in the fish- or angle-plate. The bolt is thus prevented from turning when the 
-nut is screwed on, and afterwards. Many devices have been tried, with a view to 
of preventing' the nuts from wearing loose (see lock-nut washers 
3 / J, 408) The Vulcanized Fibre Co, Wilmington, Del, furnish a vulcanized 
‘ washer which is intended to act as an elastic cushion, deadening shocks and 
vibrations’ They are said to become hard, and lose their elasticity, in time. 

The plates are frequently rolled, as in Figs 17, with a Ion tu din hi 

as wide as the head or nut of the bolt, and about % inch deep, running their entire 
ili length This groove receives either the head of the bolt, which in such cases is 
I made square or oblong and inserted first, and the nut afterwards screwed on ; or 
!!• ielse the nut is first placed in the groove, and the bolt afterwards screwed into it. 
b8 This is intended to prevent the unscrewing of the nut, but cannot be relied upon to 

;J well to have the slots in the flanges of rails or of angle-bars so spaced that 









































7 66 


RAIL-JOINTS. 


the two spike*) of a joint, driven into the same cross-tie, 

snail not he directly opposite to each other, hut “staggered,” so as to 
diminish the danger of splitting the tie. 

Joints are irequeutly laid with one fish- and one angle-plate. 

In 1888, angle- and fish-plates cost about 2 cts per lb; bolts and nuts, 3 cts p°r 
lb The cost of a complete angle-joint, with four bolts, for 7u-tb. rail, is from 8U cts 
to Si 20; of a fisji-joiut, 60 to 80 cts. See p. 764. 

Art. 12. It will be noticed that both fish- and angle-plates act by placing a 
support under the head of the rail. The Fisher bridge-joint. Figs 6 to 9 
made by Mr. Clark Fisher, Trenton, N J, applies the support under the base of the 
rail. 

The principal feature of this joint is a flanged beam, Fig 6, about 6 ins wide and 
22 ins long, which extends across, and is spiked to, the two joint-ties, as in Fig 7. 
The holes for the spikes are placed so that the two spikes in the same tie are not 
opposite to each other; and the flanges F F also are staggered, so as not to interfere 
with the driving of the spikes. The joint-ties T T are placed 7 inches apart in the 
clear. The beam has an upward camber of about one-eighth of an inch. The two 




rail-ends formtng the joint, rest upon the beam, and meet at the middle of its 
er gth. They are held down to it by a single U-shaped holt B, of 1 inch diam 

"r °\' T T,les v n ' It . s bear directly upon tl.e horizontal upper sides 

tlie lore-locks I, I., on© of which is shown separately in Fig 9. The fore-locks 
are rolled to fit accurately to the rail-flanges. The legs of the U-bolt pass first 

enf°in S M ie clrcuIar boles A In 'be beam. Fig 6; next through rounded notches 
cut in the coiners of the rail-flanges ; then through the holes in the fore-locks • and 

. aS m' V i "- 0,1S ' U r e m - tS * Be . hv, ‘ en t,ie U'bolt and the bottom of the beam is placed 
a small piece s, of spring steel,slightly cambered downward, and having two semi-cir- 
culai notches for the legs of the U-bolt, which hold it in place. This is intended to keen 

!• nt V ilSt to . take , up iiny ,oose s P ace Produced by the wear of tl.e surfaces il. 
contact, to render less abrupt the strains on the bolt, and, by keeping the threads 

lo 6 tVI^T f, r e of 'be holt, to prevent the* nuts from becoming 

loose. The joints are shipped from the factory complete, and with all the parts 
bolted together ; the nuts being screwed down to within about two threads of their 
fhe fo&cks. 0 iU the endS ° f the rail - flan ee8 can be easily slid into place und^ 

an additional precaution against creeping of the rails, the rail-flanges may be 
slotted near their ends, as in cases where flsli-plates are used, and spikes driven 

addition h6Se 8 0t , S * For ® u< r h cases 'be beams are punched, at the mill, with four 
additional square holes a little further from the edges of the beam than the others 
Unhke the angle- and fish-plate joints, the Fisher may he used with any section of 
T-rail; and the head of the rail may be made stronger by being rolled pear-shaDed 
which is inadmissible with fish-and angle-joints, because these requfre a nearly 



















































RAIL-JOINTS. 


767 


lorizontal bearing on the under side of the head. The “ Fisher” requires no diali¬ 
ng or punching of the stem of the rail. It costs about 25 per cent more than a 
isli- or angle-joint for the same rail. Its weight, complete, lor 65-lb rail, is about 
•2 lbs. 

Mr. Fisher makes also an extra strong joint with three U-bolts, 
or heavy curves and for places liable to wash-outs. It "is intended to support the 
joint, even if the ballast is removed from under the joint-ties. Either of the Fisher 
oints can be made of any desired weight. The “Fisher” is largely used on some 
>f the principal eastern roads, and with very satisfactory results. Figs 15, p 768, 
how an old form of this joint. 

Art. IS. The Gibbon boltless rail-joint, Figs 10,11, and 12, invented 
>y Mr. Thos. II. Gibbon, C E, Albany, N Y, is a supported one. Two inches in length 



if the head, at each rail-end, have to be cut off, as in Fig 10; but no further cutting, 
uid no punching or drilling, of the rail, is required. Over the ends of two rails, thus 
■ut, and placed together in their final position on the ties, the Bessemer steel “ sad- 
lie-casting,” Fig 11, is placed. The top of this casting is shaped so as to correspond 
with the head of the rail; and its length, 4 ins, just occupies the space cutaway 
from the ends of the two adjoining rail-heads. The feet of the casting fit into notches 



corresponding slot in the opposite leg of the saddle-casting. Spikes are then driven 
into the tie through the four holes in the base-plate, and the joint is 
saddle-casting and base-plate weigh, together, about 24 lbs,and cost ( 188 b) ljM. 00 . 

4rt. 14. The following are some forms of rail-joint that have fallen into disuse, 
or have* been proposed but not adopted. They illustrate early practice, and may be 
useful as hints. 

Fig 13 was an early 
form of supported 
w r « u g h t-i r o n 
chair. It was about 
7 ins square. % thick, 
and weighed 10 lbs. Fig 
14 is a later form, still 
furnished, to some ex¬ 
tent, by the Tredegar 
Co, of Richmond, Va. 




Fig.14. 


Fig. 13. 

Figs 15 show one of the earliest forms of the Fisher joint. It 

was at one time largely used on the Lehigh Valley R R. (For the present form of 
this joint, see p 766.) It was a suspended joint; and, instead of the long supporting 
beam, or bridge-plate, of the present pattern it had a plate c, 6 ins square. It had 
2 U-bolts, each of 1 inch diam, and the fore-locks 11 were 6 ins long, and had two 
holes each. The lower side of the thread on the bolt was made horizontal as at j. 
The thread in the nut was somewhat as at l. In screwing on the nut, these two 


















KAIL-JOINTS. 


768 


threads were forced into conformity, thus employing the principle of the preset 
Ilarvey lock-nut, p 408. A nut-lock B, somewhat similar to the Cambria patei 
lock, p 765, was used as an additional precaution. 






Figs 16 show a remarkable suspended fastening, called tlie Ring 

joint: highly approved of at one time on the Camden & Amboy road (now Penn 
K II, United Railroads of New Jersey Division) on which it was employed for man 
years, under a heavy traffic; to the almost eutire exclusion of others, except fo 
experimental comparison. 



This fastening consisted of a simple welded triangular ring tea, (in the end view 
or in in the side view.) ^ an inch thick, and 3]^ ins wide. This ring passed throng 
* slot, v v, (see middle fig,) 4 ins long, cut into the adjacent rail-ends. Two cast-iro 
wedges w w, 6 to 8 ins long, of a shape to fit the ring and the rail, were inserted be 
tween them ; and a thinner one s s, of plate iron, below the rail. The first were cas 
around a cylindrical rod of rolled iron, (w of the end view, it of the side view,) aboil 
inch diam ; and a little longer than the wedges; for increasing their strength 
and for preventing them from falling out from the chair in case they should break 
which they sometimes did. The rail-ends sometimes split, as shown at l and k. 

The joint was suspended between two cross-ties, 1 ft apart in the clear. 








































































RAIL-JOINTS. 


769 


Figs 17 represent the combined suspended joint-fastening introduced upon the 
tiljhilada & Reading railroad by J. Dili ton Steele, C E. 'Some of them are still 
l use (1884). 



The joint is suspended between 2 cross-ties, placed 1 ft apart in the clear. B is a 
lock (known as Trimble’s splice) of oak, 3 by 3 ins, and 3 ft long, dressed on one 
ide to fit the outside of the rail; and c is a rolled fish, 17 ins long, (shown more in 
etail at F,) placed on the inside of the rail. This fish and the oak block are bolted 
igether, through the rail, by two %-inch screw-bolts a a, 13 inches apart. Under 
lie rail is a rolled chair d, 2 ft 8 ins long, 5 ins wide, and ^ inch thick; turned up 
£ inch along each of its two sides, and fastened to the 2 wooden cross-ties by 4 hook- 
eaded spikes, inch square, by 5)^ ins long. The heads of the two screw-bolts are 
lade somewhat oblong, (about % inch by \%,) for fitting into the groove nn , seen 
long one side of the fish ; so as to prevent the tendency of the bolts to revolve under 
die action of the trains, and thus unscrew the nut at the other end. The nuts, how- 
ver, unscrew themselves, notwithstanding this precaution. The strain on the screw- 
nits is great, both vert and hor; and it becomes greater as the wooden blocks in 
ime lose (as they do) their close fit to the sides of the rails. The blocks then cease 
j act in perfect unison with the other parts of the fastening, in sustaining passing 
>ads; and when the track is not kept in good order, the various parts may plainly 
je seen to yield and move in various directions, independently of each other. The 
l olt-nuts then loosen; and the fish pieces, long chairs,and long bolts, become bent; 
nd sometimes split, or break entirely. The long wooden blocks B crush and decay 
joonest near where their tops are in contact with the rail. They, however, have an 
'verage life of 6 to 8 years, upon roads kept in tolerable order. 




Fig. 30. 


Fi<r 18 is a joint for U-rail. It was of rolled iron, 5 ft long, and rested on 3 ties. 
► was riveted loosely to one flange of the rail, as shown on the right-hand side. It 

ept the rails in position very well. . . - ^ w r v 

Fig 19 is a joint-fastening proposed many years since by Alex. W. Kae, 1/ 
f Benna. It certainly possesses merit. . , . , , 

Fi- 20 was also one of the numerous joint-fastenings suggested at an early day; 
iit like the foregoing, it never came into use. In lengths of about 8 ins, it would 
robably make an efficient fastening; especially with the addition of a broad thin 
edge between the bottom of the rail and the toot of the chair lhis would dtmin- 
h the difficulty of sliding the chairs on or off of the rail; and would thus make it 
isy to employ ‘larger ones; besides insuring a firm bearing for the base of the rail 

pou the fastening. 







































770 


TURNOUTS 


* 

TURNOUTS. 


Art. 1. To enable an engine and train to pass from one track, A It, Fig 1, 
another, A D, a turnout is introduced. This consists essentially of 



switch, <7 m p s, a frog 1 ,/, and two fixed guard-rails, g and g'. If a swit 
is made to serve for two turnouts. A D and A D', Fig 2, one ou each side of the nn 
track, A B, it is called a three-throw switch. 



Art, 2. When a train approaches a switch in the direction of either arrow, I 
1; or so that it passes the frog before reaching the switch, it is said to *‘trai’ 
the switch. When it approaches in the opposite direction, passing the switch bef 
reaching the frog, it is said to “ face” the switch. Fig 3 represents a portion 
a double-track road in which the trains keep to the right, as shown by the arro\ 
In tliits fig, Y and W are “ trailing ” switches: and X and Y are “ facing ” switch 
In order to leave the main track by a trailing switch, a train must move in a din 
tion contrary to the proper one on said track. 

Art, 2. Misplaced switches. A moving train, facing any switch, rm 
plainly go as the switch is set, whether right or wrong. If wrong, serious accidc 
may result. For instance, the train may run upon, and over the end of, a sin 
trestle siding, or may collide with a train standing or moving upon the turno’ 
Safety switches, such as the Lorenz, Arts 13, &c, and Wharton, Arts 18, i 
are so arranged that trains trailing them can pass them safely, even if the swit 
is misplaced. But in the case of the plain stub-switch. Art 4, when m 
placed, a trailing train will leave the rails at h and r, or « and u. Fig 4, and r 
upon the ties. 

Stub switches are frequently provided with “safety-castings” of in 
bolted to their sides, and reaching from their toes m and f, Fig 4. several feet tow; 
p and q. These, in case of misplacement of the switch, receive the flanges of t 
wheels of a trailing train, and guide the w heels safely on to the switch-rails qm a 
i>s. The ‘’Tyler” switch is arranged in this way. 























TURNOUTS 


771 


4* /I 1 ® common blunt-ended or sfwb-switeli consists 
tscntiaJl.y of two rails, q m and p s , Figs 1 and 4. The ends, q and p , of these 
■If, where they are fixed in line with the main track, form the •‘heel” of the 
itch. Iheir other ends, in and s, form the toe,” aud are free to move from <2 



d n (where they are in line with the main-track rails, bh and rz) to m and s 
here they are in line with the turnout-rails, ex and uy). 

On main lines of road, the switch-rails are usually from 18 to 26 feet long from 
el to toe. Formerly their heels, q and p, were fixed by being confined in the same 
airs which held the adjoining ends of the main-line rails; or by being connected 
th said rails by short fish-plates. See p 764. In either case they remained prac- 
:ally straight, even when set for the turnout. Now, however, they are generally 
ide long enough to extend 4 to 6 feet back from their heels, toward A; aud this 



ditional length is spiked unyieldingly to the ties; so that, when set for the turnout, 
3 switch-rails bend so as to form (at least approximately) a part of the turnout 
rve. Their toes, m and s, in any case, rest and slide upon iron “bead-pla,t« s.” 
P., Fig. 4, shown in detail at Fig. 5. which represents a riveted steel plate, made 
the Weir Frog Co., Cincinnati, Ohio. Base plate .1.1 inch thick. Cost, 1 S88, 

out #5.50 per pair. These head plates also receive the ends, b e and r «, Fig. 4, of 
3 main-track and turnout rails. In three throw switches, the head-plates must of 

> irse be longer, to give room for the three rail-ends side by side. 

»' frequently a plain strip of iron, about 3 inches wide by half inch thick, is fastened 
the upper surface of each tie, under the base of the rail, for the latter to slide on. 
i 'he switch-rails are connected together by from 3 to 5 transverse wrouglit-iron 
imp-rods, R R R', Fig 4, full 1% ins diam. These are fastened to the rails in 
ious ways; generally as shown in Fig 6. The clamp-bars should be placed, if 

possible, at least as low as the tops 
of the cross-ties, so as to avoid 
danger of their coming iuto con¬ 
tact with any portions of cars or 
engines that may become par¬ 
tially detached and drag on the 
track. 

One of the clamp-bars, R', is 
r the toes, in and s, Fig 4. It projects beyond the track, and is jointed, as shown 
Figs 7 and 8, and connected with the lever, L, by which the switch is moved. 

> tie, T, Fig 4, to which the head-plates are fastened, is made longer than the 
ers, in order to give room for the switch-stand, M, Figs 4,7,and 8, which is bolted 
ts upper surface. This tie should also be of larger cross-section than the others, 
fectly sound, and well bedded; because upon it coine severe strains due to the 
sage of cars and engines across the space between the rail-ends, m and e, s and u, 

, Fig 4. See second paragraph of Art 10, p 773. 

















































772 


TURNOUTS, 


t. 5. The switch-levers, and the switch-stands to which they are 
ed, are made in a great variety of forms. See Figs 7, 8,9, 14, 15, and 17. T hat 

_ ... . -- J ’ vcr 


Art 

attached,_ 0 _ „ w ....... T 

shown in Figs 7 and 8 is the “ Tumbling-lever stand” or * 4 Ground-lev 
stand,” and, in its numerous modifications, is very largely used. It is so arranged, 
that, whichever way the switch is set, the crank, C, is on the dead center, so that 
the lateral strains of passing cars or engines can exert no tendency to turn it. The 
44 Greenwood” stand, made by the Pennsylvania Steel Co, Steeltou, Pa, has 



iai 


rii 


a tumbling lever, but is so arranged that (unlike the ordinary tumbling lever) i 
can be used for either a two-throw or a three-throw switch. 

Tumbling switches are convenient because they occupy but little space. By mean e 
of a target or lantern, connected with the switch, they may be made to indicate t< 
the engine driver the position of the switch. 

When the switch is set either way, the lever is padlocked to a staple driven int( 
the tie and passing up through the 6lot in the handle of the lever. The lever is fre 
quently made with a weight of say 20 lbs on its free end, to aid in bringing it dowt 
to its proper position. Price of a tumbling-lever stand, 1888, $3.50 to $5.00, with 
out target. 

Art. 6 . Fig 9 represents a common form of the upright lever and stand, a 1 
The switch-rod, R' Figs 4, 7 and 8, is generally at tached at the lower end, A, of the lever 
The cast-iron frame, F, is fastened to the long tie, T, Fig 4, by large screws o 
Bpikes, which pass through its broad feet or flanges, B B. The top of the frame ii n 
provided with two notches, N N, and staples, to which the lever is secured by i ns 
padlock. When this stand is to be used for a tfcree-throw switch, the frame ha 1] 
three notches and three staples. The upright stand may be used wherever it wil id 
not be in the way of passing trains. The target, T, at the top of the lever, b; 
showing the position of the latter, indicates to the driver of an approaching eugim 
which way the switch is set. 



Fig- 0* 




MONKEY SWITCH 


Fig. 10. 


Art. 7. In the 44 Monkey-switch,” Fig 10, the crank, ot, is moved hor 
zontally through an arc of a circle by means of the lever, h h, about 3 ft long, whic 
fits upon the square head, s, of the vertical spindle or pin, « o. The switch-rod, R' Fij 
4, 7 and 8, is attached to the pin, i v. 

Many modifications of the monkey-switch are in use. The spindle, s o, is fr< 
quently made long enough to bring the lever to about the level of the hand; an 
the lever is permanently attached to the stand, and hinged near the spindle so i 
to hang down, out of the way, when not in use. To the top of the spindle is fr 
quently attached a vertical rod of any desired length, and carrying at its top a targi 
which turns as the spindle does, and thus indicates the position of the switch. 







































TURNOUTS. 


773 


Art. 8. All parts of the switch-stand, and the tie upon which it rests, should 
perfectly rigid, because it is very important that they should hold the ends of 
ie switch-iails exactly in line with those of the main line and turnout,. They 
perefore, in view ot the great strains to which they are subjected, must be strongly 
■nstructed, and frequently looked after. See Automatic switch-stand, Art 12, 
id I>e Vout’s switch-stand, Art 14. 






<L 



Fig. 11. 


Art. 9. In Figs 1 and 4, dqm is called the switch-angle. The dist, dm, 
gs 1, 4, and 11, required for the motion of the toes, is called the throw of the 
itch. It must be equal at least to the width, dw, Fig 11, of the top of the rail, 
addition to a width, wm, sufficient to allow the flanges of the wheels to pass 
)ng readily between b and e, Fig 1, and between r and u. The tops of the rails 
e generally between 2 and ins wide; and about 1% to 2]/ 2 ins suffice for the 

nges. The throw, d rn, however, is commonly about 5 ins. 

The gauge. Fig 6, p 771, of a railroad track, is the distance between the inner 
les G G' of the heads of its two rails. Hence these inner sides are called the gauge 
ties of the rails. 

Art. 10. The stub-switch is cheaper in first cost than the improved safety 
itches, Arts 13, 18, etc, but is less economical in the long run. 

As it is very essential that the toes of the switch-rails should never come into 
itact with the adjoining rail-ends, a space of about an iuch must be allowed 
the toes for expansion, and for “creeping” (see p 764). This renders the blows 
passing trains very severe, and injurious to rolling stock, and to the rail-ends, 
e latter are worn away rapidly and must be frequently renewed. From the same 
ise the tie under the head-plate is apt to become loose in its bed. 

In ordering fixture* for stub-switches, the exact section of rail, 
d gauge of track, should be given. The cost of a stub-switch with switch¬ 
ed, is, 1888, from $18 to $30, according to size, finish, character of stand, &c, &c. 


i 













774 


TURNOUTS 




* 


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TURNOUTS. 


775 


Art. 12. The 44 Automatic” switch.staml, made by the Penna Steel 
- / Oj is shown in hor section by Fig 14, and in vert section by Fig 15; and is designed 



Fiff. 14. 


Fig. 15. 

especially for split-switches like 
those shown in Figs 12 and 13. It 
is operated by a tumbling-lever, L, 
Fig 14. To the hor axis, A, of the 
lever, is fastened the beveled pin¬ 
ion, P. This engages in the teeth 
of the quadrant, Q, and moves it 
horizontally through a quarter of 


a circle, when the lever L is thrown from its position L to that 
of L', shown by the dotted lines. The rod, R', from the switch, 
is attached at X,or at X',to this quadrant; and the movement 
of the latter thus sets the switch, and at the same time turns 
he target, T, Fig 15, or lantern, fixed to the vert spindle, S, thus indicating the 
>osition of the switch. The gearing is enclosed in a cast-iron case, as shown in Fig 
15. If a train on the turnout, moving in the direction of the arrow, Fig 1 (or “trail- 
ng”), approaches a split-switch provided with the automatic stand, and set (through 
>versight or otherwise) for the main track, as in Fig 12, the flange of the first wheel, 
messing between rails X' and S'V', will push the switch-rails into the proper posi- 
ion, Fig 13, for the turnout; at the same time necessarily throwing the weighted 
ever over into the reverse position and turning the target so as to indicate that the 
iwitch is set for the turnout. A similar movement of the switch, in the opposite 
lirection, takes place if a trailing train on the main track approaches the switch 
.vhen set for the turnout, as in Fig 13. Hence the term “automatic,” as applied to 
j;his stand. The switch is thus made a safety switch. 

The cost of the automatic switch*stand (1888) is about $15. 



Art. 13. The Lorenz safety-switch, designed by the late Wm. 
,orenz. Esq, Ch Eng of the Phila & Reading R R, is a split-switch, in which the 
;onnecting-bar, R', nearest to the toes, is provided with a spring’. S, lig 16, placed 
mmetimes between the rails, as there shown; sometimes outside of the track. This 
spring permits the moving of the switch-rails by the wheels of a trailing train, as 
Joes the automatic switch-stand, Figs 14 and 15; but, after the passage of each wheel, 
the spring returns the switch-rails to their original position. The blow of th* 







































































































































776 


TURNOUTS. 



switch-nils against the stock-rails, thus occasioned, is injurious to both, and liable to 
break the former. On the other hand, the compression of the spring, during the 
passage of the train throueh the switch, sometimes impairs its elasticity, so that it 
then fails to return the switch rail to its proper position in contact with the stock- 
rail, and allows it to remain half an inch or mo^e away from it, and in danger ol 
being struck by the wheel flanges of approaching trains “facing” the switch. A 
similar accident may happen during the ordinary working of the switch, if an 
obstacle, as a small stone, becomes lodged between the switch-rail and the stock- 
rail; fir the spring may permit the switchman to force the switch-lever home to ite 
place without bringing the two rails properly into contact. See Art. 14. 

Art. 14. »e t out's safety switch-stand. Fig 17, made by Penna 
Steel Co, is designed to remedy this. In this stand, the spring is placed in, and se- 
cured to, a semi-cylindrical iron spring-case or box, B; to the opposite sides of which 
are tixed two hor axles. One ol these is shown at A. This axle passes through the 


6 witcli-lever, L, near its fulcrum, F. It also passes through the inverted T-shaped 
slot, H, in the tigid bar, S, which, together with the bar, W, attached to the spring- 
case, is jointed, at J, to the switch-rod, R'. W hen the switch is properly set, either 
for the main line or for the turnout, the axle. A, is in the hor part of the slot, II, 
and immediately under the vert part, so that there is no obstruction to the move¬ 
ment of the switch, and a trailing train will open a misplaced switch as explained 
in Aits 12 and 13. Cut when the lever is raised,for the purpose of setting the switch 
in the other positiou, the axle, A, rises into the vert part of the slot, as in the fig, 
lifting the spiing-case with it. It now any obstruction prevents the switch-rail 
ftom being pressed home, the rigid bar, S. by means of the axle, A, prevents the 
lever, L, from moving farther. Cost of De l out’s stand, 1888, is about $12, 
without target. 

Art. 15. Theory would require that the lengths of the switeh-rails, 

in split-switches, should vary with the l^ulius of the turnout curve, and formerly 
they were so made. Where this radius is such that a No 10 frog (see Art ‘26) is re¬ 
quited, the switch-rails should, theoretically, be 28 ft long. But in practice a uni¬ 
form length of 16 ft (just half the usual length of the steel rail from which th¬ 
aw itch-rails are cut) for all turnouts, gives the best results, combining economy o 
manufacture with greater strength, and greater ease of handling, than are possihh 
with much longer rails. 

Art 16. It will be noticed that in point-switches (as also in the Wharton switch 
Arts 18, &c) there can he no such jar as that occasioned in the stub switch by tin 
long space between the toes of the switch-rails and the ends of the adjoining rails. 

Art-. 17. It is important that the thin portions of each switch-rail should he 
carefully shaped so as to receive throughout a firm lateral support from the stock 
rail w hen in contact w ith it. Otherwise the switch-rails are in danger of bendin< 
under the lateral pressure of passing trains. This might throw r the point out frcm 
the stock-rail, endangering the train. 

The price of a 15-ft Lorenz switch, including the two point-rails witl 
connecting-bars spring spring-fixtures, sw itch-rod, slide-plates,and rail-braces, l.u 
exclusive of stock-rails, lever, and stand, is, 1888, about $30 to $40, according to weigh 
of rail, gauge, &c. 6 

. IiOrenz switches about 7’4 ft long are made for yard use Price 

including the same items as in the full-sized switches, 1888. about $25 to $33. 
w -l* 17 ~ Fl S\ 17a shows a three-throw point switch made by Th< 
Weir Irog Co., Cincinnati, Ohio. It lias the usual stock rails, C and Z, antf foui 

lai ?’ ,®» T ie 8vv i tc h rails all slide upon the same set of iror 

friction plates, which are spiked to the ties under the rails, but are not shown ii 
the figure. Rails A and B, are held rigidly together by four comiectino- 
a, a, a, a, while X and Y are similarly connected by the other four connecting bar 



























TURNOUTS. 


777 


jS ii in C 



> Ea . ch P air of 8 ^i fch rails ’ thus formed, moves independently of the other; 

he switch-rod R a operating the pair A-B, and R y operating the pair X-Y. Our 

id^nntrar 8 8w,t chas arranged for two separate switch-stands, one on the near 

he rod R « nnH h t e n r ° d •? rails XY and oae on the farther side, operating 

he lod R a and the rails A-B; but it may be arranged, instead, so that both pairs 

f rads may be operated by means of a single stand, placed on one side of the switch, 
1 he Lorenz spring (Art. 13, p. 775) may be nsed with this switch, if desired. 

The figure shows the switch set for the track X Z. 

To set it for the track A Y, the rod R y is pulled, and draws rails X-Y over, 
ringing Y into contact with Z, so that wheel K may run upon rail Y, and leaving a 


Fig. 17 5 
SECTION AT ro O 

n 



passage of the flange of wheel L, which wheel 


mce between rails X and A for the 
len runs upon rail A. 

To set the switch for the track C B, the rod R a is then pushed, and throws rails 
-B over, bringing B again into contact with Y, so that wheel K may run upon rail 
, and leaving a space between rails C and A for the flange of wheel L, which wheel 
ien runs upon rail C. 

As in all point switches, it is impossible to spike the inner flanges of the stock 
tils C and Z to the ties along that portion of their length (some 12 feet from the 
vitch point) where they come in confect with the switch-rails. As a substitute 
ley are provided with special supporting blocks S, S. on the outer side. In the 
f eir switch, each of these is made of one piece of flat bar iron, bent over and 
visted, and serving also as a sliding plate, as shown more clearly in the section n o. 

Fig. 17 b, which shows also the 


Fig. 17'c- 
SECTION AT p q . 



manner of fastening the end < f 
the flat connecting bar a to the 
base of rail A by means of a 
malleable casting M, through 
which the end of the bar a 
passes, and to which it is bolted. 
The end shown in Fig. 17 b is 
that which passes under rail X 
and is therefore hidden by it in 
Fig. 17 a. The section at p q, 
(fig. 17 c) shows the attachment 
of one of the rods x to rail X. 
le arrangement here is similar to that at n o, except that in Fig. 17 c the malleable 
sting M is bolted to the web of the rail as shown, while in Fig. 17 b it is of different 
>ape, and is riveted to the flange of the rail. 

These three-throw switches cost, 1888, about 865 each, without stand. 






















































































Wo 


TURNOUTS. 



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780 


TURNOUTS. 


I 


Art. 22. Frog'S. The frog is a contrivance for allowing the flange of tl 
wheel on the rail ex, Figl, to cross the rail rz; and that of the wheel on r z, to cro 
ex. The first contrivance lor this purpose was a bar, approx 

mately of the shape of the rail, pivoted at the point where the center lines of tl 
rails ex and r z cross each other, and free to move horizontally about this pivot, i 
that it could form a portion of ex when the train was passing to or from the tnmoi 
or a portion of r z when the train was using the main track. Sometimes the piv 



passed through one end of the bar, as in Fig 20, and sometimes through Its cent> 
p as in Fig 21. Such bars were generally moveu by a r> 

(attached at «)and lever, similar to those used for svvitche 
and they then, of course, required an attendant; but mat 
attempts have been made to use such frogs by conroctii 
them with the switch by means of rods, &c, so that the b 
should move automatically when the switch was turne 
Owing to the considerable distance (80 ft, more or less) b 
tween the frog and switch, it has been found difficult 
secuie simultaneous movements ot the switch and frog, and the contrivances referr 
to have not come into extensive use. Such bars, while they avoid the jar produci 
by wheels passing across the throat of the frog (Art 35), labor under the same d 
advantage as the stub-switch, Art 10, in requiring a iiberal allowance of space 1 
tween their ends and those of the adjoining rails, to avoid any possibility of the 
coming into contact. 




oo Ar 5* 2S * Tl,eso 1>ars were eoon superseded by rigid cast-iron frogs, Fi 
aud 2ii - Tlie se were hardened by chilling, so as better to resist the action f 



Fhm I, tnn Vliee , 1S V, b, l t r n W i th t,lis precaution they wore out so much more rapid 
than the lails, that the wings, aim and i c, and the tongue, P were cann 

with steel from Hindi to 1 inch thick, bolted or riveted to the?.- upper srnS 




















































TURNOUTS. 


781 


tijThe triangle, 
ri/«ftnd 


ll fr V ton p ue of the frog, is the meeting-point of the two rails 


een 


. - - ® support to the treads of the wheels in passing over the snares het«.' 

f ,h “ P°'" ““ d » >" d ”hich spaces,re left for the paLageo'fTe ianST* ‘ K>,W 
ll ™. ,t f 1S C “ “ ! ie n,on * h » f tile frog at o, Pig, 22 and 23: and its 

' b5»^'« ^d“ “JTtTed Hrhiel'/ Th “ l>art °' t0 " Sm b “ k of «* *». «r 

The ma , d , e t 1 ? 01 ?! 2 " ls r eep t0 prevellt tl'e flanges from touching its bottom 

if.hSS or.U‘Jif,e?’ are r f0r bol,lng "«s to the wooden crosfties 
Although one side of the frog forms a part of the turnout curve its shortness war 
rants us in making both sides, jo, S t, Fig 23, straight. ’ SUortness " ar ‘ 

'o lling * from A toS!S e R r w* ,S i’ ° r f" ard - raiIs - 9 Fig 1. Suppose wheels to he 
oiling noni A towaid B, Fig 1, on the mam track; the switch-rails being in the 

SC S ; On arriving opposite the frog, some irregularity of motion might 
utilise the flanges of the wheels running along the rail, r z to press laterally 

jaid rail. Consequently after passing the throat,^ 7 , Fig 22, they would press afainst 

he S,',’end a of p l ’Tr5 bet "r n 5 *- ">«» »«»Td tav/the track "or S.iU 

he sharp end of P, breaking it, and endangering the train To prevent this the 

’ A F ‘ S bL“ S * h “ ">]• ? * mo 2 fcom it), that the 


gainst the wing, w m, Fig 22, thus rendering the train liable to the "same kind of 
ccideut as in the preceding case. This is prevented, in the same manner as before, 
I 2*> 6 ^ Uart -rai ’ ® keeps the flanges in their proper channel, 


Ml C. 


The narrow flange-way between the guard-rail, #, Fig 1, and the rail, bh, 
iBioiilcl extend at least a loot each way from a point directly opposite the 
aunt,/ Fig 23, of the frog. In a distance of at least about 2 ft more at each of its 
nds tlie guard-rail should flare out to about 3 ins from the rail, b b, so as to guide 
he flanges into the narrow channel. The same with #'. 

Guard-rails have to resist a strong side pressure, and should 

■e very firmly secured to the wooden cross-ties. This is usually done by holtino- 
gainst th. ni two or more stout blocks of steel, or of wrought iron, which, in turm 
re bolted to the ties. 

Art. 25. The east-iron frog, as first made, had no provision for 
astening it to the rails; but was simply bolted to the cross-ties. It was 
Afterwards provided with a recess at each end, of the exact shape and size of the end 
| f the rail. The rail ends were inserted into these recesses, and the frog was thus 
ept in line with the rail. In frogs made of rails, the same purpose is served by fisli- 
r angle-plates, p 764, by which the ends of the frog are secured to those of the rails. 
Art. 26. The length, a g. Fig 23, of a cast-iron frog, usually varies from 
to 8 ft; and depends upon the angle, oft, at which the rails, e x and r z, Figs 1 and 
3, cross each other. This is called the frog-angle. This angle may be expressed 
ither in degs and mins, or in the number of times the width of the tongue on any 
ne, as o t, F'ig 23, is contained in the distance, gf, from the point./, to the center, 

, of that line. This number is called the frog number. Thus, if the angle, 
ft, Fig 23. is such that the length, gf. is 3, 4, or 10, Ac, times the width, ot, the 
•og is called a No 3, 4, or 10, Ac, frog. Fig 23 is a No 3; Fig 22, No 5. Frogs are 
sually made of Nos 4 to 12; sometimes with half numbers, as 7}^, 8%, Ac. 

Art. 27. Draw two parallel lines, b b', dd\ for the top of rail, ex. Fig 23, and 
h', kid, for that of rail, re; crossing each other at the required angle. Then the 
itersection,/, of lines dd' and h h' is the theoretical point of tlie frog. As 
iis point would be too narrow and weak for service, it is in practice rounded off 
here thetongue is about ^ inch wide,as shown. If the frog is to be simply abutted 
) the rail-ends, zx. Fig 23, as in some cast-iron frogs, the length,/#, need be only 
reat enough to give a width, to, sufficient to accommodate the rail-ends, z and x, 
nd the heads of the two spikes at v which confine them to the ties. If desired, a 
ortion of the flange of each rail-end may be cut away so that the rail-heads come 
igether; thus diminishing the width necessary for to, and, of course, the distance, 
'#. In the case of cast-iron frogs provided with recesses for holding the rail-ends, 
i in Art 25, the width, ot, and length,/#, must of course he greater. 

In frogs made of rails, the length must be such that the rail-ends, 
and x. Fig 23, are far enough apart to give room for fitting to their inner sides the 
pi ice-plates by which they are connected with the frog. Where «w#fe-plates art 
sed, this distance must be greater than in the case of /is/t-plates. 








782 


TURNOUTS 


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TURNOUTS, 


783 


, n ' f ' Z°™ ea '' I into tlie sides °f the throat-pieces and blocks. Tile clann><* 
?roatcare7s tVk ( e n l t f ™ m riveted on the flanges o Ahem?* 

tvith h fi Ve a . ,e a,, J° Inu, S surfaces in full contact 

V.5 % throu ? hout > So as t0 diminish the liability to wear 

*u C ° a ^° L make stlff fro « s in which the parts are held together hv 
t ng thrOUgh ^ the rai,8and tlie throat-pieces; and others in which there 
f ro ugh t-i ron nlatp 1111 a" WhlCh tl .‘ 1 e , four raiIs Arming the frog are riveted to a 
784t A?l nf P t! * e (A eprin 1 g * rai1 f f og ’ of the P ]ate pattern, is shown in Fig 27 
1 , of t ! ,ese t r °S® can he made of any desired length and to anv desired 
ngle. The Standard length of stiff frogs from be to dz for Nos 4 to 8 

888 U |ls e tn*^- ft ’ N °^- 9 and 10 ’- 9 lt; Nos 11 and 12 ’ 10 ft > No 15 > !2 ft. Prices’ 

* * s to $-o, according to weight of rails and the length of fro<^ 

i fiV** VL the ^? elr ( 'T eir Fro « Co., Cincinnati, Ohio), Figs. 26 A, 

6 B and 26 C, the notching of the main or long point, z, to receive the short Doint 
'» is avoided by forging the latter, under hydraulic pressure, to fit the former which 




V 



d 

z 


Fig. 26A 


i ; used as the upper die in the forging process. The rail forming the long point 2 is 
lus preserved intact throughout the space where the two points are in contact as 
lown in the section at n o, Fig. 26 B. The short point, d, is held tightly against, 


Fig.2'6B 


Fii 


!6C 




SECTION AT n O. 


1-C 


W/ z w 

INCHES. 

L^L-J-rt bd SECTION AT p q. 
12 3 4 5 6 - - 


r ‘ id under the head of, the long point 2 , by the tightening of the nnts on the bolts 
vhich are provided with nut-locks), and thus gives it additional support. 

The long fibing-blocks, t t. are forged in dies from wrought iron or steel. They 
•e then planed to fit the point rails, drilled, and bolted as shown. 

These frogs cost, 1888, from $18 to $23 each, according to length, angle, and 
eight of rail. 

Art. 35. The object in reducing- the width of the channels 

r some distance each way from the point, /, Fig 23, so as barely to admit the 
vnges freely, is to allow the treads of the wheels to have as much bearing as pos- 
ble upon the wings and tongue while moving over the broadest part of the chan- 
si near f. In frogs shorter than about No 4, it is difficult to secure sufficient 
taring for the treads, even with the utmost allowable contraction of the channel, 
.hen the width of the tires is, as usual, about 6 ins. In the earliest frogs this diffi- 
{iity was partially overcome by gradually raising the bottom of the 
' hannel between the point and wings, so that the wheels,in traversing that part, 
m upon their flanges instead of upon their treads. The jar occasioned by the 
ead, in striking against wing and point, was thus avoided. This arrangement is 
ill used in crossings, where the tracks cross at a very obtuse angle, and where, 
msequently, the wings can give little or no support to the treads. The flanges, 
nvever, soon cut gutters in the bottoms of the channels, and thus increase their 
iptlis, so that the treads strike the wings and point, as in the ordinary frog. 




































784 


TURNOUTS. 



































































TURNOUTS. 


7 85 


Ai*t. 37. In Hood s self-acting- frog-, which was largely and suceess- 
‘"•Y Y sed tor uiany years on the Camden A Amboy R R, under heavy trains moving 
high speeds, and which, like those jlist described, was made entirely of rails- the 
nnt,/, ijg 28, was firmly fixed to the cross-ties;and both of the wing-rails, t and s 
hue rigidlj- fixed at a proper distance from each other, were free to move, as a 
hole, about their heels, 6 and c, so that either t or s could be brought into contact 
ith the point,/. It thus provided an unbroken hearing to trains on the turnout, 
well as to those on the main track. There was no spring; and the frog remained 
it was set by the first pair of wheels of a train, until another train, using the 
her track, set it over the other way. 

If this frog is set as in Fig 28, and an engine moves from e toward x (so that the 
jugsi are moved by means of the guard-rail,/', and the wheel at the opposite track, 
c), the motion of the wings lias to take place while the weight of the engine itself 
resting partly on the wing-rail, et; and, consequently, their sliding transversely 
the track is attended with great friction. The same effect is produced if t is in 
ntact with/, and an engine moves from A to B. This, however, did not, in prac- 
pe, seem to interfere with the efficiency of the frog. 

Art. 38. In ordering frogs, give the frog angle or number, and the exact 
oss-section of the rail used on the road. 

For spring-rail frogs, specify also whether the turnout is to the 
ght or left hand. In the case of stiff frogs, this is not necessary. 


Art. 3d. The laying-out of Turnouts. 

The words heel and toe are used in this article with reference to the common or stub 
itch, Fig 4, in which the heels are at q and p , and the toes at m and s. In the 
barton, Lorenz, and some other switches, the positions of heel and toe will be seen 
be the reverse of this. 

The formulas given in our early editions for finding frog dist, rad of turnout, Ac, 
re based upon the old practice of regarding the straight switch-rails, qm,p s, Fig 
is forming a tangent to the turnout curve, which last was considered as beginning 
the toes, m and s, of the switch-rails. The modern practice is to curve 
e switch-rails so as to form a part of the turnout curve; the latter being supposed 
begin at the heels, q and p, of the switch. This view of the case admits of simpler 
mulas. 

in each of the Figs 29, 30, and 31, wp z represents the main track. The frog 
stance, p / is a straight line drawn from the theoretical point of frog, / to the 
el,/), of that switch-rail which, when opened, forms the inner rail of the turnout. 
rmerly, when the turnout curve was taken as starting at the toe of the switch, the 
>g dist was a straight line from the theoretical point of frog to the toe, m, Fig 4, 
the outer switch-rail, qm, when opened. 

As already remarked, frogs are usually made of Nos 4 to 12; sometimes with half 
mbers; and the turnout radii, Ac, are made to conform to them. 

Scrupulous accuracy is not necessary in these matters. Thus, a deviation, either 
iy, of say 3 per cent in the length of the turnout radius from that given by the 
.de or the formulas, will be almost inappreciable. So too, if a frog number should 
used, intermediate of those in the first column of the table, the other dimensions 
iy he found, approximately enough, by using quantities similarly intermediate, 
rail almost always has to be cut in two in order to fill up the frog dist; and the 
Jact length of the piece can be found by actual measurement at the time of cutting it. 
Hem. When the turnout leaves a straight track, as in Fig 29, the frog angle 
equal to the central angle./co. When the main track is curved, and the turnout 
rves in the opposite direction (Fig 30), it is equal to the Kttlll (vfn) of the 
ntral angles, fc n, fn n ; and when the two curve in the same direction (t ig 
), it is equal to the clitf (n/c) of the ceutral angles,/co,/ no. 






786 


TURNOUTS. 


sh 


Art. 40. To lay out a turnout, p x, Fig- 29, from a strai 
track, p z. From the column of radii in the table below, select one, co, suit 

able for the turnout; together with the correspond 
ing frog number, frog dist, p / and switch length 
Place the frog so that the main-line side of it 
tongue shall be at fz, precisely in line with thi 
inner edge of the rail, w z, and its theoretics 
point, J\ at the tabular frog dist, p / from tin 
starting-point, p. Stretch a string from q (oppo 
site p) to/; and from it lay off the three ordinate 
from the table; thus finding three points (in addi 
tion to q and./ - ) in the outer curve. Do not, how 
ever, drive stakes at these points; but as each o' 
them is found, measure off from it, inward, hal 
the gauge of the track; and there drive stakes 
Do the same from q and/. The five stakes will a 



then be in the dotted center line of the turnout, Fig 29; and will serve as guides t 
the work, without being liable to be displaced. The dimensions in the table belo\ 
are found by tlie following formulas, the main track being straight: 


Tangent of 

lialf frog angle 

Frog No. 

Or, Frog No. 

Radius c o. 

Or, Radius c o. 

Or, Radius c o. 

Frog dist i> f. 

Or, Frog dist p f..... 

Or, Frog dist p f. 

Middle ord. 

Eacli side ord.. 


=*= Gauge -i- Frog dist. 


= l A 
= % 


y Radius co-f Twice the gauge. 

Half the cotangent of half the frog angle. 

Twice the gauge X Square of frog number. 

(Frog dist pf -h Sine of frog angle) — half the gauge. 
(Gauge Versed sine of frog angle) — half the gauge 
Frog number X Twice the gauge. 

Gauge pq -i- Tangent of half the frog angle. 

(Rad co + half the gauge) X Sine of frog angle. 
x /i gmige, approx enough. 

■% mid ord = T 3 5 (or .188 of the) gauge, approx enougl 


•Switch Length 

approx enough 




Throw' in ft X 10000 


Tangential dist for chords of 100 ft, for rad coo 

turnout curve. See table, p 726 to 728. 


TABLE OF TURNOUTS FROM A STRAIGHT TRACK. Fig 29 . 

Gauge 4 ft 8)^ ins. Throw' of switch 5 ins. 

For any other gauge, the frog angle for any given frog number remain 
the same as in the table. The other items may be taken, approx enough, to vary d 
rectly as the gauge. 


5 


Frog 

Number 

Frog 

Angle 

Turnout 
Uadius 
e 0 

PoflAngof 

Turnout 

Curve 

Frog 

Dist 

Pf 

Middle 

Ordinate 

Side 

Ords 

— 

Stub 

Switel 

Lengt 


0 r 

Feet. 

0 r 

Feet. 

Feet. 

Feet. 

Feet. 

12 

4 46 

1356 

4 14 

113.0 

1.177 

.883 

34 

11 Vi 

4 58 

1245 

4 36 

108.3 

1.177 

.883 

32 

11 

5 12 

1139 

5 2 

103.6 

1.177 

.883 

31 


5 28 

1038 

5 31 

98.9 

1.177 

.883 

29 

10 

5 44 

942 

6 5 

94.2 

* 1.177 

.883 

28 

y /l2 

6 2 

850 

6 45 

89.5 

1.177 

.883 

27 

9 

6 22 

763 

7 31 

84.7 

1.177 

.883 

25 

*14 

6 44 

680 

8 26 

80.0 

1.177 

.883 

24 

8 

7 10 

603 

9 31 

' 75.3 

1.177 

.883 

22 


7 38 

630 

10 50 

70.6 

1.177 

.883 

21 

7 

8 10 

461 

12 27 

65.9 

1.177 

.883 

20 

6 3 4 

8 48 

398 

14 26 

61.2 

1.177 

.883 

18 

6 

9 32 

339 

16 58 

56.5 

1.177 

.883 

17 


10 24 

285 

20 13 

51.8 

1.177 

.883 

15 

5 

11 26 

235 

24 32 

47.1 

1.177 

.883 

14 

* 1 a 

12 40 

191 

30 24 

42.4 

1.177 

.883 

13 

4 

14 14 

151 

38 46 

37.7 

1.177 

.883 

11 


Remark. The switch lengths in the Table merely denote tl 
shortest length of Stub switch that will at the same time form part of the turi 
out curve, and give 5 ins throw. Pointed, or split-rail switches like tl 
Lorenz, Ac, require only half this throw'. In practice all kinds are frequently mar 
much shorter than the table requires, thereby sharpening the beginning of the curv 


















































TURNOUTS. 


787 


Art. 41. Turnout from a curved main track. The following con- 
enmnt approximate method was, we believe, first published by Mr. E. A. 
deseler, C. E., of New York, in 1878. There are two cases. 

Case 1, Fig 30; when the two curves deflect in opposite directions. 

Case 2. Fig 31; when the two curves deflect in the same direction. 

Having determined approx upon a radius for the turnout curve, take from the 
ible, p 726, its corresponding direction angle, and that for the main curve. In 

Case 1, find the min of these two angles. In Case 2, 
find their difference. In the table, p 786, find the 
deflection angle (not the frog angle) nearest to the 
sum or diff just found. The frog number, 
switch length and frog distance pfl in 
the table, opposite the defleciion angle thus se¬ 
lected, are the proper ones for the turnout. 
Theoretically we should, in Case 1, add to the tab- 




Ms 


lar frog distance pf about half an inch per 100 feet for each degree of deflec- 
ion angle of the easier of the two curves; and, in Case 2 , deduct 1 1; but this 
efinement is unnecessary. 

tellection angle of turnout curve , . . 

(in Case 1 ) = Tabular deflect ion angle (p 786)—deflection angle of mam curve, 
(in Case 2) = Tabular deflection angle (p 786) + deflection angle of main curve. 
For radius co of turnout curve, having its deflection angle, see 

a Ex. P Rad of main curve, 2865 ft. Rad selected approx for turnout, 716.8 ft. 
[ere the defl angles (table, p 726) are, respectively, 2° and 8 °. 

In Ease 1 : 8 ° + 2° = 10°. Nearest defl angles in table, p 786, 10° 50' and 9° 31 . 
If we select 10° 50', we have Frog No, 7^! Switch L’gth, 21 ft; Frog Dist= 70.6 
; + twice \i in = say 70.7 ft; Defl Angle for turnout = 10° 50 — 2° = 8 oO ; Rad 
o (table, p 726), 649 ft. If we select 9° 31', we have Frog No, 8 ; Switch L gth, 

2 ft: Frog Dist = 75.3 ft + twice % in — say 75.4 tt; Defl Angle — 9 31 2 — 

"in Case 2 * = 6 °. Tabular defl angles, p 786, 6 P 5' and 5° 31'. 6 ° 5' 

ives Frog No, 10; Switch L’gth, 28 ft; Frog Dist = 94.2 ft- twice ]4 >» = »y ^ 
Defl Angle = 6 ° 5' + 2° = 8 ° 5'; Rad, say 709 ft. 5° 31' gives h rog No, 10^ 
witch L’gtli, 29 ft; Frog Dist = 98.9 ft — twice % ia — 8a J ^.3 e 

3 31' + 2° = 7° 31'; Rad, say 763 ft. 

The frog’ dist p f may also he found thus: 

Gauge X Frog No 

Tangent of half fn o = |^; U8 — 

Frog I>ist p f = np X twice the sine of half fno. 

Place the frog with the main-line side of its tongue at/*, in line with the inner 
Ige of the frog-rail, p z, of the main line, and with its theoretical point,/, at the 
ist. p / (found as above), from the heel, p, of the inner switch-rail. Stretch a string 
■om/to the heel, 7 , of the outer switch-rail. Measure the dist, qfl divide it into 
>ur emial parts, and lay off three ordinates, found thus. .. 

Middle ord = (Square of half qf) = twice the rad of turnout curve. Each side 

rd = three-fourths of middle ord. , 

These three ords. and the points 7 and/, give us 5 points of the out ® r °. f the 
irriont curve; and from these we measure, inward, half the gauge, and duve 5 ceu- 
»r guide-stakes, as in Art 40. 


54 











7«8 


TURNOUTS. 


Art. 42. To find frog clists. Ac, by means of a drawing t« 
scale. The frog dist can generally be found near enough for practice, from 
drawing on a scale of about 34 or x /$ inch to a foot. And so in the many cases wher 
turnouts cross tracks in various directions, in and about stations, depots, &c. 

Tigs 32 and 33 are intended merely to furnish a few general hints in regard t 



such drawings. Por instance, the curves of a main track, as well as those of a turn 
out, generally have radii too large to admit of being drawn on a scale of 34 inch t 
a foot, by a pair of dividers or compasses. But they may be managed thus: Drav 
any straight line, ah, Fig 32, to represent by scale a 100-ft chord of the curve, divid 
it into twenty 5-ft parts, a 1, 1 2, 2 3, &c, and lay off by scale the 19 correspond^ 
ordinates, 11, 2 2, 3 3, &c, taken from the table on page 730. By joining the end 
ot these, we obtain the reqd curve, a c 6, of the main track; and of course can dra\ 



the inner line, y l , distant from it by scale the width of track, say 4 ft 834 ins. Nov 
iet acb and y t, Fig 33, be a curved main track so drawn ; and let any point m b 
taken as the starting-point of the turnout, mv, &c. On each side of m measure of 
any two equidistant points, n and n , in the same curve; and through m draw's r 
parallel to n n. Then is m.? a tang to the curve, y m t, at m. Having determine 
on the rad of the turnout curve, m v e , draw that curve by the same process as before 
first laying off the angle, g m i, equal to the tangential angle of the curve, take 
from the table, p 726. Then, beginning at m, lay off 5-feet diets along m i ; and froi 
them, as in Fig 32, draw the ords corresponding to the turnout curve. Throuo-h th 
ends of these ords draw the curve, mve. itself. Then the frog dist will be tl 
straight dist from c to v, and can be measured by the scale, with hi a few inches - . i 
neai enough foi practice. The middle ord of the arc, mv, cannot be found correcti 
>> so small a scale as 34 inch to a foot, but should be calculated thus: From th 
square of the rad take the square of half the chord, m v. Take the sq rt of the ren 
subtract this sq rt from the rad. If two other ords should be desired, half wav b< 
tween m and y and the center one, they may each be taken as % of the center on. 
Make the switch-rail long enough to leave 2*4 ins at its toe between mn and m u. 

Ihe frog angle at v will be equal to the angle, rvd, formed between the tang, vi 
to the curve, acb; and the tang, v cl, to the curve, mve. These tangs are found i 
the same way as mg; namely,for the tang,r r, lay off from u two equidistant point. 
h and on the curve, acb; and through v draw v r parallel to h h. Also for v . 
lay off from » any equidistant points, u and u, on the curve, mve , and through 
draw v rt parallel to them. This angle may be measured by a protractor. Or, if o 
the two tangs we make v 4 and v 4 equal to each other, anddraw the dotted line 4 4 
and from its center at 6 draw 6 v; then 6 v divided by 4 4 will give the No of the fro< 
With care, and a little ingenuity, the young student will be able, by similar pro! 
esses, to solve graphically any turnout case that may present itself. The metlio 
by a drawing has great advantages over the tedious and complicated caleulatior 
w hich otherwise become necessary in cases where curved and straight tracks inte 
sect each other in various directions. The drawing serves as a check against seriot 

mIn!!’jfi' C i h W ?" d f . be detected , at or »ce by eye. None of the graphical measur 
nifints will be strictly accurate; but with care, none of the errors need be of nra. 
tical importance. The ordinates for bending rails so as to suit turnout curves ca 
be found from the table, p (61. All of Art 42 may be done on the ground. 






















TURNOUTS. 


789 


“43. An experienced track-layer, with a good eye, can place his own guide- 
takes by trial on the ground; and by them lay his turnouts with an accuracy as 
radically useful as the most scrupulous calculations of the engineer can secure. 

The following exaniple, Fig 34, of a turnout from a straight track, Y Z, exhibits a 
ommou case, in which all the work may be performed on the ground, without pre- 



ious calculation. Let ivo be the tongue of a frog, with which the assistant has 
een directed to make a turnout from Y Z; and that he has received no instructions 
tore than that the turnout must start at d, and terminate in a track, W, to be laid 
arallel to Y Z, and distant from itrxorrx, equal to 6 ft. 

Place the tongue of the frog by guess near where it must come, having its edge, 
imprecisely in line with the inner or flange edge of the rail, b r. Then stretch 
a piece of twine along the edge, ov, of the frog, and extending to dg. Try 
y measure whether v e is then equal to ed; and if it is not, move the frog along 
le line, br, until those two dists become equal. Then is v the proper place for the 
oint of the frog ; b v is the frog dist; one-half of c e is the length of the middle ord 
f the turnout curve, d v ; and if two intermediate ords are needed at s and s, each 
f them will be % of said middle one. 

The frog being now placed, proceed thus: Place two stakes and tacks, x and x, at 
le reqd inter-track dist, rx and r x, of G ft from the rails, br. Then range by 
ieces of twine xx and v/, to find the point, n, of intersection. Then measure nv, 
id make n m equal to it. Then is m the end of the reverse curve, v m, of the turn- 
nt. The ords of this curve may be found as before; one-half of nk being the 
liddle one, Ac. 

Rem. It may frequently be of use to remember that in any arc, as v m, of a circle; 
n and m n being tangs from the ends of the arc; one-half of the dist, kn, is the 
liddle ord, kz, of the curve; near enough for most practical purposes, whenever the 
ngth of the chord , v m, of the arc is not greater than one-half the rad of the circle 
f which the arc is a part. Or, within the same limit, vice versa, if we make k n equal 
> twice kz, then will n be very approximately the point at which two tangs from 
fie ends of the arc will meet. Also, the middle ord of the half arc, vz or zm, may 
e taken as 34 of the middle ord, k z, of the whole arc. 













790 


TURNTABLES 



SELLERS’ 60 EX. CAST-IRON TURNTABLE, p. 791, and BALDWIN “DECAPOD" LOCOMOTIVE, p. 807, 



















































































































































































































TURNTABLES. 


791 


TURNTABLES. 


Art. 1. A turntable is a platform, usually from 40 to 60 ft long and 
about 6 to 10 ft wide (see Fig 1,) upon which a locomotive and its tender may 
be run, and then be turned around hor through any portion of a circle; and thus be 
transferred from one track to another forming any angle with it. The table is sup- 
V, ported by a pivot under its center; and by wheels or rollers under its two ends. 
Frequently other rollers are added between the center and ends. Beneath the plat* 
f< >rm is excavated a circular pit about 4 or 5 ft deep, having its circumf lined with 
a wall of masonry or brick about 2 ft thick, capped with either cut stone or wood. 
The diam of the pit in clear of this lining is about 2 ins greater than the length of 
)the turntable. The lining is generally built with a step, as seen in Fig 1, for sup- 
i porting the circular rail on which the end rollers travel; or, instead of this step, a 
■ detached support may be used for this circular rail, as at w, Fig 11. At the center 
I of the pit is a solid well-founded mass of masonry or timber, for the pivot to rest 
on, as seen in Fig 1. This, as well as the step for the end rollers, should be very 
firm, and perfectly level; otherwise the platform will be hard to work. The plat¬ 
form is frequently floored across for a width of 6 to 10 ft to furnish a pathway 
across the pit, without stepping down into it; especially when under cover of a 
building. At first they were floored over so as to cover the entire circular pit; but 
this increased not only their cost but their wt, so as to make them difficult to turn; 
besides causing much expense for repairs; with greater trouble in making them. 
It is therefore rarely done at present, except where want of space sometimes ren¬ 
ders it necessary in indoor turntables. 

For the minimum length of a turntable, add from Vy^ to 2 ft to 

the total wheel base (p 805) of the longest locomotive and tender for which it is to 
be used; but a turntable should be several feet longer than is necessary for merely 
allowing the engine and tender to stand on it; for the increased length enables the 
.< engine-men to move them a little backward or forward, so as to balance them chiefly 
upon the central support; and thus relieve the end rollers By this means the fric- 
|l tion while turning is confined as much as possible to the center of motion; and is there* 

N 





Fig. 5 




Fig. e 















































































































































































































































































































































































































































792 


TURNTABLES. 


fore more readily overcome than if it were allowed to act at tlie circumf. The 
engine-men soon learn, by teeling, the proper spot for stopping the engine so as thus 
to balance the platform. 

Art. 2. Figs l to 6 represent the Sellers cast-iron turntable of Win 

Sellers A Co., Pliila. It consists of two cast-iron girders of about 1% ins average 
thickness, perforated by circular openings to save metal. One of these girders is 
shown in lig 1; and parts of one in Figs 3 and 4. Bach girder is in two separate 
pieces, which are fastened, as shown in Figs 1, 3, and 4, to a hollow cast-iron “cen¬ 
ter-box,” A B, Figs 2, 3. 4, and 6, by means of 2% inch screw-bolts, at/, Fig 3; and 
by hor bars, o o, of rolled iron about 3% ins square, fitting into sunk recesses on 
' of the boxing, and tightened in place by wedges, i i, screw-bolted beneath. 

Th e sides of the center-box are about ins thick. It is sus¬ 
pended from the steel cap, C, by 8 screw-bolts 2 ins diam. On its lower side 
this cap has a semi-cylindrical groove extending across it, transversely of the 
tiack, as shown in Fig 6. This groove fits over a corresponding semi-cylindrical 
ridge on the top ot the cast-iron “socket,” s (so called), on which the cap thus 
rests The socket, in turn, rests upon the upper one, w, of two annular steel plates 
u and r, which form a circular box containing 15 steel conical anti¬ 
friction rollers, d d, Figs 2, 5, and 6. These are about 3 ins in length, and 
in greatest chain. They have no axles, but merely lie loosely in the lower part, r 
ot the circular box; filling its circumf with the exception of about inch left for 
play. In the direction of their axes they have % inch play. The lid, w, of the cir¬ 
cular box, rests upon the tops of the rollers, which separate it from v by about \4> 
fl Y' ests } ,p on the top of the hollow cast-iron post, P, which, by means 
ot its flanges, is bolted to the cap-stone, M, of the foundation pier. 

in order to insure a perfect bearing of the revolving surfaces upon each other, 
and thus diminish the liability to abrasion, the rollers, d d , and the insides of the 
box in contact with them, are accurately finished, as are also the top and bottom of 
Ijiy yy-’ a “ tl , t J^ e surfaces of the socket, s, and post, P. in contact with them. 
The rollers are oiled by means of the spaces shown by the arrows in Fi- 2. 

Adjustment of the height of the table By tnrnino- th#* 
nuts, N N, of the 8 screw-bolts which support the table, the latter may be raisedor 
loweied 1 or 2 ins; the cap, C, socket, s, and roller-box, u v, remaining stationary 
on top of the post, P All turntables should have the means of making ",mb 
mhustnient. Betore the nuts, N N, are finally tightened up, the blocksAv w 
^ a r?* W004 *; CUt to * he P''°per thickness for the desired ht of the table are 

'T1 1 o hi « 7°# H k i°’ a » d th i t ,° I 1 ) of the center-box, A B, as shown. 

°f / lh ® table should be such that each of the wheels at its outer 
ends shall beinch, in the clear, from the circular rail on which they travel 

♦ e . aC ' 0t the ?' ,ter ends of t,,e table, the two girders are connected 

i H V yyly heavy cast-iron beams, called “ cross-girts.” These nroiect 
bejond the girders, and carry the cast-iron Cltd-xviteels. 20 ins diam 9 It l. 
end ot the platform. The treads of these wheels are but about 3 or 4 ins below the 

,tt ^?? P ^ #ndthe ^ eeh therefore do not require auy Sonsidemble 
depth of pit for their accommodation. In order that they 1 may roll freely their 
reads are coned, and their axles are made radial to the circular turntable pit I , 
termed,ate transverse connection between the main girders is semired bv tl" 
c, ’ oss "t #os notched upon them to support the rail’s and freouentlv in 
or 12 it long, for giving a wide footway across the pit. A levei 8 or 10 ft V,mi 

f ° r ^ ‘ a '" e ’ “ 0t on Si 

takes place when a locomotive comes upon it or leaves it; but prevents it from tin 
taU-.nd-jock.t joint with a casting U jE„ whichS ^le“ to ^ » 

Art. 8. The girders of turntables are now very generally made of 




TURNTABLES 


793 


rolled-iron plates, each girder being in one piece; and the two main girders 
ire then connected with each other, near their centers, by two cross-girders 
of I or channel beams, or of plate-iron, one on each side of the central pivot. These 
cross-girders, and the sides of the main girders between them, form a sort of rectan¬ 
gular box, corresponding to the cast-iron center-box of the Sellers table, Arts 2 and 
1. The arrangement of the center bearing apparatus, and the manner in which it 
is connected with the cross-girders, differ in the tables of the several makers. 

Art. 9. In the plate-iron turntables made by the Edge Moor Iron Co, Wil¬ 
mington, Del, the cross-girders, GG, Fig 8, are of plate-iron ; and, at their ends, have 
flanges. F F, of angle-iron, by which they are riveted to the main-girders, E. In the 
54-ft tables the cross-girders are 26 ins apart in the clear. 



Fig. 7 


Enlarged transverse section of roller- 
box Ac, on center line of Fig 8. 



Art. 10. The bolts, H, by which the table is suspended from the cap, C, ara 
iix in number, and are arranged in two rows of three bolts each. one. row being on 
jach side of the post, P, and between it and one end of the turntable. To each cross¬ 
girder is riveted a short hor bar of angle-iron, J J. Through the hor flange of each 
if these two bars pass the 3 bolts, H, of one row. The hor flange, bearing all the 
n't of its girder and load, rests upon the heads of these 3 bolts. To prevent the 
lange from yielding under its load, vert struts of angle-iron are riveted to the web 
if the cross-girder between the bolts. Two of these struts are shown at K K, Fig 7. 
Their ends abut against the upper flange of the cross-girder, and against J J. 

! The 6' bolts, IIII, pass up through the flanges of the cap, C (three bolts through 
hacli flange), and their nuts rest upon its top. 

Art. 11. The cap, C, is held in place on the socket, s, by means of 
flanges which extend down from it on both sides, as shown in Fig 7. 

The rollers, d d, and the roller-box, uv, are those made by Wm. Sellers & Co, 
Phila, Art 3. 

The ht of the table may be adjusted, within a range of 2 or 3 ins, by 
means of the nuts, N N, as in the Sellers table, Art 4. 

The lower part, v, of the roller-box, instead of resting directly upon the post, P, 
is in the Sellers table. Art 3. rests upon an iron casting, L, which, in turn,rests upon 
the post, P, and is held in place upon it by a lug which projects down into it. 

The post is built up of plate- and angle-irons riveted together, and may be a hol¬ 
low truncated square pyramid, as shown, or of other shapes. Those presenting the 
shape of a cross in hor section have the advantage of being accessible for painting 

and inspection. _ _ , , . 

4rt. 12. The main girders are braced by hor dmg rods, whose ends 
are shown at Q, Fig 8. These rods are provided with sleeve-swivel turn-buckles, by 

which they may be tightened or loosened. 

The remarks iu Art 5 on the end wheels of the Sellers table apply also to 
those of the Edge Moor, except that iu the latter the wheels are 25 ins iu diarn, and 
the cross-girts which carry them are of angle-irou. The wheels are fasteued to their 
axles which turn iu brass bearings enclosed in cast-irou journal-boxes. Each box 
is provided with means for raising or loweriug it; and the brasses are rounded on 
top so as to insure a uniform bearing, no matter what inclination may be given to 
the’axle by the vert adjustment of the journal-boxes. The brasses may be readily 









































































794 


TURNTABLES. 


replaced when worn ont. The lower part of the journal-box is filled with oil aDd 
Wiistefor lubrication. The several parts are made large, in order to lessen the fric¬ 
tion per square inch, ami to withstand the shocks to which they are subjected. 

The Edge Moor tables are made 50, 54. 5(3, 
and 60 ft long, and are designed to turn the 
heaviest “Consolidation ” locomotive and ten¬ 
der without subjecting any part of the iron 
work of the table to a greater tensile or com¬ 
pressive strain than 10000 lbs per square inch. 

Art. 13. The approximate weights 
nml prices of Edge Moor turn¬ 
tables are given on the accompanying table. 


Length. 

Finished wt. 

Price, 1888, 
on cars at 
Edge Moor. 

50 ft 

54 ft 

56 ft 

60 ft 

18900 ll>s 
22600 lbs 
24:400 lbs 
26400 lbs 

81150 

81400 

81450 

81550 


Average cost, 1888, say 6 cts per lb. 

Art. 14. Fig 9 shows the center bearing arrangement of the tnrntable patented 
ami furnished by Mr. C. O. M. Fritzsche, 71 Broadway, New York. The left 
side of the fig is in section; the right side is in elevation. 



Here each cross-girder, G, is a heavy rolled iron I beam, to each end of 
which are riveted flanges, F, of angle-iron. These flanges are bolted to the webs of 
the main girders, E, in order that iron “> ad justing-plates ” of any require* 
thickness may be inserted between either angle-flange and the web of the mail 
girder which it supports. This permits a transverse adjustment of the table, so tha 
the center of gravity of its transverse cross-section may be brought immediately 
over the vert axis of the post, l\ 

Art. 15. Four rolied-iron plates, of which two (ora pair) are shown at Z Z 
extend from one cross-girder bo the other, and are fastened to the webs of the lattei 
by angle-pieces, Y Y. The lower edges of these plates rest upon rolled-iron washers 
X. Each washer supports two of the four plates, Z Z, and rests, in tnrn, upon on. 
of the nuts, N N, of the single inverted U-bolt, II. The upper part of this U-bol 
rests in a cast-iron saddle, S; the neck, V, of whieli enters the top of the cast-iroi 
post, P, and rests upon the upper one of two steel discs, d, which, finally, rest on th< 
bottom of the cylindrical cavity in the top of the post. The object in th« 
use of the single F-bolt is to reduee, as far as possible, the number of 
points of support, and thus to reduce, also, the uncertainty as to the proportion ol 
the total load sustained by each. Its shape, and the manner in which it rests in it 
saddle, S, render the table a rigid mass revolving on the center post. 
















































































































































TURNTABLES. 


795 


Art. 16. Oil is applied through openings in the side of the saddle-casting, 
These openings communicate with a vert hole through its center, and thus with 
similar hole through each disc. The two faces of the discs in contact with each 
ther, form a segment of a sphere; and each face has three radial gutters,extending 
•om its cen to its circumf. Channels are cut in the sides, and around the lower 
Ige, of the cylindrical cavity in the top of the post, 'l'lius all the parts which 
jvolve upon one another can be kept bathed in oil. 

Art. 17. Each leg of the U-bolt passes through, and is held in place by, a cast- 
ron box, B. These boxes are held in position by the edges of the four inner 
ngles (corresponding to Y Y, but not shown in the fig) between the plates Z Z. 

J Sometimes the two washers, X, four plates, Z Z, and the two boxes, B, are omitted, 
nd two cast-iron boxes of another shape are substituted in their places, one 
>r each leg of the U-bolt. These boxes are not fastened to the cross-girders, but the 
pper flanges of the latter rest upon the upper corners of the boxes, which are fitted 
!> them. The cross-girders, in such cases, are prevented from spreading apart, by 
folts passing through both of them, close to the cast-iron boxes. 

Art. 18. The following gives the weights of iron in several of these tnrn- 
ibles now in use; with the \vt of locomotive anti tender which they 
urn ; the max load on the end carriage; and that on the pivot pier. This last, 
f course, includes the wt of track and cross-ties in addition to the other wts 
anted. * 


Piam of 

Wt of iron in 

Wt of loco 

Max load on 

Max load on 

turntable. 

turntable. 

and tender. 

end wheels. 

centre pivot. 

ft. 

lbs. 

tbs. 

lbs. 

lbs. 

50 

17500 

121000 

57000 

149000 

50 

20500 

150000 

64000 

177000 

55 

22500 

124000 

61000 

154000 

55 

25500 

161000 

68000 

194000 

60 

27500 

169000 

74000 

206000 


Prices, accompanied by strain sheets based upon the wheel loads of loco and ten- 
ler, furnished upon application. 

Art. 19. Tlie Greenleaf turntable, Clements A. Green leaf, M E, Indi¬ 
anapolis, Ind, patentee and manufacturer; is made in a variety of forms. Its dis- 
iuguishingr feature consists in a series of 27 cylindrical steel roll¬ 
ers, R R, Rig 10, 2/ g ins diam, with their axes vert. These rollers are arranged 
n a circle’around the post, P, P, near its base. They have no axles, but are held in 
i circular cast-steel box attached to the cross-girder, as in the fig, or in a circular 

I groove in the center-box when this last is of cast-iron. The post is made cylindrical 
externally,near its foot, as shown, in order to give a full bearing to the vert sides of 
he rollers’. Their object is to prevent the tipping of the turntable, while 
, anting, even although the locomotive is not exactly balanced on the table (Art. 1) 
mil thus to prevent the end w heels from bearing on their circular rail. 

Art. 20. C’onical rollers, cl d, in a roller-box, u v, are used, as in 
he Sellers (Art 3) and other tables. The cap, C, rests and fits upon a cast- 
iron hemisphere, S, a cylindrical dowel on the bottom of which enters and 
[ its the space in the center of the roller-box. Oil is supplied to the roller-box 
| through the vert passage, Q, and its branches. 

j Art. 21. The post here shown is made of eight 60-lb steel 
, f rails, P P P, &c, with their flanges outward. To these flanges is riveted an 
j iron-plate, 0 0, % inch thick, forming a conical shell or covering for the post. 
, i'he steel casting, L (at the top of which is the path for the vert rollers, R), is hehl 
to the wrought-iron base-plate, K, by long rivets which pass between the feet of the 
rails, P, of the post. A wwought-iron “ tire,” T, is shrunk around the ends of the rails. 
When this form of post is used, the arrangement for hanging- tlie table 
to the cap is similar to that in the Edge Moor table, Art 10. Hollow cast- 
iron posts, however, are more largely used, and are recommended as being bet- 


er and cheaper. , , , ,. . 

With cast-iron posts a cast-iron center-box is used, closely surrounding the post, 
nd furnished with flanges, by which it is riveted to plate-iron cross-girders; but 
he arrangement of the cap, C, hemisphere, S, and rollers, is the same as in Fig 10. 

Art. 22. In all cases the table is suspended from the cap by 8 

olts arranged in a circle. The holes in the cap are made a little larger 

> diam than the bolts, to allow the slight hor movement of the center-box, caused 
y the tipping of the table when an engine comes upon it, or leaves it, because the 



















796 


TURNTABLES, 



vert rollers, R R, prevent this motion from taking place near the foot of the pos 
as it aoes in other tables. ^ 

„ A. rt * 28 -* ,' rhese tables are made both of wrought- and of cast-iron,and are large 
used (especially on Western roads) with very satisfactory results. 

ts 11 V* r ° a turnt f b i e ’ 60 ft lon S’ weighs 28000 lbs, an 

costs, 1888, ©1650, say b cts per lb; cast-iron, same length, 32000 lbs, $1300, sa 

about $625 ^ complete, with cast-iron center-post and center-box, cos 

n The wroiight-iron turntable made by the Union Bridge C 

(late Kellogg * Maurice), of Athens. Pa, also has vert roller 

a . r .°’ ,nfJ 1 the P° st near 'ts base; but the rollers are only two in numbe 
Iliey aie held in place by vert axles, and are fixed one on each side of the post or « 

t !e turnml le 6 ’ T. ,n,n fi the,r ifi right angles to the longitudinal axis c 

the turntable. 1 hey thus aid in preventing sideways tipping, but not in balancin 

iittundilton?* 1 a,Iy * They thei ' ef0re brin » 1,0 great strains upon the po! 

, ^ r * • T. Stock, Chicago, Ill, furnishes a turntable in whic 

the main girders are heavy rolled I-beams, trussed on their upper sides with a cei 

tial vert strut and inclined tension-rods like Fig 52, p 514 inverted Price of a 
table, 1888, $800; 60 ft, $1500. S 1 , inverted. Price ot a 


40- 
































































































TURNTABLES. 


797 


Wooden turntables, with none but two common wheel rollers at each 
nd of the platform, are sometimes resorted to from motives of original cost, 
hey are, however, much harder to turn, generally requiring two men, aided 
y wheelwork ; and are more liable to get out of order; and more expensive to 
epair. They are made of a great, variety of patterns, both as regards the 
irders, and the central pivots, end rollers, &c. Frequently an addition is made 
f 8 to 12 small rollers travelling on a circular rail of 6 to 12 feet diameter, 
round the pivot as a center. These are intended to sustain the whole weight ; 
le end rollers being so adjusted as to touch their rail only when the platform 
ocks or tilts as the engine enters or leaves it. Therefore, there is less resistance 
•om friction than when, as in Figs 11, there are only the end rollers r. In this 
ist case, the engine and tender cannot be balanced so precisely upon the 
ender central pivot, as to prevent a great part of the weight from being 
irowu upon the end rollers; thus materially increasing the frictional re- 
stance. 


In plan, these wooden platforms are sometimes in shape of a cross; that is, in 
ddition to the main platform for the engine, there is another transverse or at 
light angles to it, also extending across the pit; and having end rollers travel¬ 
og on the circular rail. Thus, in Figs 11, (which show one of the many modes 
f framing a table which has only a central pivot/, and end rollers r,) the main 
latform rests on the girders c, which are strengthened below by braces a; while 
ie transverse one rests on the timbers o, o, which must be imagined to extend 
cross the pit. One-half, or one arm, of this transverse platform is intended to 
irry the wheelwork Rxx, for turning the platform; and the other arm 
brves merely as a balance to it: therefore, neither of them requires to be 
ery strong. It is important to connect the four ends of the two platforms by 
aur beams, as the whole structure is thereby materially stiffened. In the 
gures the wheelwork Rxx is for convenience improperly shown as if it stood 
pon the main platform. 


TR. SEC.AT CENTER 


SIDE-VIEW 



The figures need but little explanation. They represent an actual 45-foot 
latform. which has been in use for some years. The convex foot / of the 
entral pivot, about 6 inches diameter, should be faced with steel: and should 
est, on a steel step ss. This should be kept well oiled; and protected from 
ust by a leather collar around p, and resting on ffff. Its upper part, about 4 
nches diameter, is cut into a screw with square threads aboutinch thick, 
or a distance of about 15 inches. It works in a female screw in the strong 
ast-iron nut yy\ and serves for raising the whole platform when necessary. 
Vhen not in use for this purpose, it is keyed tight to the platform, (by a key at 








































































































793 


TURNTABLES. 


its liead n,) so as to revolve with it. Strong screw-bolts ii connect the severa' 
timbers at the center of the platform. 


• 

R is a light cast-iron stand supporting two bevel wheels about 1 foot diameter 
winch give motion by means of an axle d, \% inches diameter to two similar 
ones below, shown more plainly at W and Y. These last give motion by the 
axle x to the pinion e, (6 inches diameter, and 2J4 inches face,) which turns the 
platform by working into a circular rack t , (teeth horizontal, 1 inch pitch; HU 
inches face,) which surrounds the entire pit. This rack is spiked to the undei 
side ot a continuous wooden curb H, which is upheld by pieces F, a few fee 
apart, which are let into the wall J J, which lines the pit. The short beam 
M iS, (about G leet,) which carries the lower wheelwork, is suspended strongly 
from the beams of the transverse platform above it. Instead of the two lower 
bevel wheels W Y, and the horizontal axle x, a more simple arrangement is to 
place the pinion e at the lower end of the vertical axled; and let it work into 
a lack with vertical teeth at u, on the inner tace of the stone foundation of the 
circular rail, for this purpose the stand R should be directly over u. There 
are t wo cast-iron rollers r, 2 feet diameter, 3 inch face, under each end of the 
main platform ; and one under each end of the secondary one. 


Figs. 12. 


Although this kind of platform necessarily has much friction, yet one man 
can generally turn a 4o-foot one by means of the wheelwork, when ioadad 

hatJ'nMv en , g -', ne f," d tender - Indeed, he may do it with some difficulty 
by hand only, while all is new and in perfect order; but when old and the 

wRh entire a “ d d ‘ rty ’ requires two Ulen at the winches to do it 

? e f fore r o"! ark 'o d ’ f n e resistance to turning is diminished by employing 
set of from 8 to 12 rollers or wheels r, * l j o 

Figs 12, about a foot to 15 inches in diam¬ 
eter, so arranged as to form a circle 8 to 12 
feet diameter around the pivot. When this 
is done, the main girders of the platform 
are placed 8 to 12 feet apart; and long 
cross-ties are used for supporting the 
railway track. Also, the main girders 
are sometimes trussed by iron rods, as 
in the swing bridge on page593; but’in- 
stead of one post ac, it is best to have 
two, 6 or 8 feet apart at foot, and meeting 

at top The width of platform must then be sufficient to allow the engine tc 
pass the posts on either side of it. Ten feet will suffice. engine tc 

Fig 12 shows the arrangement of these rollers r, which revolve unon a cir- 
cular track while the platform rests on their’tops by the track " The 
rollers r are held between two wrought-iron rings o.o, about 3 inches deep 
J4inch thick, which also are carried by the rollers. From each roller a radial 

c.o'ir 1 V*' nCh * iTn ter ’ extends to » "-Inch surrounds the pivot » 

closely, but not tightly, so as to revolve independently of it. These tie-rods 
keep the rings oo at, their proper distance from the pivot, so that the rollers 
cannot leave the rails s and u. Between each two rollers the rines oo should 
be strengthened by some arrangement like a , to prevent change offhape The 
pivot, p may be as in Figs 11. Thf>re must, of course, he the usual two rolled 
under each end of the platform, for sustaining the engine as it goes on or off* 
but, during the act of turning the platform, the whole weight should rest on 
the central rollers. Such a platform of 50 feet length can if carefully mailp 
be turned, together with an engine and tender, bv one man by means of a 
wooden lever 12 to 15 feet long,inserted in a staple for that purpose and there 
fore may dispense with the transverse platform for sustaining wheelwork 



Such rollers as have just been described, in connection with friction rollers 
Fur o, form perhaps the best arrangement for a larirc tirnE 

At r aSt o 116 end of a P |atforin must, be provid«>d wit^ a eatch oi 
stop for arresting Us motion at the moment it has reached the proper spot 












ENGINE-HOUSES, ETC. 


799 


common mode is shown at Fig 13. It consists of a wrought-iron bar m n, 4 
et long, 3 inches wide, and % thick; hinged at its end m, which is confined to 

the top of the platform. Its outer end n is 
formed into a ring V for lifting it. A strong 
casting ee (or in longitudinal section at tt,) 
about 15 inches long, 3 inches wide, and 1 
inch thick, is also firmly bolted to the top of 
the platform ; and the stop-bar mn rests in its 
recess r, while the platform is being turned. 
A similar casting a a is well bolted to the 
wooden or stone coping cc, which surrounds 
the top of the lining wall of the pit. When 
the stop-bar reaches this last casting, as the 
platform revolves, it rises up one of its little 
inclined planes tt, and falls into the recess of 
a a, bringing the platform to a stand. When 
the platform is to be started again, the bar is 
ted out of its recess by the ring F, until it passes the casting; when it is again 
id upon the coping cc, and moves with the platform; or, if required, the 
nge at m allows it to be turned entirely over on its back. When there is a 

I 'ansverse platform, the proper place for the stop is at that end which carries 
e turning gear; as it is there handy to the men who do the turning. If there 
only a main platform, the stop may be placed midway of the rails. Some- 
nes a spring 1 catch is placed at each end of the platform; and each catch 
loosened from its hold at the same instant by a long double-acting lever. All 
e details of a platform admit of much variety. 

Instead of the friction rollers. Fig 5, friction 
balls 5 or 6 inches diameter, of polished steel, are 
sometimes used. The pivots also are made in 
many shapes. 



w 



Fig. 14. 


Platforms like ©«, Fig 14, revolv¬ 
ing around one end o as a center of mo¬ 
tion, are sometimes useful. The shaded space is 
le pit. If an engine approaching along the track W, is intended to pass on to 
ly one of the tracks 1, 2, 3, 4, the platform is first put into the required posi- 
on, and the engine passes at once without detention. If the platform is long, 
will be necessary to have roller-wheels not only under the moving end a, but 
, one or two other points, as indicated by the roller rails c c. 


Engine houses, of brick, cost from $1000 to $1200 per engine stall, exclli¬ 
ve of the foundations. 


The cost of a complete set of shops of brick, for the thorough re- 
ir of about 20 locomotives, and of the corresponding number of passenger 
id other cars; together with suitable smith shop, foundry, car shop, boiler 
op, copper and brass shop, paint shop, store rooms, lumber shed, offices, &c , 
mpletelv furnished with steam power, lathes, planing machiirues, a, ' d 

l other necessary tools and appliances, will be about from $/5000 to $100000 ex- 
rsive of ground. A large yard, of at least, an acre, should adjoin the buildings, 
moderate establishment, for the repairs of a few engines only, may be bu It 
id furnished for $25000. 


























WATER STATIONS. 


800 


WATER STATIONS. 

Water stations are points along a railroad, at which the engines ^op to 
take in water. Their distance apart varies (like that of the fuel sta¬ 
tions, which accompany them,) from about 6 miles, on roads doing a very large 
business; to 15 or 20 miles on those which run but few trains. Much depends, 
however, upon where water can be had. It has at times to be conducted in 
pipes for 2 or .3 miles or more. The object in having them near together is to 
prevent delay from many engines being obliged to use the same station. To 
prevent interruption to travel, they are frequently placed upon a side track. 
A supply of water is kept on hand at the station, usually in large wooden tubs 
or tanks, enclosed in frame tank-houses. The tank-house stands near the track, 
leaving only about 2 to 4 feet clearance for the cars. It is two stories high; the 
tank being in the upper one ; and having its bottom about 10 or 12 feet above the 
rails. In the lower story is usually the pump for pumping up the water into the 
tank; and a stove for preventing the water from freezing in winter.* 

The tanks are usually circular; and a few inches greater in diameter at the 
bottom than at the top, so that the iron hoops may drive tight. Tlieir 
capacity generally varies from 6000 to 40000 gallons, (rarely 80000 or more,) 
depending on the number of engines to be supplied. A tender-tank holds 
from 1000 to 3000 gallons; and an engine evaporates from 20 to 150 gal¬ 
lons per mile, depending on the class of engine; weight of train ; steepness of 
grade, &e. Perhaps 40 gallons will be a tolerably full average for passenger, and 
80 for freight engines. The following' are the contents of tanks 
of different inner diameters, and depths of water. U. $. gallons of 231 cubic 
inches; or 7.4805 gallons to a cubic foot. 


Diam. 

Depth. 

Contents. 

Diam. 

Depth. 

Contents. 

Ft. 

Ft. 

Gallons. 

Cub. Ft. 

Ft, 

Ft. 

Gallons. 

Cub. Ft. 

12 

8 

6767 

905 

24 

12 

40607 

5429 

14 

9 

10363 

1385 

26 

13 

51628 

6902 

16 

9 

13535 

1810 

28 

14 

64481 

8621 

18 

10 

19034 

2545 

30 

15 

79310 

10603 

20 

10 

23499 

3142 

32 

16 

96253 

12S68 

22 

11 

31277 

4181 

34 

17 

115451 

15435 


Cypress or any of the pines answer very well for tanks. The staves 
may be about 2% inches thick for the smaller ones; to 4 or 5 inches for the 
largest. The bottoms may be the same. The staves should be planed by ma~ 
clnneiy to suit the curve precisely. Nothing is then needed between the staves 
to produce tightness. A single wooden dowel is inserted between each two near 
the top, merely to bold them in place while being put together. The bottom is 
dowelled toget ner ; and simply inserted into a groove verv accurately cut, about 
an inch deep, around the inner circumference of the tub’at a few inches above 
the bottoms of the staves. 


One of 20 feet diameter, and 12 feet deep, may have 9 hoops of good iron ; placed 
se . v ® , ; i,1 0 1 ." ch u es nearer together at the bottom of the tank than at the top Their 
width 3 inches; the thickness of the lower two.^inch; thence graduallv dimin¬ 
ishing until the top one is but half as thick. The lower two are driven close 
toget hei. These dimensions will allow for the rivet-holes for riveting together 
the overlapping ends; and for a moderate strain in driving the hoops firmlv 
into place.f Three rivets of ^ inch diameter, and 3 inches apart, in line, are 
sufficient for a joint of a lower hoop. One of 34 feet diameter, 17 deep, may 

lVwe^hoirHoi’n? 6 l0W6r 0063 4 iUCheS by With three %' inch rivets to a 

oAear^rs 0111 PlankS ° f th * tank mUSt bear finn,y upon their supporting joists, 

A tank must have an inlet-pipe hv which the water mav enter it; a waste- 
pipe tor preventing overflow; and a discharge or feed-pipe 7 or 8 
'• or near the bottom; through which the water flows out to 
the tender. The inner end of the discharge-pipe is covered by a valve, to be 
opened at will by the engine man, by means of an outside cord and lever. To 


A frame tank-house, 18 feet square, with stone foundations for both house 
and tank, will by itself cost $100 to $600. A brick or stone one somewhat more 
TSuch a tank, put. up in its place, will cost, about $400. Geo. J. Burkhardt’s 
nnrl^'n h ®road St below Cambria, Philadelphia, make tanks their specialty • 
and are provided with machinery which secures perfect accuracy of joint’in 
every part. Their work is sought from great distances J 

























WATER STATIONS. 


mi 


its outer end is generally attached a flexible canvas and gum-elastic hose about 
7 or 8 inches diameter, and 8 or 10 feet long, through which the water enters the 
I tender-tank. Or, instead of a hose, the feed-pipe may be prolonged by a metal¬ 
lic pipe, or nozzle, sufficiently long to reach the tender; and so jointed as, when 
Inot in use, to swing to one side, or to be raised to a vertical position, (iu the last 
•ijcase it is called a drop,) so as not to be in the way of passing trains. 

The same tank may supply two engines on different tracks, at once. The 
tanks are very durable. 

t The patent frost-proof tank of John Bnrnhain, Batavia, 
Illinois, is simply au ordinary tank, in which the water is prevented from 
^freezing by means, 1st, of a circular roof which protects a ceiling of joists, be¬ 
tween which is a layer of mortar; 2d, by an air-space obtained by a similar ceil¬ 
ing beneath the timbers on which the tank rests. Although the sides are en- 
I tirely unprotected, no house is necessary; but merely strong posts and beams 
an a stone foundation, for the support of the tank.* The supply pipes are in 
boxes made of boards and tar-paper. 

Tanks are frequently made rectangular, with vertical sides of 
l posts lined with plank, and braced across in both directions by iron rods. They 
hare more apt to leak than circular ones. They have been made of iron; but 
wood seems to be preferred. 

The water for supplying tlie tanks, may be pumped by hand, steam, 
i horse, wind, hydraulic ram, or otherwise, from a running stream; from a pond 
^made by damming the stream if very small or irregular; from a cistern below 
• the tank; or from a common well. Many roads doing a business of 10 or 12 
{engines daily in each direction,depend entirely upon wells; and pump by hand ; 
i generally two men to a pump. Those doing a very large business, when the 
supply cannot be obtained by gravity, mostly use steam. The windmill is 
the most„economical power: and when well made, is very little liable to get out 
of order/ Of course it will not work during a calm ; but this objection may be 
obviated in most cases by having the tanks large enough to hold a supply for 
'several days.f Steam, however, is most reliable. 

The following table will give some idea of the power required in 
a steam engine for the pumping. In ordering an engine, specify not 
iits number of horse-powers, but the number of gallons it must raise in a given 
number of hours, to a given height; with a given steam pressure, (say about 60 
to 80 lbs per square inch.) The pump should be sufficiently powerful not to have 
to work at night; and should be capable of performing at least 25 per cent, more 
than its required duty. 

A fair average horse should pump in 8 hours the quantities 
contained in the first 3 columns; to the height in the 4th column; or sufficient 
to supply the number of locomotives in the 5th column, with about 2000 gallons 
each. Two men should do about oue-tbird as much.J 


Cub. Ft. 

Lbs. 

Gals. 

Ht. Ft. 

No. of 
Locos. 

Cub. Ft. 

Lbs. 

Gals. 

Ht. Ft. 

No. of 
Locos. 

1600 

100000 

11968 

100 

6 

4571 

285714 

34194 

35 

17 

20il0 

125000 

14960 

80 

7A 

5333 

333333 

39893 

30 

20 

2667 

166666 

19946 

60 

10 

6400 

400000 

47872 

25 

24 

3200 

200000 

23936 

50 

12 

8000 

500000 

59840 

20 

30 

3555 

222222 

26596 

45 

13^ 

10667 

666667 

79787 

15 

40 

4000 

250000 

29920 

40 

15 

16000 

1000000 

119680 

10 

60 


A reservoir, w ith a stand-pipe, or water column, is preferable 
to the ordinary tank, when the locality admits of it; being less liable than the 
pump to get out of order; and being cheaper in the end. The reservoir is sup¬ 
posed to be filled by water flowing into it by gravity; and to have its bottom at 


*The U S Wind-Engine and Pump Co, of Batavia, Ill, make a specialty of the 
instruction and erection of these tanks, complete in every detail, ready for use. 

rhey also make windmills. . 

t Andrew J. Corcoran, No. 76 John St, New York, furnishes excellent machines 
Te also, when desired, provides pumps, &c, complete. The cost of windmill 
done, for railway stations, varies iroin about $450 for 18 feet diameter, lo $lo00 
or 36 feet diameter, at the factory. „ 

+ The co*t of a direct acting steam pump, with its boilers, Ac, 
ixed iii place, readv for work, and capable of the duty of the above table,.may 
ie roughly set down at about $450; twice the duty, $600; 4 times, $750; 6 times, 
5900 • 10 times,$1300; 20 times.$2000. Add costof engine-house. Made by Henry 
“ Worthington, 86 Liberty St. New York ; Gej. F. Blake Manufacturing Co,44 
Washington St, Boston ; and by many establishments m most of our large cities. 

































802 


WATER STATIONS. 


least about 8 feet above the rails; or at any greater height whatever that the 
ground and the height of the water may require. It may be excavated in the 
ground; lined with brick or masonry in cement; with a bottom of concrete; 
or it may be built above ground, according to the locality. It may be roofed 
and covered in, or not; and it. may be near the tracks, or at a considerable dis¬ 
tance from them, according to circumstances.* From its bottom, an iron pipe 
from 8 to 12 itic-hes diameter, is carried (generally underground,) to within a few 
feet of the track. At that point it turns vertically upward to about 8 or 10 feet 
above the track, forming a stand-pipe, or water-column; from the 
upper end of which the water flows (through either a hose or a jointed nozzle,) 
as in the case of a tank. Several such pipes, or one larger one, may be laid, for 
the supply of two or more engines at once, through as many stand-pipes. Where 
the pipe makes its bend, and becomes vertical, is a valve for opening and closing 
it; and which may be worked by a hand-wheel placed at such a height as to be 
easily reached by the engine rnan.f A valve on the principle of those for street 
pipes, page 301, is best. 

On some of the more important lines, the tenders of fast trains scoop 
up water, while running, from a long trough, or “ track tank” 

laid between the rails. The tanks are about K mile long. They must of course be 
level, and they therefore require a level track. 

As originally introduced in England, by Ramsbottom, the trough was of cast- 
iron, in lengths of about 6 ft. These were bolted together by means of flanges at 
their ends. The ends were not in contact with each other, but were separated by 
a strip of vulcanized rubber. 




Ml, 


-1-9-incheg- 



Our figure is a vertical cross-section of the standard track tank of the Pennsyl¬ 
vania Railroad, 188-1. It is of ^ inch rolled plate-iron, the sheets of which are 62 
ins long. The lengths overlap each other 2 ins ; leaving 5 ft as their shoiving length, 
i lie sheets are cut slightly tapering, so that at one end of each length the trough is 
t 3 g in.deeper than at the other,and the tops are thus kept flush with each other through¬ 
out. The joints are double riveted with % inch rivets, about 1 K ins from center to 
center, and staggered. At each end of the trough, the bottom slopes upward, and, 
in a length ot 6 ft, comes to the level of the tops of the sides. The cross-ties are 
notched, as shown, to receive the trough, which is loosely held to them by two 
spikes, S and S, in each tie. The heads of the spikes fit over the horizontal flanges 
ot the X IK i,lch angle bars, A and A. M and M are mouldings of lbk X K 
inch bar-iron. The angles and the mouldings are in lengths of 15 ft, and are riveted 
to the sides of the trough continuously throughout its length. 

The scoop on the tender is lowered into the trough, and raised from it, 
by means ot a lever on the fireman's platform, and is not permitted to touch the 
bottom of the trough. 

. trough is supplied with water by means of pipes leading from an ad¬ 
jacent tank. The supply is regulated by a man in charge. 

To prevent the water from freezing in winter, steam is led to the trough 
lrom the boiler of the pumping engine, through iron pipes laid under ground along¬ 
side of the track. These pipes are provided with branches which introduce the 
steam to the trough at every 40 ft of its length. The steam-pipes are protected by 
wooden boxes, and are furnished with valves for regulating the supply of steam. 



































FENCES, ETC. 


803 


Evaporation from Locomotives. In addition to what is said on page 
*0°, i" the passage preceding the table, we may state that the evaporation is 
lsually from 6 to 7 &s of water to 1 !b of fair coal. Hence if we take the average 

I it 6% lbs, or say .8 ot a gallon of water to 1 lb of coal, and assume, as on page 
Y>00, that a passenger engine evaporates an average of 40 gallons per mile, and 
■i freight engine 80 gallons, we shall have very nearly 2% tons of coal consumed 
>er 100 miles by the former; and 4 % tons by the latter. The evaporation from 
i heavily tasked powerful engine may amount to 150 gallous or more per mile; 
>ut such is an exceptional case. 

Theoretical thickness near bottom of sheet-iron water 
auks, single riveted; safety 4 ; ultimate strength of the iron 40000 lbs per 
quare inch, but reduced say one-half by punching the rivet holes. Although 
afe against the pressure of the water, some are plainly far too thin for handling. 


Depth in 
Feet. 

5 

II 

10 

vrNER 

15 

DIAMETER IN FEE 
20 j 25 j 30 

T. 

35 

40 

THICKNESS IN INCHES. 

1 

5 

10 

15 

20 

25 

30 

.0026 

.0130 

.0260 

.0891 

.0521 

.0651 

.0781 

.0052 

.0260 

.0521 

.0781 

.1042 

.1302 

.1562 

.0078 

.0391 

.0781 

.1172 

.1562 

.1953 

.2344 

.0104 

.0520 

.1042 

.1562 

.2084 

.2604 

.3124 

.0130 

.0651 

.1302 

.1953 

.2604 

.3255 

.3906 

.0156 

.0781 

.1562 

.2344 

.3125 

.3906 

.4687 

.0182 

.0911 

.1823 

.2734 

.3645 

.4557 

.5470 

.0208 

.1042 

.2083 

.3125 

.4166 

.5208 

.6250 


I Railroad track scales are made by Riehle Bros., office, 413 Market St., 
'hila., and by Fairbanks & Co., St. Johnsbury, Vt. Price-list of Riehle track 
ailes. Discount, 1888, about 45 per cent. The capacities are in tons of 2000 
is or 2240 lbs, as may be desired. 


Capac¬ 
ity. • 

Length. 

ft. 

Price. 

$ 

Capac¬ 

ity. 

Length. 

ft. 

Price. 

$ 

10 

12 

350 

65 

40 to 65 

1850 

15 

12 to 15 

400 

75 

40 to 85 

2200 

20 

12 to 16 

600 

100 

50 to 112 

2800 

30 

20 to 32 

850 

150 

60 to 123 

3200 

40 

30 to 40 

975 

150 

100 to 150 

3700 

50 

40 to 50 

1100 





Post-and-rail fences, in panels 8^ ft long; 5 rails; usually cost between 
) to 100 cents per panel, including the putting up ; or from $512 to $1280 per mile 
f road fenced on both sides, with 1280 panels. 

Fence-posts are usually of chestnut,cedar, or white oak. They last about 10 years 
a an average. The usual size is 2 to 3 ins thick X 6 to 7 ins wide, 8 ft long, 5 ft 
love ground. Their cost varies greatly; say from 5^ to 25 cts each; average, 10 
> 15. 

Worm fences seven rails high, with two rails on end at each angle, cost about 
'th less. Labor $1.75 per day. The scarcity or abundance of timber chiefly in- 
uences the price; as is also the case with ties. 

Harked Steel wire fence costs per mile of single row of fence, put up, 
(eluding the wooden'posts anil all labor, from $150 to $250, depending on the 
eight of fence, the varying market price of w'ire, labor, &c. 

A way-station house, 30 X CO feet, surrounded by a platform 12 feet wide, 
rotected by projecting roof; for passengers and freight; will cost from $6000 to 
10,000, according to finish aud completeness, at eastern city prices. 

55 




































804 


COST OF RAILROADS. 


Approximate average estimate for a mile of single-track 

railway. Labor $1.75 per day. 

Grubbing and clearing , ( average, of entire road,) 3 acres at $50.$ 150 

Grading; 20000 cub yds of earth excavation , at 35 cts . 7000 

“ 2000 cub yds of rock excavation, at $1.00 . 2000 

Masonry of culverts, drains, abutments of small bridges, retaining-walls, etc; 

400 cub yds, at % 8, average . 3200 

Ballast; 3000 cub yds broken stone, at $1.00.... 3000 

Cross-ties; 2640, at 60 cts, delivered . 1584 

Rails; (60 /6s to a yard ;) 96 tons, at $30, delivered . 2S80 


Spikes . 15<1 


Rail-joints . . 300 

Sub-delivery of materials along the line ... 

Laying track .. 

Fencing ; (average of entire road,) supposing only x /, of its length to be fenced.. 

Small wooden bridges, trestles, sidings, road-crossings, cattle guards, etc, dc . 

Land damages . 

Engineering, superintendence, officers of Co, stationery, instruments, rents, 
printing, law expenses, and other incidentals .. 

Total ...$26000 

Add for depots, SllOps, Engine-houses, Passenger and Freight Stations, Platforms, 

n ti L' j on/1 mi mna Tiiln/rm »\Vi U n /vi n na F* /. \I T n. CaaIaa 


■Wood Sheds, Water Stations with their tanks and pumps, Telegraph, Engines, Cars, Weigh Scales, 
Tools, 4c, 4c. Also for large bridges, tunnels, Turnouts, 4c. 


























LOCOMOTIVES 


805 


ROLLING STOCK. 


LOCOMOTIVES. 

Dimensions, Weights, <fcc, of Locomotives. 

The following lists of the dimensions, weights, &c, of some of the principal sizes 
f locomotives made by the Baldwin Locomotive Works, Phila; Burnham, Parry, 
V illiams & Co, proprietors; will give an idea of the present usual proportions 
*f locomotives ami tenders as made in the United States. 

In the designation of the class, the first number (8, 10, <Src) is the total number 
f wheels of the locomotive. The second (20, 26, &c) is an arbitrary number indi- 
ating the diameter of the cylinders. The letter (C, D, or E) indicates the number 
t, 6, or 8, respectively) of driving-w heels. 

Tile wheel-base is the distance from center to center of the front and hack 
'heels. For minimum length of turntable, add to 2 ft to the total 
■heel-base of locomotive and tender, which is = wheel-base of locomotive -f- wlieel- 
jase of tender+distance between centers of front tender wheel and hind engine wheel. 

Under ‘‘Service,” P means passenger ; F, freight; M, mixed; S, switching. 

For gauge of 4 ft 8 1-2 ins. 

Dimensions. 






Cylin¬ 

ders. 

Driving 

Wheels. 

Wheel-base. 

3 

Nto . 
—< 0 j- 


• 

Height 

of top of 

IJlass. 

© 

.2 

► 

© 

Q« 

a 

C3 

© 

O 



a 

a 

Locomotive. 

Ten- 

Loco 

and 

<D cS 

3 0 q 

£ S2 

© 

a 

© 

u 

n 

*£ 

chimney 
above 
top of 





5 

«-> 

m 



s 

Driv’rs 

Total. 

der. 

tender. 

X 

W 


X 

w 


rails. 





Ins. 

Ins. 



Ins. 

Ft. 

In. 

Ft. 

In. 

Ft. 

In. 

Ft. 

In. 

Ft. 

In. 

Ft. 

In. 

Ft. 

In. 

i-14—C 

p ? 

u > 

© . 

( 10 

20 

4 

45 

to 51 

5 

6 

16 

4 

5 

1 

27 

6 

41 

3 

7 

9 

12 

4 

-20- 

-0 

P F > 

3 S 

< 13 

22 

4 

49 

to 57 

7 

0 

20 

6 

14 

2 

42 

0 

47 

8 

8 

4 

13 

0 

-30- 

-c 

“ > 

I ^3 

l 18 

22 

4 

61 

to 66 

8 

6 

22 

5 

14 

0 

44 

4 

54 

6 

8 

6 

13 

6 

-26—D 

F Ml 

r 

fl6 

24 

6 

51 

to 56 

12 

6 

22 

8 

13 

5 

42 

2 

54 

9 

9 

4 

13 

13 

0 

-28- 

-D 

“ t 

2 « 

I 17 

24 

6 

51 

to 56 

12 

10 

23 

0 

13 

1 

42 

11 

55 

1 

8 

11 

0 

-32- 

-D 

“ ) 

2 .fl 
£ 

l 19 

24 

6 

54 

to 60 

13 

6 

23 

8 

13 

11 

44 

10 

55 

9 

9 

0 

13 

6 

-18- 

D 

F ) 


(12 

18 

6 

37 

to 41 

10 

0 

16 

0 

10 

11 

33 

0 

41 

8 

8 

4 

12 

6 

-26- 

D 


S 3 

<16 

24 

6 

45 

to 51 

14 

2 

21 

6 

13 

5 

42 

2 

54 

6 

9 

0 

13 

0 

-32- 

D 

“ 5 

« to 

l 19 

24 

6 

54 

to 60 

15 

2 

22 

0 

13 

11 

43 

10 

56 

2 

9 

2 

13 

8 

-34- 

E 


fl 

C 20 

24 

8 

48 

to 50 

14 

9 

22 

10 

14 

4 

46 

2 

58 

6 

9 

6 

14 

0 

-36— 

E 

“ 5 

OS 0 

i 21 

24 

8 

48 

to 50 

14 

9 

22 

10 

14 

4 

46 

2 

57 

3 

9 

8 

14 

0 

-26— 

C 

s 

* 8 H 

16 

24 

4 

48 

to 54 

7 

6 

7 

6 

13 

5 

34 

6 

47 

2 

9 

0 

13 

0 

-28— 

D 

s 

J 

17 

24 

6 

45 

to 49 

11 

0 

11 

0 

13 

1 

35 

5 

48 

4 

9 

0 

13 

4 


IVeights, d'c. 





lass. 

© 



*> 

© 

04 


© 



GO 

E- 

.11.0 

p ) 

£> ; 

© . 

20. C 

PF > 

S 2 

.30.C 

“ J 

<1 

26.D 

FM) 

t 

.28. D 

“ t 

2 ^ 

.32.0 

“ ) 

z -a 
* 

18.D 

F ) 

0 2 

.26.14 

“ t 

x£ 

.32.0 

“ ) 

c to 

.31.E 

“ } 

d «5 : . 

© ^ -S 

.36. E 

“ 5 

or 0 

.26.0 

s 

- O ■- 

.28,0 

s 



Greatest on 
1 pair of 
drivers. 


Weights in working order 

Locomotive. 


IDs 

(10500 
1 17500 
( 27000 

(18000 
-> ioooo 
( 22000 

(11000 
( 21000 
( 25000 

( 23000 
} 23000 


a 
o 

♦3 

4.6 

7.3 

12.0 

8.0 

8.5 

9.8 

4.9 

9.4 

11.2 

10.3 
10.3 

28000 i 12.5 
24000 10.7 


On all 
drivers. 


IDs 

21000 

35000 

54000 

58000 

63000 

72000 

33000 

63000 

S0000 


a 

o 

9.4 

15.6 

24.1 

25.9 

28.1 

32.1 

14.7 

28.1 

35.7 


Total. 


tbs 


3600016.1 
56000 25.0 
8200036.6 

78000 34.8 
8400037.5 
96000,42,9 

40000)17.9 

75000J33.5 

94000i42.0 


Tender 

loaded. 


tbs 


26000,11.6 
34200 15.3 
55500 24.8 

47800 21.3 
51600 23.0 
59300 26.5 

21600 9.6 
47800 21.3 
59300 26.5 


88000 39.3 104000 46.4 59300 26.5 
96000 42.9 112000 50.0 63400,28.3 

56000 25.0 5600ol25.0 47800 21.3 
69000 30.8 69000 30.8 51600,23.0 



Capacity of tender. 

Total 





loco and 


T3 

O 

Water. 

tender. 

0 e 

O 





O 




IDs 

tons 1 

1 

IDs cords 

Gals of 
231cu in 

IDs 

62000 

27.7 

8000 

1.25 

1000 

8300 

90200 

40.3 

9000 

1.60 

1400 

11700 

137500 

61.4 

12300 

2.90 

2400 

19900 

125800 

56.2 

11800 

1.70 

2000 

16700 

135600 

60.5 

12000 

1.80 

2200 

18400 

155300 

69.3 

12500 

2.00 

2600 

21700 

61600 

27.5 

8500 

1.50 

1200 

10000 

122800 

54.8 

11800 

1.70 

2000 

16700 

153300 

68.4 

12500 

2.00 

2600 

21700 

163300 

72.9 

12500 

2.00 

2600 

21700 

175400 

78.3 

13000 

2.20 

2800 

23300 

103800 

16.3 

11800 

1.70 

2000 

16700 

20600 

53.8 

12000 

1.80, 

2200 

18400 

































































































806 


LOCOMOTIVES 


For ^aug'e of 3 ft. 
Dimensions. 


Class. 

Service. 

Cylin¬ 

ders. 

Driving 

Wheels. 

Wheel-base. 

1 Extreme l’jjth 

1 loco and ten- 

! der. 

• 

as 

S-2 
£ £ 

X 

« 

Height 

of top of 
chimney 
above top 
of rails. 

a 

C9 

5 

Stroke. 1 

© 

a 

n 

5 

Locon 

Drivers 

jotive. 

Total. 

Tender. 

Loco 

and 

tender. 



Ins. 

Ins. 


I us 

Ft. 

Ins. 

Ft. Ins. 

Ft. 

Ins. 

Ft. Ins. 

Ft.In. 

Ft.Ins. 

Ft. Ins. 

8—18—0 

P 

12 

16 

4 

43 

7 

2 

18 3 

11 

4 

35 8 

44 7 

7 4 

12 0 

8 —22—C 

44 

14 

18 

4 

45 

8 

2 

20 1 

13 

2 

40 5 

48 6 

7 10 

12 4 

8 —20—D 

F 

13 

18 

6 

37 

12 

0 

17 10 

11 

4 

55 3 

47 1 

7 4 

12 2 

8 —22— D 

44 

14 

18 

6 

39 

12 

0 

18 4 

13 

2 

38 8 

45 5 

7 8 

13 0 

10—24—E 

44 

15 

18 

8 

37 

11 

4 

17 10 

13 

5 

39 11 

49 5 

8 1 

12 5 

10—26—E 

44 

16 

20 

8 

37 

11 

9 

18 1 

13 

0 

40 9 

50 0 

8 4 

13 7 

4—18—0 

S 

12 

16 

4 

37 

6 

0 

6 0 



6 0 

37 0 

7 10 

13 0 

6 —22—D 

44 

14 

18 

6 

37 

9 

6 

9 6 



9 6 

43 6 

7 4 

11 1 


Weights, Ac. 


Class. 


8—18—C 
8—22—C 
8—20—D 
8—22—D 
10—24—E 
10—26—E 
4—18—C 
6—22—D 


c n 


Weights in working order. 



Locomotive. 






Greatest 
on 1 pair 
of drivers. 

On 

drive 

all 

rs. 

Tot 

al. 

Ten 

load 

der 

ed. 

loco 

tend 

uid 

er. 

fi)S 

09 

p 

B>s 

99 

a 

fi>s 

09 

a 

Bis 

09 

a 

Bis 

CO 

p 

13000 

5.8 

24000 

10.7 

37000 

*9 

16.5 

31000 

-*-> 

13.8 

68000 

30.4 

16000 

7.1 

33000 

14.7 

48000 

21.3 

35000 

15.6 

83000 

37.0 

13000 

5.8 

39000 

17.4 

46000 

20.5 

31000 

13.8 

77000 

34.4 

14000 

6.2 

42130 

18.8 

50330 

22.5 

35000 

15.6 

85330 

3S.1 

12000 

5.4 

48000 

21.3 

56000 

25.0 

37000 

16.5 

93000 

41.5 

15000 

6.7 

58000 

25.9 

67000 

29.9 

38500 

17.2 

105500 

47.1 

15000 

15000 

6.7 

6.7 

28000 

48000 

12.5 

21.3 

28000 

48000 

12.5 

21.3 



28000 

48000 

12.5 

21.3 


Capacity of tender 

or tank. 


T3 
U © 


B>s cords 


9000 

9775 

9000 

9775 

10300 

9775 

1800 

2000 


1.3 

1.4 

1.3 

1.4 

1.5 
1.4 

.5 

.5 


Water. 


Gals of 
231 cu in 


1200 

1400 

1200 

1400 

1500 

1600 

400 

500 


lbs 


1000 

1170 

1000 

1170 

1245 

1328 

332 

415 


Standard passenger (Class N) and freight (Class I) locomotive! 


of Pennsylvania Railroad, 1885. 

Dimensions. 


Gauge, 4 ft 9 ins. 


Class. 

Cylin¬ 

ders. 

Driving 

Wheels. 

Wheel-base. 

Extreme l’gth 
loco and ten¬ 
der. 

Extreme 

width. 

Height 

of top of 
chimney 
above top 
of rails. 

Diam. 

Stroke. 

O 

| Diam. 

1 

Locomotive. 

Tender. 

Loco 

and 

tender. 

Drivers 

Total. 


Ins. 

Ins. 


Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

N 

17 

24 

4 

62 

8 6 

23 2 

14 10 

45 2 

54 7 

9 0 

15 0 

I 

20 

24 

8 

50 

13 8 

21 6 

15 4 

47 7 

56 0 

* 

9 3 

15 11 


Weights, Ac. 


Class. 

Weights in working order. 

Capacity of tender. 

Locomotive. 


Locomo¬ 
tive and 
tender. 


Water. 

Grea 
on 1 
of dri 

test 

>air 

rers. 

On 

drive 

all 

jrs. 

Toti 

ll. 

Tend 

er. 

Cc 

al. 

N 

1 

lbs 

30800 

22200 

09 

P 

O 

-4-» 

13.8 

9.9 

lbs 

57700 

80500 

09 

P 

O 

4J 

25.8 

35.9 

lbs 

91300 

92700 

09 

P 

O 

40.8 

41.4 

lbs 

50500 

56650 

CO 

P 

O 

22.5 

25.3 

lbs 

141800 

149350 

00 

P 

O 

4-1 

63.3 

66.7 

lbs 

8000 

8000 

00 

p 

o 

+5 

3.6 

3.6 

Gals of 
231 cu ins. 

2400 

3000 

lbs 

2000 < 

2500< 















































































































































LOCOMOTIVES. 807 


The following are the principal dimensions, «fcc, of five recent and ex¬ 
ceptionally larg-e types of locomotives: 



“ Decapod.”# 
Largest loco 
built by the 
Baldwin Wks 
up to 1887. 

“ El Goberna- 
dor,” Ceutl 
Pacific K R. 
1881 , cvls 21 
X 36. 

“Class K." 
Engine “ No. 
10, &c, Penna 
R R 

“ Class P.” 
Penna R R. 

Phila & 
Reading R. R. 
Baldwin Wks, 
1888. 

Service. 

Freight. 

Freight. 

Fast Pass'r. 

Passenger. 

Fast Pass’r. 

Gauge. 

Driving wheels: 

4 ft 8% ins 

4 ft 8% ins 

4 ft 9 ins 

4 ft 9 ins 

4 ft 8% ins 

Number. 

10 

10 

4 

4 

4 

Diam. 

i Truck wheels: 

45 ins 

57 ins 

78 ins 

68 ins 

68 % ins 

Number. 

Wheel-base: 
Locomotives: 

2 

4 

4 

4 

4 

Drivers. 

17 ft 0 ins 

19 ft 7 ins 

7 ft 9 ins 

7 ft 9 ins 

7 ft 6 ins 

Total. 

24 ft 4 ins 

28 ft 11 ins 

22 ft 7% ins 

22 ft 7% ins 

22 ft 1 in 

Tender. 

Loco and ten- 

14 ft 5 ins 

15 ft 2 ins 

15 ft 4 ins 

15 ft 4 ins 

16 ft 0 ins 

der. 

Extreme length: 
Loco and ten- 

49 ft 2 ins 

52 ft 7 ins 

47 ft 8 ins 

47 ft 8 ins 

49 ft 5 inB 

der. 

Height: 

Top of chimney 
above top of 

50 ft 9% ins 

63 ft 8 ins 

58 ft 6 ins 

58 ft 7 ins 

60 ft 8 ins 

rail. 

Weight in work¬ 
ing order, lbs 
Locomotive: 
Greatest on 1 
pair of dri- 

14 ft 6 ins 

16 ft 2 ins 

15 ft 0 ius 

15 ft 0 ins 

14 ft 6 ins 

vers. 

On all dri- 

27000 

25800 

33600 

34700 

41500 

vers. 

134000 

121600 

59000 

59450 

82500 

Total. 

148000 

152000 

92700 

100600 

114500 

Tender. 

Loco and ten- 

82000 

85650 

56300 

56300 

76000 

der. 

Capacity of ten¬ 
der : 

230000 

237050 

149000 

156900 

190500 

L Coal: tbs. 

Water: 

Gals of 231 

1C000 

10000 

12000 

12000 

14000 

cubic ins... 

3600 

3000 

2400 

2400 

3500 


From the above lists it will be seen that tlie weight of road locomo¬ 
tives per foot of their total wheel-base varies from .9 ton in light 
passenger eugiues for 3 ft gauge, to 2.5 tons in very heavy freight engines for 
standard gauge. 

The cost of locomotives and tenders is about from 9 to 12 cts per 
pound of locomotive alone, varying with the details of the specifications, &c. 
j Steel driving tires are usually of open hearth steel, and are from 2% 
'to 3% inches thick; occasionally 4 inches. Flanges average 1% to 1% inches 
deep,‘l% inches thick. Prices, deliver!d in New York: rough, 5 cts perlh; 
bored, 5% cts. Standard Steel Works, makers, Lewistown, Pa; oflice, 220 S 4th 
St, Phi la. The inner diameter of the tire is usually made (or inch per 
foot) less than diameter of the wheel center upon which it is to go. Heating the 
tire increases its inner diameter so that it can be placed upon the wheel center. 
Its contraction, or teudency to contract, in cooling, binds it fast. 

The treads are re-turned in a lathe as often as from x % to % inch wears off from 
their thickness. (=% to % inch in diameter) and are abandoned when worn down 
to about inches thick. On passenger engines, they run about 60000 miles 
between turnings; on freight e ngines, 35000. _ 

* Built 1886 for Northern Pacific R R, Nos 500, 501; Extreme width 10 ft. 
The original “ Decapod,” built by same works in 1884 for Dorn Pedro II R R, Brazil 
(5 feet 3°inches gauge), and described in our 1886 edition, was a little lighter. Tne 
newer engine, above d« scribed, is illustrated in Hg. 1, p. 790. 









































808 


LOCOMOTIVES 


Driving axles are of iron, or of a softer steel than the tires. They are usu¬ 
ally from 5 to 7% ins diam. t , *-j 

Passenger engines usually carry fuel and water sufficient for 40 or 
50 miles; some, 50 to 60. Freight trains, enough for 20 to 25 miles. Roads, or 
divisions, with steep grades require the fuel and water stations to be nearer together 
than where the grades are easy. 

Performance of Locomotives. 

The following gives the loads (exclusive of locomotive and tender) which the 
above described Baldwin engines will haul, at their usual speed,onustraiglit 
track and on different grades varying from a level to 3 ft per 100 ft, or 158.4 
ft per mile. The loads are based upon the assumption that the so-called “adhe¬ 
sion” of the locomotive is one-fourth of the weight on all the drivers, and that 
the condition of road and cars is such that the frictional resistance of the 
cars does not exceed 7 lbs per ton of 2240 lbs of their weight. These are ordinarily 
favorable conditions. The adhesion is seldom less than oue-tifth, or more than one- 
third, of the weight on the drivers. 

The resistance of cars to motion, on a level track, and with cars and 

track in fair order, is usually taken at about from 6 to 8 lbs per ton of 2240 lbs. 
With everything in perfect order, it may fall as low as 5, or even 4 lbs per ton. On 
the other hand, if the wheels are not truly round, and if the journals are not well 
lubricated, it may greatly exceed 10 or 12 lbs. See p 374 e. 

Gauge 4 ft 8 1-2 ins. 

Loads in tons of 2240 lbs (exclusive of locomotive and tender). 


Class. 

* 

Service. # 

j Type. 

On a grade of 

0 per 
center 
0 ft per 
mile. 

16 per 
cent ~ 
26.4 ft 
per 
mile. 

1 per 
cent — 
52.8 ft 
per 
mile. 

116 per 
cent = 
79. *2 ft 
per 
mile. 

2 per 
cent~ 
105.6 ft 
per 
mile. 

216 per 
cent = 
132 ft 
per 
mile. 

3 per 
cent = 

158.4 ft 
per 
mile. 

8—14—C 

P 1 



( 500 

220 

130 

85 

60 

45 

35 

8—20—C 

PF 


“ Ame- 

■{ 1000 

435 

255 

170 

125 

95 

75 

8—30—C 

“ . 



( 1550 

675 

395 

265 

200 

150 

115 

10—26— D 

F M 



l 1685 

740 

435 

305 

220 

175 

135 

10—28—D 




■{ 1830 

780 

460 

320 

235 

185 

150 

10—32-D 

“ . 



( 2090 

915 

540 

375 

270 

215 

175 

8—18—D 

F ) 



l 930 

405 

240 

160 

120 

95 

75 

8—26—D 

it 


“ Mo- 

ml *• 

■< 1830 

800 

475 

330 

245 

195 

165 

8—32—D 

“ , 



( 2330 

1026 

610 

412 

315 

245 

195 

10—34—E 

“ 1 

L 

“Con- 

J 2560 

1130 

670 

465 

350 

275 

220 

10—36—E 

“ 

■ 

solida- 

(2740 

1205 

720 

495 

370 

290 

235 

4—26—C 

S 



1635 

720 

430 

300 

225 

175 

140 

6—28 —D 

ii 



1865 

820 

490 

340 

255 

200 

160 


Gauge 3 ft. 

Loads in tons Of 2240 lbs (exclusive of locomotive and tender). 


Class. 

* 

Service. » 

Type. 

On a grade of 

0 per 
cent = 
0 ft 
per 
mile. 

16 per 
cent = 
26.4 ft 
per 
mile. 

1 per 
cent = 
52.8 ft 
per 
mile. 

116 per 
center 
79.2 ft 
per 
mile. 

2 per 
cent = 
105.6 ft 
per 
mile. 

2 16 per 
cent = 
132 ft 
per 
mile. 

3 per 
cent — 

158.4 rt 

per 

mile. 

8—18—C 

P 

“ Ame- 

650 

252 

148 

103 

73 

55 

43 

8—22—C 

ii 

rican ' 

790 

308 

182 

123 

90 

68 

53 

8—20—D 

F 

“ 10- 

900 

353 

215 

145 

112 

83 

66 

8—22—D 

ii 


990 

392 

235 

162 

120 

92 

74 

10—24—E 

ii 

“ Mo- 

<rn1 ” 

1160 

460 

275 

191 

142 

116 

89 

10—26—E 

ii 


1450 

580 

350 

243 

181 

142 

114 

4—18—0 

S 


825 

335 

205 

145 

110 

90 

75 

6—22—D 

ii 

tion ” 

1320 

540 

330 

235 

180 

145 

120 


* See foot note, p. 809. 



















































LOCOMOTIVES. 


809 


The mean 
tractive force 

of a locomotive, 
in pounds 


Square of diam of ^ Single length of A vera S e steam pres- 

one piston in ins. ^ stroke in ins. ^ ? UI £ 111 cylinders 
_ in lbs per sq inch. 


Diameter of driving-wheel in inches 


Deduct 20 to 30 per cent for internal friction, etc. Then, if the result exceeds 
the “ adhesion,” the tractive power is only equal to the adhesion. 

The initial steam pressure in tlie cylinders is always less than the 
boiler pressure; and the disproportion increases with the speed. Thus, at 8 or 10 
miles an hour, the boiler pressure may be about 110 lbs per square inch; and the 
cylinder pressure from 90 to 100 lbs, while at a speed of 30 or 40 miles, the propor¬ 
tion may be as 110 to 00 or 70 lbs. The average cylinder pressure is ascertained by 
means of an indicator applied to the cylinder; and its proportion to the initial 
pressure depends upon how early in the stroke the supply of steam from boiler to 
cylinder is cut off; or, in other words, upon the extent to which the steam is used 
expansively. 

The power and speed of locomotives, and their consump¬ 
tion of fuel and water, vary greatly with circumstances, such as grades and 
curvature; condition of track and rolling stock; number of cars in train: diam¬ 
eters, number and distance apart, of car wheels; manner of coupling the cars; skill 
of locomotive runner and fireman, &c, &c. The following records of actual perform¬ 
ance will serve as indications : # 

Baldwin engines. Anthracite passenger engine, class 8—28—C, 
“American ” type, hauls 9 loaded passenger cars, 216 tons besides weight of engine 
and tender, 52 tons, up a grade 1771 ft long, averaging 107 ft per mile, at 10 miles 
per hour. At the foot of the grade is a curve of 225 ft radius, on which the grade 
is 100 ft per mile. A similar engine hauls 4 passenger cars, 1 parlor car and 1 bag¬ 
gage car, 145 tons, engine and tender, 59 tons, 59 miles over a nearly level road, with 
few and easy curves, in 1% hours. Boiler pressure about 125 lbs per square inch. 
Four such runs consumed 10000 lbs of coal. Bituminous passenger en¬ 
gine, class 8—30—'C, “American” type, hauls 4 passenger cars, 1 sleeper, and 3 
baggage and mail cars, 150 tons, engine and tender, 61.4 tons, 11,55 miles in 28 min¬ 
utes, up continuous grades, mostly of about 70 ft per mile, and over nearly continu¬ 
ous reversed curves of from 1° to 7°. Freight engines, class 10—30—D, “10 
wheel’’type, haul the following trains: with bituminous coal, 18 cars, 337 tons, 
engine and tender, 65 tons, up a grade of 79.2 ft per mile, with a curve of 819 ft 
radius, 885 ft long; 48 cars. 386 tons, engine and tender, 65 tons, up a grade of 62 ft 
per mile, with 4° curves; 40 cars, 785 tons, engine and tender, 65 tons, up a grade 
of 21 ft per mile, on 5° curves. Freight engines, class 8—26—D, “Mogul” 
type, haul 45 cars, 300 tons, engine and tender, 55 tons, up grades of 83 ft per mile, 
with a 2° curve ; starting on the grade; boiler pressure about 130 lbs; also, 37 cars, 
185 tons, engine and tender, 55 tons, up a grade of 85 ft per mile, with curves of 9° 
and 10°. Freight engines, class 10—34—E, “ Consolidation ” type, haul 90 
cars, about 2000 tons, engine and tender, 68 tons, 45.5 miles in 4 hours 21 minutes, 
over a nearly level road with easy curves, consuming 1.8 to 2.7 lbs of bituminous 
coal per loaded car per mile; also,with anthracite coal,33 cars, 264 tons,engine and 
tender, 68 tous, up a grade of 96 ft per mile, 12 miles long, with many curves of 573 
ft radius, at from 10 to 20 miles per hour; also, with anthracite, 25 loaded 4-wheel 
coal cars, 235 tons, engine and tender, 68 tons, up a grade ot 126 ft per mile. 

Central Pacific R R, 1883. Engines with 4 drivers, 56 ins diam, cylinders 
17X24 ins. Weight on drivers, 47550 lbs = 21.25 tons; weight of engine and ten¬ 
der, in working order, 133000 lbs = 59.4 tons. Average train, 51 fright cars, 860 tons. 
Maximum grades, 52 ft per mile. Sharpest curve, 5°. Runs of 84 and 145 miles. 
Total distance run, 3000 miles. Average speed about 10 miles per hour. Maximum 
boiler pressure 125 lbs per square inch. Bituminous coal consumed 73 lbs per train 
mil&; or 30.8 train miles per ton of coal; or 11 ton miles per lb of coal. Water 
evaporated 41 gallons per mile; 4.71 lbs per lb of coal. 

On the Boston «fc Albany R R, passenger engines with 4 drivers 
|68% ins diam, cylinders 18 X 22 ms, with 6 passenger cars, run between Boston and 
Springfield, 97 miles, in about 4% hours, consuming 8000 lbs coal, 400 lbs wood for 
lighting, and evaporating 47800 lbs of water; or say 6 lbs per lb of coal. Boiler 
pressure 130 to 150 lbs. 


* In the Baldwin Works classification (“8—30—C” etc) the first number (8, 10 etc) 
is the total number of wheels of the loco. The second indicates arbitrarily the diam 
of the cyls; thus. 12 means 9 ins; 14, 10; 16, 11; 18, 12; 20, 13; 22, 14; 24, 15; 26, 
16- 28 i7- 30, 18; 32, 19; 34, 20; and 36, 21 ins diam. The letter (C, D, or E) indi¬ 
cates the number (4, 6, or 8, respectively) of dnving wheels. In the “ Service ” col¬ 
umn of opposite table, P means passenger; F, freight; M, mixed; and S, switching. 










810 


LOCOMOTIVES. 




Average of main line of Pliila dc Reading: R R, 1884. 


Train®. 


Passenger, t.i . 

Freight. 

Coal: 

Up (empty). 

Down (foil) ....... 


Average 

speed. 

No. 

of cars. 

Weight of traiD. 

Coal consumed. 

Cars. 

L. & T. 

Total. 

Kind. 

per. 

train-mile 

Per 

ton-mile 

miles 


tons. 

tODS. 

tons. 


lbs. 

lbs. 

per hour. 
30 

8 

157 

66 

223 

A nth 

68.4 

.31 

20 

40 

835 

72 

907 

Waste 

103.8 

.11 

15 

165 

578 

72 

660 

“i 

138 


12 

145 

1320 

72 

1392 

“ $ 




“ Coal consumed” includes amount used in firing up. “ Ton ”= 2240 lbs. The 
“waste” is finely broken anthracite, or “culm,” the refuse of collieries, Ac 
burned on the special grates designed for it by Mr. John E. M ootten. 

On the Pbila & Reading R R, in 1883, the cost of fuel per freight train 
mile was, for anthracite, T2.4 cts; for “ culm,” 2.3 cts. The fast passenger engmei 
ran 55 miles in 76 minutes, with 15 full passenger cars, consuming 53 lbs culm pel 
minute = 73 lbs per mile, aud evaporating 55 gallons of water per minute = 16 gal 

Ions per mile. . _ _ _ 

The average consumption of bituminous coal by passengei 

engines, on 9 roads, in 18S2, was 52.41 lbs per mile run ; greatest, 6. , least, 34. 

Wood fuel. A ton (2240 lbs) of good anthracite or bituminous coal is ubou 
equal to 1U cords of good dry, hard, mixed woods (chiefly white oak); or to 2 cordi 
of such soft ones as hemlock, white, and common yellow pine. Much of the ltiferio 
bituminous coal ot Illinois is hardly equal (per ton) to a cord of average wood. . 

A cord is 4 X 4 X 8 ft, or 128 cub ft. A cord of good dry, white oak, (next to luck 
ory, the best wood for fuel,) weighs 3500 lbs or 1.563 tons. Dry hemlock, white, o 
common yellow pine, (all of them inferior for fuel,) about .9 toD. Perlectly gieei 
w.xids generally weigh about g to ^ more than when partially dried for locomotiv 
use; in other words, a cord of wood, in its partial drying, loses from to % ton o 
water, and still contains a large quantity of it. Since this water causes a grea 
waste of heat, green wood should never be used as fuel. The values of woods as fut 
are in nearly the same proportion as tlieir weights per cord when perfectly dry. 

The run of freight ami passenger trains throughout the 1 
States is 20 to 40 miles per cord, and 30 to 60 miles per ton ; or say 40 to 80 pound 
of coal per mile; but very heavy freight trains will burn from 100 to 200 pound 
or more of coal per mile ; = 22.4 to 11.2 miles per ton. Much depends upo 
the adaptation of the engine to the kind of fuel used. A good coal-burner may be ba 
for wood,and vice versa; so that trials with the same engine may give very erroneov 
results as to the comparative merits of the two kinds of fuel. When wood is used, abo 
.2 cord; or when coal, about % cord ot wood, must be used for kindling, and gettii , 
up steam ready for running; and this item is the same for a long run as for a shoi j~ 
one; so that long roads have in this respect an advantage over short ones, in ecoil 
omy of fuel. Wood has the disadvantage of emitting more sparks ; and is, moreove ( 
nearly twice as heavy as coal, for the performance of equal duty; and is, therefor 
more expensive to handle. It also occupies 4 or 5 times as much space as coal. 

Up grades greatly increase the consumption of fuel. Thus, on a road 95 mib 
long, with grades mostly of less than 6 ft per mile, and with very few exceeding : 
ft per mile, with coal trains of 734 tons descending, and 291 tons (empty) ascendin 
at about 10 miles per hour each way, the coal consumption per 100 miles for eat 
ton of total train (including engine aud tender) wus 14.5 lbs descending, and 36.6 11 
ascending. 

On first-class roads a passenger engine will average about 3500 
miles per year, or say 100 miles per day; a freight engine 25000 miles per ye; 
or say 70 miles per day. 

Locomotive expenses per 100 miles run, will average about as follows: 


Fuel.... 

Water. 

Oil, waste, &c. 


Putting away, cleaning, and getting out 
Locomotive superintendence. 


Passenger. 

Freight. 

3.00 

6.00 

1.00 

2.00 

.70 

.90 

4.00 

8.00 

5.00 

6 00 

1.50 

2.00 

.30 

.50 

$15.50 

$25.40 










































CARS. 


811 


Tins is all that is usually stated in annual reports of expenditures; but inasmuch 
as an engine in active service, even under a judicious system of repairs, generally 
becomes worthless, (except as old iron,) in say 16 years on an average, an additional 
allowance of about 6 per ct on the first cost, or about $500 to $600 = say $2 per 100 
miles, should be made annually for depreciation of each engine. 

CARS. 

Usual dimensions, weights, and capacities. Approx prices 
in 1886. For 4 ft 8^ in gauge. 


1» 

!C 

Passenger. 

Ilj Parlor. 

Sleeper.... 

Baggage, mail, and 

express.... 

Box and cattle. 

Gondola. 

Platform. 

3oal, 8 wheels. 

“ 4 wheels. 

5ump. 


Length 

of body, 
ft. 


44 to 52* 
50 to 66* 

“ * 

45 to 55* 


22 to 35 
12 to 16 
14 


Width. 

Height 

Weight, 

Nominal ca¬ 
pacity, iu 

passengers, or 

Approx cost. 

above rail. 

euiply. 

1886. 


ft. 

ft. 

lbs. 

lbs. 

$ 


9)4 to 10 

14 

40000 to 50000 

50 to 60 

4000 to 

6000 

9)4 to 11 

44 

50000 to 60000 

30 to 40 

8000 to I 2000 


a 

60000 to 72000 

25 to 50 

10000 to 15000 

9% 

44 

40000 to 50000 


2000 to 

4000 

8 to 9)-£ 

11 to 13t 

20000 to 28000 

40000 to 50000 

400 to 

550 

8 to 9)4 

6 to 7)4t 

16000 to 22000 

U 

300 to 

450 

8)4 to 9 

3)4 to 4)4 1 

14000 to 19000 

30000 to 40000 

*« 


8 to 9 

7 to 8)4J 

18000 to 22000 

35000 to 50000 

400 to 

450 

6 to 8 

6)4 to 7 } 

5600 to 9600 

11000 to 20000 

200 to 

250 

8 

6)4 

9000 

18000 

100 


* Add 6 ft in all for the two platforms. These are usually from 3 ft 6 ins to 4 ft above the rails 
t Add from 1 to 2 ft for projecting brake rod and handle. 

} Add about 1 ft for projecting brake rod and handle. 


On narrow gauge (3 ft and ft) roads, there is but little uniformity in 
;ar building. The dimensions and capacities of the cars are now not much less than 
hose of corresponding cars for standard (4 ft 8^ ins) gauge ; while the weights and 
osts are usually from 15 to 20 per cent less. The following are about the averages : 
Passenger, 45 ft long, ft wide, $3000. Parlor, 45 ft long, 8 x / & ft wide, $9000, 
laggage, Ac, 35 ft long, ft wide, $1800. Freight and coal, 24 to 30 ft long, 7 to 
ft wide, $200 to $400. 

dimensions, Arc, of iron-frame ears, built by the United States Tube 
^tolling Stock Co; office, No 3 Broad St, New York. 


“I 

jrl 

»|. 

e'.tox and cattle. 


a lox and cattle . 
, Jondola. 

*" >! 


H 


’latform. 

!oal, 8 wheels. 


latform. 

loal, 8 wheels. 


Length. 

ft. 

Width. 

ft. 

Depth 

of body, 
ft. 

W 7 elght, 

empty. 

lbs. 

Nominal 

capacity. 

1 bs. 


For 4 ft 8ins gauge. 

33 a 

8)4 

6)4 

21500 

60000 

34 to 34)4 7)4 to 8)4 

1)4 

16000 to 19000 

50000 to 60000 

34 

8 


14000 

60000 

34)4 

8 

2)4 

18900 

60000 


For 3 ft gauge. 


28 

7 

6 

18000 

40000 

44 

44 

2 

10500 

it 

<4 

ti 


9500 

14 

4* 

44 

2 

10400 

ll 


Cost. 


" ts ojca 
£• * c <v 

□ o 
i- a o it 

tz I 

t n 

. 

•*» «S w co 
3 0) u, 

© t- v- *-» cj 
£ ttOQ u 

«<s 


For data respecting cars, Ac. we are indebted to Jackson & Sharp Co and Dure A Co, 'Wilmington, 
)e!; Allison Mtg Co, makers of cars and dealers in wheels and axles, Phila; Harrisburg (Pa) Car 
4fg Co ; Erie (Pa) Car Works, Lint ; Youngstown (O) Car Mfg Co; Michigan Car Co, Detroit; Mid- 
lletown (Pa) Car Works ; J G Brill A Co, Phila.; Gilbert Car Mfg Co, Troy, N Y. 

The average life of a passenger car is about 16 years. Average annual 
repairs, including painting, $300 to $700; for mail and express cars, $150 to $300; 
reight cars, $75 to $150; 4-wneel coal cars, $20 to $30. On the Phila & Reading, in 
883. repairs were as follows: passenger cars,$0,156 per 100 passenger-miles; freight 
:ars. $0,133 per 100 ton-miles; coal cars, $0,078 per 100 ton-miles. 

Allowing 125 lbs per passenger, a full car-load of passengers (50 to 60 in number) 
vould weigh but from 6250 to 7500 lbs, or say 3 tons; while the cars themselves 
veigh say 20 tons, or nearly 7 tons of dead load to 1 paying* ton of 
lasseiigers. But, as a general rule, passenger trains are not more than half- 
llled; thus making the proportion about 13 to 1. The foregoing table shows that 































































812 


CARS. 


when freight cars are loaded to their nominal capacity, there is hut about l-5,f 
ton of dead load per ton of paying’ load ; or, with cars half loaded^ 

From the table, p 814, it will be seen that the average cost, in the United State; 1 
of moving a passenger one mile is 2^ times that of moving a ton ot 4ieight on 
mile, while the receipts per passenger-mile are not quite double those per freigh 

ton-mile. L 

The resistance of cars to motion, on a level track, and with cars an F 
track in fair order, is usually taken at about from 6 to 8 lbs per ton of 2240 lb. 
With everything in perfect order.it may fall as low as 5, or even 4, lbs per toi^ 
On the other hand, if the wheels are not truly round, aud if the journals are no 
well lubricated, it may greatly exceed 10 or 12 lbs. See p 374e. 

The wheels for passenger and freight cars are usually about f 
to 33 ins diam ; aud those of coal cars 26 to 28. The cast-iron wheels made by Messjfl 
A. Whitney & Sons, of Philada, weigh as follow's, per single wheel. Cone, 1 in 32 



Usual Gauge. 

Diameter 

of 

Wheels. 

Single Plate. 

Double Plate. 

4 in 
Tread. 

in 

Tread. 

4 in 
Tread. 

4 % in 
Tread. 


fi>3 

fits 

lbs 

fibs 


260 

270 



22 in. 

24 in. 



(345 

(355 

360) 
370 j 

365 

380 

26 in. 

(360 

(375 

380) 
395 j 

380 

400 

28 in. 

(405 

(440 

425) 
460 ( 

440 

460 

30 in. 

33 in. 

(445 

(485 

(510 

(545 

585 

685 

470) 
510 j 

540 ) 
575) 

640 

480 

(525 
■{ 545 
j 560 

505 

555) 
575 > 
590 j 









Narrow Gauge. 


SiDgle Plate. 


SK in 
Tread. 


fits 


(200 
(250 
245 
(265 
< 320 
(330 
(285 
1 335 
(350 
(320 
1 365 
(400 
(405 
(445 


34* in 
Tread. 


tbs 


205) 
255 j 
255 
275') 
330 y 
340 ) 
3001 
350 y 
360) 
3351 
380 > 
415) 
425 ( 
465 j 


Double Plate. 


314 in 
Tread. 


fibs 


340 

345 


380 

460 


3^ in 1 
Tread t- 

— 1 


tbs 


350 

360 


395 

470 


The weights by other makers do not differ from these materially. 

The diam of car or engine wheels does not include the flanges; but is t 
least diam from tread to tread. 

Price of cast chilled wheels, in Phila, in 1888. about 2 cts per lb. 

Good chilled cast wheels. 33 ins diam, will run about 50.( 

miles; usually about 40,000; rarely 60,000 or much higher. 42-inch wheels (r,u<| 
used) average about one-third higher. On first-class roads about 1 to l l /£ per c( 
of all the car-wheels are cracked or broken annually. 

In the steel-tired “paper car-wheels” of the Allen Paper Car-Wheel <i 
office 240 Broadway, New York, the hub is of cast-iron or steel; the “center,” 
main body of the wheel, of compressed straw-hoard confined between two circu 
plates of rolled iron; and the tire is of rolled steel. The bolts, which confine i 
iron plates to the paper center, pass also through flanges on the outside of the li 
and the inside of the tire. To secure elasticity, the bolt-holes in the flange of i 
tire are slightly elongated, and the circular iron plates are made of a little less di 
than the inside of the tire, so that the plates and tire do not come into contact, 1 
the weight of car and load is transferred from the hubs to the tires through i 
paper centers only. These wheels are now largely used on passenger, parlor, * 
sleeping cars, and on the trucks of locomotives. The principal sizes for 4 ft 8^4 





































































CARS. 




auge are 33 inch and 42 inch ; treads, for either diam, 3% and 4% ins. A 42-inch 
>aper wheel costs, 1888, from $73 to $80, according to width of tread; and 
'eighs 1085 lbs; of which 165 lbs are paper; 550 tbs tire; 160 lbs side-plates; 180 
>s hub; and 30 lbs bolts. The steel tires.on 42-inch wheels run about 100,000 miles 
etween turnings, and about 400,000 miles before having to be abandoned. A new 
re is then placed upon the old center, at a cost of about $55. Allowance must be 
ade for tlie fact that these wheels are generally under sleeping or parlor cars or 
rat-class passenger cars, on through trains which make few stops; and that they 
e therefore subjected to less of the destructive action of the brakes than arecom- 
on wheels. Besides, the great majority of the latter are used under freight cars, 
here they have rough usage, due to the inferior character of the springs on such 
trs, &c. The average cost of turning steel-tired wheels is about $1 to $1.25 per 
mum per pair. Axles ave said to run several times longer with paper wheels than 
ith cast-iron ones. 

Wheels of cast- and of wrought-iron with steel-tires are being largely used ex- 
?rimen tally under high-class cars. They run longer than chilled cast-iron wheels, 
ut are more costly. 

Axles. Standard dimensions adopted by the Master Car Builders’ and 
aster Mechanics’ Associations in 1879: Length, total, 6 ft 11% ins; between 
- ubs, 4 ft 0 % in ; each wheel-seat. 7 ins ; each journal, 7 ins. I>iam, at middle, 
% ins; at hubs, 4% ins; at journals, 3% ins. Weight, finished, 347 lbs per 
xle. The diam at middle was increased to 4% ins by the Master Car Builders’ 
'.ssociation in 1884. .This change of course increased the weight slightly. 

Cost of hammered iron axles, about 2% cts. per ib.; hammered steel axles, about 
cts. per lb. (1888 ) 

For Standaid Railway Time, see p. 396. 


I 






814 


RAILROAD STATISTICS, 


RAILROAD STATISTICS. 


Art. 1. In the following, most of the figures for 1880 are based upon the U. S. 
Census for that year; those for 1884, upon Poor’s Manual. 


IX THE EXITED STATES. 


Plant. 


Miles built in one year . 

(In 1881, 9789; in 1882, 11596.) 

Miles in operation..... 

Gau <r e. Percentage of all, 1880. 

3?t. 5 . 

4 ft 8% ins.66. 

4 ft 9 ins.II. 

5 ft (Southern gauge).11. 

Cost of road, exclusive ot rolling stock, 

per mile, in dollars. 

total, in millions of dollars. 

Rolling stock in operation. 

Number of locomotives.. 

“ passenger cars. 

“ baggage, mail, and express cars. 

“ freight cars. 

“ other “ . 

Cost of rolling stock, 

per mile of road, in dollars. 

total, in millions of dollars. 

Cost of road and equipment, 

per mile, in dollars. 

total, in millions of dollars. 


m , 
§ . 


ufl 


.5 

c 7i 


u 

a 

u 

ii 


Operation. 

For one year. 

Passengers carried one mile, per mile of road.. 

Tons of freight carried one mile, per mile of road. 

dross earnings, 

per mile of road, from passengers, dollars. 

“ “ “ freight. “ . 

“ “ “ mails, &c. 

“ “ total. 

per passen ger-m il e, from passe n gers, 

ton-mile, from freight. 

passenger earnings total earnings. 

freight “ “ 

mail, &c, “ -t- “ . 

gross earnings total investment. 

Expenses. (For details, see Art 3.) 

per mile of road.dollars. 

cost of moving freight, per ton-mile. “ 

(Penna R R, 1883, $.0056.) 
cost of moving passengers, per passenger-mile.. “ 
(Penna R R, 1883, $.0163.) 

expenses -4- gross earnings. 

Xet earnings. 

Net earnings -4- total investment.. 


1880. 1884. 


7174 

87801 


5191 

71403 

12335 

12282 


46800 

4112 


17412 

12330 

4475 

375312 

80138 


4761 

418 


51561 

4530 


65392 

368514 


1641 

4740 

230 

6611 

.0251 

.0129 

.2483 

.7169 

.0348 

.1136 


4019 

.0076 


.0171 


.6078 

.0504 


3977 

125152 


24587 

17993 

5911 

798399 


55330 

6925 


70143 
357367 1 


1653 

4018 

429 

6100 

.0236 

.0112 

.2709 

.6588 

.0703 

.1102 




397C 


.650! 


.038! 













































































RAILROAD STATISTICS. 815 


Art. 2. UNITED STATES BY DIVISIONS, 1884. 



Eastern 

States. 

Middle 

States. 

Southern 

States. 

Western 

States. 

Pacific 

States. 

Total, 
U. S. 

Plant. 







liles in operation. 

6405 

18256 

19826 

72704 

7961 

125152 

ostof road and equipment 







per mile, dollars. 

52166 

92306 

42338 

48418 

68549 

55330 

Operation. 







For one year. 







Lross earnings per mile, $.. 

9142 

12177 

3524 

5199 

4348 

6100 

Expenses per mile, $. 

6564 

7951 

2322 

3339 

2615 

3970 

•xpenses-t-Gross earnings. 

.718 

.653 

.659 

.642 

.601 

.651 


Art. 3. Items of total annual expenses for maintenance and opera- 
on of all the railroads of the United States iu 1880. 



$ per 
mile 
of road. 

per cent 
of total. 

per cent 
of earn¬ 
ings. 

epairs of road-bed and track. 

451 

11.23 

6.82 

enewals of rails (total $17243950). 

“ “ ties (total $10741577). 

197 

4.89 

2.97 

122 

3.04 

1.85 

epairs of bridges. 

102 

2.55 

1.55 

“ “ buildings. 

87 

2.17 

1.32 

“ “ fences, crossings, <li:c. 

17 

.42 

.25 

elegraph expenses... 

41 

1.01 

.62 

152 

3.77 

2.29 

laiutenance of road and real estate. 

1169 

29.08 

17.67 

.epairs, &c, of locomotives. 

249 

6.19 

3.76 

“ “ passenger, baggage, and mail cars. 

120 

2.99 

1.82 

“ “ freight cars. 

257 

6.40 

3.89 

•epairs, &c, of rolling stock. 

626 

15.58 

9.47 

(Including renewals and additions.) 

assenger train expenses... 

137 

3.41 

2.07 


330 

8.21 

4.99 

uel for locomotives.. 

374 

9.31 

5.66 

Tater supplv, oil, and waste. 

70 

1.74 

1.06 

7ages of locomotive runners and firemen. 

310 

7.72 

4.69 

gents and station service and supplies. 

451 

11.23 

6.82 

alaries of officers and clerks. 

139 

3.46 

2.10 

dvertising, insurance, legal expenses, stationery, and 

123 

3.06 

1.87 

images to persons and property. 

40 

.98 

.60 

250 

6.22 

3.78 


2224 

55.34 

33.64 


4019 

100.00 

60.78 






Each of these items is, however, subject to great variation, not only on diff roads, 
ut on the same road, from year to year. A road witn many bridges,deep cuts, high 


























































































RAILROAD STATISTICS. 


816 



smbkts, &c, to keep in repair, will have heavier maintenance of way than one which 
has hut few; and this item may he but small one year, and twice as great the next. 
Fuel may be cheap on one road, and dear on another; thus materially affecting the 
item of motive power. And so with the other items. Sometimes maintenance of 
way exceeds motive power and cars together; at others, conducting transportation 
is fully half the total expense. 

The total annual expenses on railroads in the United States 

usually range between 65 and 130 cents per train mile; that is, per mile actually 
run by trains. Also, between 1 and 2 cents per ton of freight, and per passenger, 
carried one mile. When a road does a very large business, and of such a cliaractei 
that the trains may be heavy, and the cars full, (as in coal-carrying roads,) the ex¬ 
pense per train mile becomes large; but that per ton or passenger small; and vice 
versa, although on coal roads half the train miles are with empty cars. 

Art. 4. (ilroKS annual earning* per mile, per passenger 
mile, and per ton mile, of some of the principal U S rail- 
‘ roads in 1880. 



Length 

miles. 

From 
passrs per 
mile of 
road. 

From 
passrs per 
passr 
mile. 

From frt 
per mile 
of road. 

-- 

From frt 
per ton 
mile. 

Pennsylvania R R. 

1806 

$4700 

$.0242 

$15615 

$.0089 

New York Central & Hudson River... 

994 

6651 

.0200 

21794 

.0086 

lhiltimore & Ohio. 

1487 

1812 

.0206 

10310 

.0089 

Central Pacific..:.. 

2447 

2237 

.0303 

4577 

.0249 

Chicago. Burlington, & Quincy. 

1805 

1532 

.0240 

7202 

.0111 

Philadelphia & Reading. 

780 

3429 

.0201 

17200 

.0161 

Union Pacific. 

1215 

2624 

.0320 

7154 

.0199 

Wabash, St Louis, & Pacific. 

1730 

1220 

.0271 

4382 

.0080 

Atchison, Topeka, & Santa Fe. 

1398 

1144 

.0606 

3974* 

.0209 

Average of United States. 

87801 

1641 

.0251 

4740 

.0129 


Art. 5. Annual earnings and expenses of some of the prin¬ 
cipal railroads of the United States in 1880. 



Length 

miles. 

Gross 
earnings 
per mile 
of road. 

F.xpenses 
per mile 
of road. 

Kxpenset 
t- gross 
earnings 

Pennsylvania R R. 

1806 

$20,315 

$12,267 

.585 

New York Central & Hudson River. 

994 

28,445 

17.969 

.609 

Baltimore & Ohio. 

1487 

12,122 

7,035 

.571 

Central Pacific. 

2447 

6,814 

3,340 

.470 

Chicago, Burlington, & Quincy. 

1805 

8.734 

4,454 

.497 

Philadelphia & Reading. 

780 

20,629 

11,754 

.568 

Union Pacific. 

1215 

9,778 

4,507 

.426 

Wabash, St Louis, & Pacific. 

1730 

5,602 

3,942 

.678 

Atchison. Topeka, & Santa Fe. 

1398 

5.118 

2.408 

.458 

Total, United States. 

87801 

6,611 

4,019 

.608 


The following table of expenses in past years will serve for comparison with tin 
above. 





















































817 


RAILROAD STATISTICS. 


Table of Annual Expenses of some II S Railroads.* 


Names of Companies. 


eliigh Valley, I860. 

“ “ 1862 . 

“ “ 186S and 1869 .about ... 

“ “ 1872 . . 

taltimore <fe Ohio, main stem, 1859 . 

“ “ “ I860. 

“ “ “ 1865. 

“ “ “ 1866.. 

“ “ “ 1872. 

last Tennessee & Georgia, 1872. 

lemphis & Charleston, 1860. 

eorgia Central, 1872. 

enna Central, main line from Phila to Pittsburg, 358 miles, 

1859, exclusive of State tonnage tax. 

1860, “ “ “ “ . 

1861, “ “ “ “ . 

1868, tonnage tax repealed. 

1869, “ “ “ .about... 

1872, “ “ “ . 

hila & Reading, 1859. 

“ “ 1860 . 

“ “ 1868, 365 miles of main road and branches... 

“ “ 1869 .. . 

“ “ 1872 . 

orth Pennsylvania, 1860, 54 miles long. 

“ “ 1862. 

“ “ 1867. 

“ “ 1868. 

“ “ 1872. 

onnecticut; average of all the railroads, 1861. 

lassachusetts; “ “ “ “ 1861.. 


a 

i% 

ii 


a 

u 

a 

it 

tt 


averages of 19 years previous 


.average. 


“ 1867 . 

alena & Chicago, 1859.. 

“ 1860 . 

'hila, Wilmington & Baltimore, main stem, 1859. 

1 “ “ “ “ 1860. . 

‘ “ “ “ “ “ 1861.'.. 

« “ “ “ “ 1867. 

ew York; all the It R in the State, average,! 1859. 

(l u u u u a 1861 

“ “ “ “ “ “ 1867.!!.*!!!!.*.!!!!! 

ew Jersey R R and Transportation, 1861. 

ouisviile & Nashville, 1861. 

hila & West Chester, 1861, 27 miles. 

“ 1862. 

“ 1872.;. 

hila, Germantown & Norristown, 1861, 20 miles.. 

“ “ 1862.. 

“ “ 1867. 

ew York & Erie, 1861. 

“ 1867, with its branches, 784 miles in all. 

ew York Central, 1861. 

1867, with its branches, 696 miles in all.... 

nglish R It, averages for 1856-7-8 . 

•otch “ “ “ “ “ “ . 

jgj U U M (« «( u. 


Per 

Mile of 
Road 


4434 

4254 


2617 

4180 

7848 


32000 


17200 


3213 

3240 

9534 


3781 

3785 

2700 

to 

4300 


3102 

4586 

7100 

7785 

17380 

4964 

5100 

13856 

12213 


Per 

Train 

Mile. 

cts. 


103 

178 


54 

49 


89 % 

105 

87 


91 

144 


76 

68 


95 

85 

70 

to 

110 


2274 

2282 

7030 

6405 

6405 

18208 

6461 

14545 

8360 

15620 


93 

73 

116 

167 


187 

106 


84% 
85 
59 

104 
56 
55% 
128 


177 

170 
66 
56 
52 


Per ct. 
of 

Recpts. 


50 


46 
41 
56 
58 

£8 

46% 

50 

53% 

56 

50 
69 

70% 

63 % 

54% 

51 
71 

61% 

66 % 

45 

45 

58 

57 
55 
57 
60 


71 

60 

55% 
41 
58 
51 
68 
683 
65 
76 

41% 

38 

47 

44 

62 

50 

33 


79% 

$ 

50 

44 

40 


* Annual reports often omit the lengths of the roads and branches; and as these frequently vary 
>m year to vear, it is possible that the table may contain some errors in the first column. 

I 2528 miles in operation. Total exps equalled 1.5fi cts per passenger or ton carried 1 mile. Dead 
iight of cars, equal to 1.19 tons per passeuger; aud to 1.74 tons per ton of freight. 










































































































818 


RAILROAD STATISTICS 


Art. 6. Statistics of several U. S. narrow-gauge railroads for IS 
from Poor’s Manual. 


o 

to 

p 

a 

C 


Bridgton & Saco River, Maine. 2 

Profile & Franconia Notch, N H. 3 

Camden, Gloucester & Mt Ephraim, N J. 3 

Bradford, Bordell & Kinzua, Pa. 3 

Denver & Rio Grande, Col.' 3 


Roiling Stock. 


to 

□ 

V 

1-1 


16 2 
141 3 

6 .... 
qo; & 

1685 239 




T3 ! 
P ; 

eS ' 

« < 
P 


P 

c- 




cc S 


K T3 
bt K 

C P 


0) 

w . 
P T 3 
O CC 

P. o 
X *- 

W <*- 
o 

*3 a 

p *- 

Og. 


2 1 16 $12167 $1112 $ 834 

7 - 6 15430 1346 640 

... 136451 2868 1 2642 

69 14922 1793 1717 

115 71 5676 35000 3519 2573 


I 

•i* 

V. 

1 

•J. 

I 

« 


The weights of the steel rails used on narrow gauge roads vary from 30 to 40 
per yard. 

Art. 7. Miles of railroad in the world at the close of 1883. Amei 

(U S, 125152), 143335. Europe, 114313. Total, 270856. 

In threat Britain, in 1883, there were 18000 miles of railroad. Gross ea 
ings for half year, $10130 per mile. Expenses for half year, $5364 per mile, 
penses -j- Gross earnings = .53. 

... „ . , .. ,. • 





































GLOSSARY OF TERMS, 


819 


GLOSSARY OF TERMS. 


Abacus; the flat square member on top of a column, 
i Absciss or abscissa; any portion of the axis of a curve, from the vertex to any point from which 
' the axis at right angles, aud extends to meet the curve itself; said iiue heiug called an 

, rdinate. An absciss and ordinate together are called co-ordinates 
Acclivity ; an upward slope, or ascent of ground, <fcc. 
i‘ : j Adit; a horizontal passage into a mine, Ac. 

- Adze; a well-known curved cutting instrument, for dressing or chipping horizontal surfaces. 

; 1 Alternating motion ; up and down, or backward and forward, instead of revolving, &c. 

Angle-bead, or plaster bead; a bead uaiied to projecting angles in rooms, to protect the plaster on 
,ieir edges from injury. 1 

•' Angle-block; a triangular block against which the ends of the braces and counter* abut in a Howo 
> I ridge. 

■ Angular velocity. See p 365. 

- 1 Anneal; to toughen some of the metals, glass, Ac, by first heating them, and then causing them ta 
ol very slowly. I his process however lessens the tensile strength. 

Anticlinal axis ; in geology ; a line from which the strata of rocks slope away downward in oppo- 
te directions, like the slates on the roof of a house; the ridge of the roof representing the axis. 
Apex; a point iu either chord of a truss, where two web members meet. 

Apron; a covering of timber, stone, or metal, to protect a surface against the action of water fiow- 

- g over it. Has many other meanings. 

Arbor. See Journal. 


Architrave; that part or an entablature which is next above the columns. Applies also when there 
e no columns. Also, the mouldings around the sides and tops of doors and windows, attached to 
Uther the inner or outer face of the wall. 

Arris; a sharp edge formed by any two surfaces which meet at an angle. The edges of a brick are 
rises. 

Ashler; a facing of cut stone, applied to a hacking of rubble or rough masonry, or brickwork. 
Astragal; a small moulding, about semi-circular or semi-elliptic, and either plain or ornamented by 
.rving. 1 

Axis; an imaginary line passing through a body, which may be supposed to revolve around it: as 
e diarn of a sphere. Any piece that passes through and supports a body which revolves: iu which 
se it is called an axle, or shaft. 

Axle-box. See Journal-box. 

Axletree; au axle which remains fixed while the wheel revolves around it, as in wagons, Ac. 
Azimuth. The azimuth of a body is that arc of the horizon that is included between the meridian 
•cle at the given place, and another great circle passing through the body. 

Backing; the rough masonry of a wall faced with finer work. Earth deposited behind a retaining- 
ill, Ac. 

Balance-beams; the long top beams of lock-gates, by which they are pushed open or shut. 

Balk; a large beam of timber. 

Ballast; broken stone, sand or gravel, Ac, on which railroad cross-ties are laid. 

Ball-cock; a cistern valve at one end of a lever, at the other end of which is a floating ball. The 
11 rises and falls with the water in the cistern; and thus opens or shuts the valve. 

Ball-valve. See Valve. 

Bargehoards; boards nailed against the outer face of a wall, along the slopes of a gable end of a 
use, to hide the rafters, <fcc ; and to make a neat finish. 

Bascule bridge; a hinged lift-bridge furnished with a counterpoise. 

Batter, (sometimes affectedly batir.) or talus ; the sloping backward of a face of masonry. 

Bay; on bridges, Ac, sometimes a panel; sometimes a span. 

Bead; an ornament either composed of a straight cylindrical rod ; or carved or cast in that shape 
any surface- 

Bearing ; the course by a compass. The span or length in the clear between the points of support 
a beam, Ac. The points of support themselves of a beam, shaft, axle, pivot, Ac. 

Bed-mouldings ; ornamental mouldings on the lower face of a projecting cornice, Ac. 

Bed-plate; a large plate of iron laid as a foundation for something to rest on. 

Beetle; a heavy wooden rammer, such as pavers use. 

Bell-crank. See Crank. 

Bench-murk; a level mark cut at the foot of a tree for future reference, as being more permanent 
mi a stake. 

Berm. or herme: a horizontal surface, as if for a path wav, and forming a kind of step along the face 
sloping ground. In canals, the level top of the embankment opposite and corresponding to the 
wpath is called the berm. 

Bessemer steel is formed by forcing air into a mass of melted cast iron ; by which means the excess 
carbon in the iron is separated from it, until only enough remains to constitute cast steel. The 
rbon is chemically united with the steel, but mechanically with the iron. 

Beton; concrete of hydraulic cement, with broken stone and bricks, gravel, Ac. 

Bevel; the slope formed by trimmiug away a sharp edge, as of a board, Ac. Edges of common 
awing rulers and scales are usually bevelled. See 13. p 613. 

Bevel gear; cog-wheels witn teeth so formed that the wheels can work into each other at an angle. 
Bilge; the uearly flat part of the bottom of a ship on each side of the keel. Also, the swelled part 
a barrel, Ac. To bilge is to spring a leak in the bilge, or to be broken there. 

Bitts; the small boring points used with a brace. 

Blast-pipes; in a locomotive; those through which the waste steam passes from the cylinder into 
e smoke-pipe, aud thuscreal.es an artificial draft in the chimney, or smoke-pipe. 

Boasting: dressing stone with a broad chisel called a boaster, and mallet. The boaster gives a 
toother surface after the use of the point, or the narrow chisel called a tool. 

Body ; the thickness of a lubricant or other liquid. Also, the measure of that thickness, expressed 
the number of seconds in which a given quantity of the oil, at a given temperature, flows through 
jiven aperture. 


56 






820 


GLOSSARY OF TERMS, 


Bolster; a timber, or a thick iron plate, placed between the end of abridge and its seat on th' 
abutment. 

Bond; the disposing of the blocks of stone or brickwork so as to form the whole iuto a firm struc 
ture, by a jsdicions overlapping of each other, so as to break joint. Applies also to timber, &c, ii 
various ways. 

Bonnet; a cap over the end of a pipe, Ac. A cast-iron plate bolted down as a covering over ai 
aperture. 

Bore; inner diameter of a hollow cylinder. , - ll 

Borrow-fil ; a pit dug in order to obtain material for an embankment. 

Bose ; an increase of the diameter at any part of a shaft for any purpose. A projection in shape 
of a segment of a sphere, or somewhat so, whether for use or for ornament; often carved, or cast. 

Box-drain; a square or rectangular drain of masonry or timber, under a railroad, Ac. 

Brace ; a kind of curved handle used for boring holes with bitts. The head of the brace remain. 1 
stationary, being pressed against by the body of the person using it, while the other part with thf 
bitt is turned round, by his hand. Also, an inclined beam, bar, or strut, for sustaining compression 

Bracket; a projecting piece of board, iio, frequently triangular, the vertical leg attached to th< 
face of a wall, and the horizontal oue supporting a shelf, Ac. Often made in ornamental shapes foi 
supporting busts, clocks, Ac. Also, the supports for shafting; as pendent, wall, and pedestal brackets 

Brake ; an arrangement for preventing or diminishing motion by means of friction. The frictiot 
is usually applied at the circumference of a revolving wheel, by means of levers. On railroads, th< 
car-brakes-should We worked by steam, as those of Loughridge, Westinghouse, and Creamer. Also 
such a handle as that of a common pump. 

Brass is composed of copper and zinc. 

Brasses; fittings of brass in many plummer-blocks, and in other positions, for diminishing th< 
friction of revolving journals which rest upon them. 

Braze; to unite pieces of iron, copper, or brass, by means of a hard solder, called s; elter solder 
and composed, like brass, of copper and zinc, but in other proportions. 

Break joint; to so overlap pieces that the joints shall not occur at the same place, and thus pro 
duce a bad bond. r 

Breast-summer a beam of wood, iron, or stone, supporting a wall over a door or other opening 
a kind of lintel. 

Breast-wall; one built to prevent the falling of a vertical face cut into the natural soil; in dis 
tinction to a retaining-wall or revetment, which is built to sustain earth deposited behind it. 

Breech ; the hind part of a cannon. &c. 

Bridge, or bridge-piece, or bridge-bar; a narrow strip placed across an opening, for supporting 
something without closing too much of the opening. 

Bronze is composed of copper and tin. 

Bulkhead; on ships, &c, the timber partitions across them. Also, a long face of w harf paralle 
to the stream. 

Buoy; a floating body, fastened by a chain or rope to some sunk body, as a guide for fiuding th< 
latter. Sometimes also used to indicate channels, shoals, rocks, Ac. 

Burnish; to polish bv robbing; chiefly applies to metals. 

Bush; to line a circular hole by a ring of metal, to prevent the hole from wearing larger. Also 
when a piece is cut out, and another piece neatly inserted into the cavity, the last piece is sometime: ' 
said to be bushed in; sometimes it is called a plug. 

Butt-joint; ene in which the ends of the two pieces abut together without overlapping, and an 
joined by one or more separate pieces called covers or welts, which reach across the joint and art 
fastened to both pieces. 

Buttress; a vertical projecting piece of brickwork or masonry, built in front of a wall t( 
strengthen it. 

Caisson; a large wooden box with sides that may be detached and floated away. 

Caliber; the inner diameter, or bore. 

Calipers ; compasses or dividers with curved legs, for measuring outside and inside diameters. 

Calk, or caulk; to till seams or joints with something to prevent leaking. 

Calking iron . a tool for forcing calking into a joint. 

Camb, or cam, or i viper; a piece llxcd upon a revolving shaft in such a manner as to produce an 
alternating or reciprocating motion in something in contact with the cam. An eccentric. 

Camber; a slight upward curve given to a beam or truss, to allow for settling. 

Camel; a kind of barges or hollow floating vessels, which, when filled withWater, are fastened t> 
the sides of a ship; and the water being then pumped out, they rise by their buoyancy ; and lift th' 
shin so that she can float in shallower water. 

Cantilevers; projecting pieces for supporting an upper balcony, Ac. 

Cants, rims, or shroudings ; the pieces forming the ends of the buckets of water-wheels, to preven 
the water from spilling endwise. 

Capstan: a long hollow rope-drnm snrrounding a strong vertical pivot, upon the head of which 
rests, and around which it turns. Its ton is a thick projecting circular piece, having holes around it 
ont»r edge nr circumference, for the insertion of the ends of levers; or capstan-bars. It is a kind oi 
vertical windlass. 

Case harden ; to convert the outer surface of wrought iron into steel, bv heating it while in contac 
W’th charcoal. 

Casemate; in fortification ; the small apartment in which a cannon stands. 

Castors ; rollers usually combined with swivels; as those used under heavy furniture, Ac. 

Causeiray ; a raised footway or roadway. 

Cavetto; a moulding consisting of a receding quadrant of a circle. 

Cementation; the process of converting wrought iron into steel, by heating it in contact with char 
coal. This process produces blisters on the steel bars; hence blister steel. These are removed, an 
the steel compacted, by reheating it, aud then subjecting it to a tilt-hammer. It is then tilted steel 
or shear steel. Or if the blister steel is broken up; remelted; aud theu ruu into ingots or blocks - i 1 
is called cast, or ingot steel; which is harder and closer-grained than tilted steel. It mav be softened 1 
and thus become less brittle, by auuealiug. The ingots may be converted iuto bars by either rollin 
or hammering, the same as shear and blister. 

Center; the supports of au arch while beiug built. 

Center of gravity. See p. 347, Ac. 

Center of gyration. See Radius of Gyration p 440. 





GLOSSARY OF TERMS 


821 


i Center of oscillation, or of vibration. See Rem 2, of Pendulums, p 365. 

Center of percussion, in a moviug body, is that point which would strike an opposing body with 
greater force than any other point would. If the opposing body is immovable, it will receive all the 
orce of a rigid moving body which strikes with its center of percussion. See Pendulum, page 365. 
Cesspool; a shallow well for receiving waste water, filth, &c. 

Chamfer; means much the same as bevel; but applies more especially when two edges are cut away 
o as to form either a chamfer-groove, (see 14, p 613, of Trusses,) or a projecting sharp edge. 

Cheeks; two flat parallel pieces confining something between tnem. See w, at 15, of Figs 21J4i of 
trusses, p 583. 

Chilling chill-hardening, or chill-casting; giving great hardness to the outside of cast-iron, by 
•ouring it into a mould made of iron instead of wood. The iron mould causes the outside or skin of 
he casting to cool very rapidly; and this for some unknown reason increases its hardness. This pro 
1 ess is frequently confounded with case-hardening. 

Chock ; any piece used for filling up a chance hole, or vacancy. 

Chuck; the arrangement attached to the revolving shaft, arbor, or mandril of a lathe, for holding 
he thing to be turned. 

Churn-drill; a long iron bar, with a cutting end of steel; much used in quarrying, and worked by 
aising it and letting it fall. When worked by blows of a hammer or sledge it is called a jumper. 
Cima, or cyma : a moulding nearly in shape of an S. When the upper part is concave, it is called 
cima recta; when convex, a cima reversa. See page 151. 

Clack valve. See Valves. 

Clamp; a piece fastened by tongue and groove, transversely along the end of others, to keep them 
rotn warping. A kind of open collar, which, being closed by a clamp-screw, holds tight what it sur- 
ounds. bee Cramp. 

Clap boards ; short thin boards, shingle-shaped, and used instead of shingles. 

Claw , a split provided at the end of an iron bar, or of a hammer, &c, to take hold of the heads of 
tails or spikes for drawing them out; as in a common claw-hammer. 

Cleat; a piece merely bolted to another to serve as a support for something else ; as at 7, 8, 10, 

iC, p. 613, of Trusses. Often used on shipboard for fastening ropes to, as at 11. Also a piece of 
nard nailed across two or more other boards, for holding them together, as is often done in tempo- 
ary doors. &c. 

Clevis. See Shackle. 

Click. See Ratchet. 

Clip; a fastening like that on the tops of the Y’s of a spirit level; being a kind of half collar opening 
>y a hinge. 

Clutch ; applied to various arrangements at the ends of separate shafts, and which by clutching or 
latching into each other cause both shafts to revolve together. A kind of coupling. 

Cock; a kind of valve for the discharge of liquids, air. steam, <fcc. 

Coefficient; or a Constant of friction, safety, or strength, &c, may usually be taken to be a num¬ 
ber which shows the proportion (or rather the ratio) which friction, safety, tensile strength, &c, bear 
o a certain something else which is not generally expressed at the time, but is well understood. Thus, 

vhen we say that the coeff of friction of one body upon another is y^y, &c, it is understood that the 
’riction is in the proportion of yLth of the pressure which produces it. A coeff of safety of 3, means 
hat the safety has a proportion or ratio of 3 to 1 to the theoretical breaking load. A coeff of 500 lbs, 
ir oT 20 tons, &c, of tensile strength of any material, denotes that said strength is in the proportion 
if 500 lbs, or of 20 tons, &c, to each square inch of transverse section. &c. Same as Modulus. 

Coffer-dam ; an enclosure built in the water, and then pumped dry, so as to permit masonry or 
ither work to be carried on inside of it. 

Cog; the tooth of a cog-wheel. 

Collar; a flat ring surrounding anything closely. 

Collar-beam ; a horizontal timber stretching fiom one to another of two rafters which meet at, top; 
)ut above the main tie-beam. See 21, p 613. 

Concrete; artificial stone formed by mixing broken stone, gravel. &c, with common lime. When 
hydraulic cement is used instead of lime, the mixture is called beton. The terms “ lime concrete •’ 
And ‘‘cement concrete” would be convenient. 

Connecting-rod; a piece which connects a crank with something which moves it, or to which it 
ijivea motion. 

Console; a kind of ornamental bracket, somewhat in shape of an S; much used In cornices, &c, 
Ifur supporting ornamental mouldings above it. 

Coping ; flat plates of stone, iron. &c, placed on the tops of walls exposed to the weather. 

Corbel; a horizontal projecting piece which assists in supporting one resting upon it which protects 
still farther. . . . . . 

Core; anything serving as a mould for anything else to be formed around. A term much used In 

foundries. . , 

Cornice; the ornamental projection at the eaves of a building, or at the top of a pier, or of any other 

structure. , . , , . 

Cutter-bolt, or key-bolt; a bolt which, instead of a screw and nut at one end, has a slot cut through 
it near that eud, for the insertion of a wedge shaped key or cotter, for keeping it in its place. Some¬ 
times the ends of these keys are split, so as to spread open after being iuserted, so as not to be jolted 

out of place. . 

Counterfort; vertical projections of masonry or brickwork built at intervals along the back of a wall 
to strengthen it; and generally of very little use. 

Counter shaft; a secoudary shaft or axle which receives motion from the principal one. 
Countersunk. See Ream. 

Counter-weight; or counter-balance; auv weight used to balance another. 

Couplings; a term of very general application to arrangements for connecting two shafts so that 
they shall revolve together. 

Cover; see “ butt-joiut.” „ , , , , . „ . , 

Cover • in re-rolling iron and steel from piles of small pieces, a large bar or slab, called a cover, of 
the same width and length as the pile, is employed to form the bottom of the pile, and a similar slab 
for the top. The covers serve to hold the pile together; and, after rolling, they form unbroken top 
and bottom surfaces of the finished plate, bar, rail, I beam, &c. 



822 


GLOSSARY OF TERMS, 


Ciab; a short shaft or axle, which serves as a rope-drum In raising weights ; and is revolved either 

■ " iee * s ’ tt or by levers or handspikes, inserted in holes around its circumference like a 

windlass, or capstan, of which it is a variety. It may be either vertical or horizontal, it is often 
■et in a frame, to be carried from place to place. Also the t vhi-le machine is culled a crab. 

Cradle ; applied to various kinds of timber supports, which partly enclose the mass sustained. 

Cramp ; a short bar o( metal, haviug its two euds bent downward at right angles for insertion into 
two adjoiuing pieces of stone, wood, Ac, to hold them together. Much used at the ends of coping-stones. 
Also a similar bent piece, with a set-screw passing through oue of the bent ends, for holding things 
tight between it and the other end. This last is also called a clamp. 

Crane; a hoisting machine consisting of a revolving vertical post or stalk; a projecting jib ; and 
a stay for sustaining the outer end of the jib. The stay may be either a strut or a tie. There are 
also cog-wheels, a rope drum or barrel, with a winch, ropes, pullevs. &c. In a craue the post, jib, 
and stay do not change their relative positions, as they do in a derrick. 

Crank; a. double bend at right angles, somewhat like a Z, at the end of a shaftor axle, and forming 
a kina ot handle by which the axle may be made to revolve. Sometimes, as in common grindstones, 
this crank is lornied ot a separate piece removable at pleasure. That part of this piece which has the 
square opening in it for fitting it to the square end of the axle, is called the c rank-arm; and the other 
part the crank-handle. A bell-crank cousists of 4 bends at right angles at the center of an axle, form¬ 
ing in it a kind of U. A double crank consists of two bell cranks arranged thus, The bend in 

the U forms the crank-wrist. The term bell-crank is applied also to those used in tixing common dwell¬ 
ing house bells: and to larger ones on the same principle. A crank-pin is a pin projecting from a re¬ 
volving wheel, disk, or other body, and serving as a crank-handle. A crank shaft is a shaft which 
has a crank in it. or at its end. A cranked shaft has it in it only. A ship or other vessel is said to 
be crank when its breadth is so small in proportion to its depth as to make it liable to upsel easily ; or 
when the same liability is caused by want of sufficient ballast. 

Crest; that top part of a dam over which the water pours. - 

Cross-cut saw; a large horizontal saw worked by two men, one at each end. 

Cross-head; a piece attached across the end (or near it) of another piece, and at right angles to it, 
so as to form a kind of T or cross. Often seen on piston rods, which they serve to keep in place by 
resting on the slides, or guides. 

Crowbar; a bar of iron used as a lever for various purposes; often pointed at one end. 

Crown, or contrate wheel; a cog-wheel in which the teeth staud not upon its outer circumference as 
usual, but upon the plane of its circle. 

Curb ; a broad flat circular ring of wood, iron, or stone, placed under the bottoms of circular walls, 
as in a well, or shaft, to prevent unequal settlement; or built into the walls ut intervals, for the same 
purpose. Has many other meanings. 

Cut off: an arrangement for cutting off the steam from a cylinder before the piston has made its 
full stroke. Also a channel cut through a narrow neck of land, to straighten the course of a river. 

Cutwater, or starling; the projecting ends of a bridge pier, Ac, usually so shaped as to allow water, 
ice, Ac, to strike them with but little injury. 

Damper; a door or valve to regulate the admission of air to a furnace, stove, Ac. 

Dead load; the cars, engiue, Ac, in a train ; non-paying load. 

Dead-load; in a bridge, the weight of the bridge itself, with flooring, roof, Ac; as distinguished 
from the live load of passiug trains, vehicles, pedestrians, Ac. 

Dead points; those two points in the revolution of a crank, where the crank arm is parallel wish 
the rod which connects it with the moving power; and at which said rod exerts no tendency to turn 
the crank. 

Declination, of the sun, or of a star, is its angle north or south of the earth’s equator at the time 
of observation. 

Declivity; a downward slope or descent of ground. Ac. 

Dentils; blocks constituting ornaments in a cornice; placed at short intervals apart, they resemble 
teeth. When, instead of mere blocks, they are handsomely carved in various shapes, they are called 
modillions. 

Derrick; a kind of crane, differing from common ones, chiefly in the fact that the rope or chain 
which forms the stay may be let out or hauled in at pleasure, thus raising or lowering the inclination 
of a jib; thereby enabling the raised load to be placed vertically at the required spot. This cannot 
be done with a crane, which, therefore, is not as well adapted for laying heavy niasonrv, especially 
at great heights. 

Diaphram; a thin plate or partition placed across a tube or other hollow body. 

Die; that part of a stamp that gives the impression. Dies are also two flat plates of hardened steel, 
en an edge of each of which is hollowed out a semicircular half of a short female screw. When these 
plates are put in contact they form a complete female screw like that in a nut; and being strongly 
held together by an iron boxing called the die-stocks, which have long handles for revolving them.thev 
constitute a mould or cutter for forming threads on a male screw. Also the main body of a pedestal. 

Dip; in geology, either the angle which the slope of a stratum forms with a horizontal; or the 
direction by compass, toward which it slopes. In surveying, the inclination at which an unbalanced 
eompass-needle rests on its pivot after being magnetized. 

Disk; a flat circular piece. 

Dock; an artificial enclosure, either partial or total, in which ships and other vessels are placed 
for being loaded or unloaded, or repaired. The first is a icet dock ; the last a dry one. 

Deg iron; a short bar of iron, forming a kind of cramp, with its ends bent down at right angles 
and pointed, so as to hold together two pieces into which they are driven. Often used for temporary 
purposes. It is also called a dog-iron when only one end is bent down and pointed for driving, the 



GLOSSARY OF TERMS. 


823 


Other end being formed into an eye or a handle by which the piece into which the other end is driven 
may be hauled or towed away. 

Donkey-engine ; a small steam engine attached to a large one, and fed from the same boiler. It is 
used for pumpiug water into the boiler. 

Double crank. See Crank. 

Double keys. See K. of Trusses ; page 613. 

Dovetail; a joint like 20. page 613 ; it is a poor one for timber when there is much strain, 

beiug then apt to draw out more or less. 

Dowel; a straight pin of wood or metal, inserted part way into each of two faces which it unites. 

Draft; the depth to which a floating vessel sinks in the water; in other words the water it draws. 

Draught; a drawiug. A narrow level stripe which a stonecutter first cuts around the edges of a 
rough stone, to guide him in dressing off the face thus enclosed by the draught. 

Draw-plate; a plate of very hard steel, pierced with small circular holes of different diameters, 
through which in succession rods of iron are drawn, and thus lengthened out iuto wire. Sometimes 
the holes are drilled through diamond or ruby, &.C, instead of steel. 

Drift; a horizontal or inclined passage-way, or small tunnel, in mines, &c. To float away with a 
current. Trees, &e, carried along by freshets. 

Drip; a small channel cut under the lower projecting edge of coping, &c, so that rain when it 
reaches that point will drip or fall off, instead of finding its way horizontally beneath to the wall, 
which it would make damp. 

Drop; short pieces of nearly complete cylinders, placed at small distances apart, in a row lik* 
teeth, as an ornament to cornices, &e. 

Drum; a revolving cylinder around which ropes or belts either travel or are wound. 11 hen nar¬ 
row and used with belts they are called pulleys. 

Dry-rot; decay in such portions of the timber of houses, bridges, &c, as are exposed to dampness, 
especially in confined warm situations. The timber in cellars and basement stories is mere liable to 
it than in other parts, owing to the greater dampness absorbed by the brickwork from the ground. 
Contact with lime or mortar hastens dry rot. The ends of girders, joists, &c, resting on damp walls, 
may be partially protected by placing pieces of slate or sheet iron under them. The painting or tar¬ 
ring of unseasoned timber expedites internal dry rot. A thorough soaking of timber in a solution of 
28 grains of quicklime to 1 gallon of water is said to be a preventive of dry-rot; but the best process 
for that purpose, is saturation with creosote or carbolic acid. . _ . 

Dyke; mounds of earth, &c, built to prevent overflow from rivers or the sea. A kind of geological 
irregularity or disturbance, consisting of a stratum of rock injected as it were by voleauie action, be¬ 
tween or across strata of rocks of another kind. A levee. 

Eccentric; a circular plate or pulley, surrounded by a loose ring, and attached to a revolt ing 
shaft, and moving around with it, but not having the same center; fur producing an alternate motion. 
Often used instead of a crank, as they do not weaken the axle by requiring it to be bent, 'lhere are 
many modifications. 

Escarpment; a nearly vertical natural face of rock or soil. 

Escutcheon; the little outside movable plate that protects the keyhole of a lock from dust. 

Eye; a circular hole in a flat bar, &c, for receiving a pin, or for other purposes. 

Eye and strap; a hinge common for outside shutters, <fec, one part consisting of an iron strap one 
end of which is forged into a pin at right angles to it; and the other part, of a spike with an eye, 
through which the pin passes. When the eye is on the strap, and the pin on the spike, it is called a 
hook and strap. Such hinges are sometimes called “ backflaps." 

Eye-bolt; a bolt which has an eye at one end. . ... „ 

Face-wall; one built to sustain a face cut into natural earth, in distinction to a retaining-wall, 
which supports earth deposited behind it. 

Fall; the rope used with pulleys in hoisting. . 

False-works; the scaffold, center, or other temporary supports for a structure while it is being 
built. In very swift streams it is sometimes necessary to sink cribs filled with stone, as a base for 
false-works to foot upon. ....... . 

Fascines: bundles of twigs and small branches, for forming foundations on soft ground. _ 

Fatigue; of materials; the increase of weakness produced by frequent bending; or by sustaining 

heavy loads for a long time. , . , . ._, _. 

Faucet; a short tube for emptying liquids from a cask, &c; the flow is stopped by a spigot. The 

wider end’of a common cast-iron water or gas pipe. , , , . ... . . . 

Feather • a slightly projecting narrow rib lengthwise of a shaft, and which, catching into a corre¬ 
sponding groove in anything that surrounds and slides along the shaft, will hold it fast at any required 
part of the length of the feather. Has other applications. 

Feather-edqe; when one edge of a board, &c, is thinner than the other. . . . . 

Felloe, or felly; the circular rim of a wheel, into which the outer ends of the spokes lit; and which 

Is often surrounded by a tire. ... , . . 

Felt: a kind of coarse fabric or cloth made of fibres of hair, wool, coarse paper, Ac, by pressure, 

aE Fender J a piece^for protecting one thing from being broken or injured by blows from another; 
frequently vertical timbers along the outer faces of wharves, to prevent injury from the rubbing of 

vessels. . ... 

Fender-piles: piles driven to ward off accidental floating bodies. 

Ferrule; a broad metallic ring or thimble put around anything to keep it from splitting or breaking. 
X small sleeve. 

Fillet ■ a plain narrow flat moulding in a cornice, &c. See Platband. 

Fish; to join two beams, &c, by fastening other long pieces to their sides. 

Flags; broad flat stones for paving. 

^Fl'ashingl; 'broad^sti'/ps'^f'sheet'lead, copper, tin, &c, with one edge inserted into the joints of 
brickwork or masonry an inch or two above a roof, &c; and projecting out severa. inches, so as to be 
flattened down close to the roof, to prevent rain from leaking through the jo.nt between the roof and 
the brick chimney, &c, which projects above it. 

Flasks; upper and lower; the two parts of the box which contains the mould into which mel„ed 

Flatting ; causing painting to have a dead or dull, instead of a glossy finish, by using turpentine 
instead of oil in the last coat. 

Fliers ; a straight flight of steps in a stairway. 

Floodgate ; a gate to let off excess of water in floods, or at other times. 






824 


GLOSSARY OF TERMS. 


Flume; a ditch, trough, or other channel of moderate size for conducting water. The ditches o' 
culverts through which surplus water passes from an upper to a lower reach of a canal. 

Flush ; forming an even continuous line or surface. To clean out a liue of pipes, sewers, gutters 
Ac, by letting on a sudden rush of water. The splitting of the edges of sloues under pressure. 

Fluxes; various substances used to prevent the instantaneous formation of rust when welding tw< 
pieces of hot metal together. Such ru.->t would cause a weak weld. Borax is used for wrought iron 
a mixture of borax aud sal ammouiac for steel; chloride of ziuo for zinc, sal ammoniac for coppei 
or brass; tallow or resin for lead. 

Fly-wheel; a heavy revolving wheel for equalizing the motion of machinery. 

Foaming; an undue amount of boiling, caused by grease or dirt in a boiler. 

Follower; any cog-wheel that is driven by another; that other is the leader. See also p 645. 

Forceps; any tools for holding things, as by pincers, or pliers. 

Forcbay, or penstock; the reservoir from which the water passes immediately to a water-wheel. 

Forge; to work wrought iron into shape by lirst sorteuiug it by heat, aud then hammering it int» 
the required form. 

Forge-hammer; a heavy hammer for forging large pieces; and worked by machinery. 

Foxtail; a thin wedge inserted into a slit at the lower end of a pin, so that as the pin is driver 
down, the wedge enters it and causes it to swell, and hold more firmly. 

Frame; to put together pieces of timber or metal so as to form a truss, door, or other structure 
The thing so framed. 

Friction-rollers; hard cylinders placed under a body, that it may be moved more readilv than bv 
sliding. j j 

Friction-ivheels ; wheels so placed that the journals of a shaft may rest upon their rims, aud thus 
be enabled to revolve with diminished friction. See page 374 

Frieze; in architecture, the portion between the architrave and cornice. The term is often applied 
when there is no architrave. 

Fulcrum; the point about which a lever turns. 

Furrings; pieces placed upon others which are too low, merely to bring their upper surfaces up tc 
a required level; as is often done with joists, when one or more are too low ; a kind of chock. 

Fuze, or fuse; to melt. A slow match, which, by burning for some time before the fire reaches the 
powder, gives the men engaged in blasting time to get out of the way of Hying fragments of store. 

Gasket; rope-yarn or hemp, used for stuffing at the joints of water-pipes, Ate. 

Gearing; a train of cog-wheels. Now much supplanted bv belts. 

Gi6; the piece of metal somewhat of this shape, I-I, often used in the same hole with a wedge- 

shaped key for confining pieces together. In common use for fastening the strap to the stub-eud of 
the connecting-rod of an engine. 

Gin ; a revolving vertical axis, usually furnished with a rope-drum, and having one or more long 
arms or levers, by means of which it is worked by horses walking in a circle around it. Used for 
hoisting. Cotton-gin, a machine for separating cotton from its seeds. 

Girder; a beam larger than a common joist, and used for a similar purpose. 

Glacis; in fortification, an easy slope of earth. 

Gland. See Stuffing-box. Also, a kind of coupling for shafts. 

GUe; a, cement for wood, prepared chiefly from the gelatine furnished by boiling the parings of 
nides. Cood glue will hold two pieces of wood together with a force of from 400 to 750 lbs per sq in. 

Governor; two balls so attached to an upright revolving axis as to fly outward by their ceutritbgal 
force, and thus regulate a valve. 6 

Grapnel; a kind of compound hook w ith several curved points, for finding things in deep water 
™ ° f !' etwork of .timbers laid crossing each other at right angles; 'frequently placed 

outlie neaus of piles, for supporting piers of bridges, aud other masonry. See p 641, 

»i=/T°l , V« fn l rn !f 1 two segmental arches or vaults intersecting each other at right angles. 

Also, a kind of pier built from the shore outward, to intercept shingle or gravel 

.hSK <•« >■ »'"*■ • 

Ground-swell; waves which continue after a storm has ceased ; or caused by storms at a distance 

Groat; thin mortar, to be poured into the iuterstices betweeu stones or bricks. 
speeds^* 0 ”*' **** metal j° urna ' s a horizontal shaft, such as that of a water-wheel. For moderate 

Diam, ins V Weight in ttis on one gudgeon 
if of cast-irou J jq 

For wrought-iron, add one-twentieth. 

Gun-metal, or bronze: a compound of copper and tin, sometimes used for cannon, 
of cast iron fit for the same purpose. 

Gussets; plain triangular pieces of plate iron, riveted bv their vertical and horizontal legs to the 
sides, tops, and bottoms or box-girders, tubular bridges, Ac. inside, for strengthening their angles. 

Guys ; ropes or chains used to prevent anything from swinging or moving about. 

Gyrate; to revolve around a central axis, or point. 

Halving; to notch together two timbers which cross each other, so deeply that the joint thickness 
shall equal only that of one whole timber. 

Hammer dress; to dress the face of a stone by slight blows of a hammer with a cutting edge. The 
patent hammer for such purposes has several such edges placed parallel to each other, each of which 
mav be removed and replaced at pleasure. 

Hand lever; in an engine, a lever to be worked bv hand instead of bv steam. 

Handspike; a wooden lever for working a capstan or windlass; or other purposes. 

Hand-wheel; a wheel used instead of a spanner, wrench, winch, or lever of any kind for screwing 
nuts, or for raising weights, or for steering with a rudder, Ac. 

Hangers, or pendent brackets; fixtures projecting below a oeiling, to support the journals of long 
lines of shafting ; and for other purpose. Should be “ self-adjusting." 

Hasp ; a piece of metal with an opening for folding it over a staple. 

Hatchway; a horizontal opening or doorway in a floor, or in the deck of & vessel. 

Haunches ; the parts of an aroh from the keystone to the skewback. 

Head-block ; a block on which a pillow-block rests. 

Header; a stone or brick laid lengthwise at right angles to the face of the masonry. 

Heading ; in tunnelling, a small driftway or passage excavated in advance of the main bodv of the 
tunnel, but forming part of it; for facilitating the work. 3 

Headway; the dear height overhead. Progress. 

Heel-post; that on which a lock gate turns on its pivot. 

Helve ; the handle of an axe. 


Also, a quality , 








GLOSSARY OF TERMS, 


825 


Hinge; those commonly used on the doors of dwellings are called butts, or butt hinge*. (Eye and 
Sirup, .) Rising hinges are such as cause the door to rise a little as it is ojaixted.aud thus cause 
the door to shut itself. 

Hip roof, or hipped roof; one that slopes four ways: thus forming angles called hips. 

Hoarding} a temporary close fence of boards, placed around a work in progress, to exclude 
stragglers. 

Holding-plates, or anchors ; strong broad plates of iron sunk into the ground, and generally sur¬ 
rounded by masonry ; for resisting the pull of the cables of suspension bridges ; and for other simi¬ 
lar purposes. 

Hook and strap. See Eye and strap. 

Horses ; the sloping timbers which carry the steps in a staircase- 

Housings ; in rolling mills, &c, the vertical supports for the boxes in -which the journals revolve. 

Huh, or nave ; the central part of a wheel, through which the axletree passes, and from which 
the spokes radiate. 

Impost; the upper part of a pier from which an arch springs. 

Ingot; a lump of cast metal, generally somewhat wedge-shaped. A pig of cast iron is an ingot. 

Invert ; an inverted arch frequently built under openiugs, in order .to distribute the pressure more 
evenly over the foundation. 

Jack; a raising instrument, consisting of an iron rack, in connection with a short stout timber 
which supports it, aud worked by cog-wheels and a winch. A screw-jack is a large screw working 
in a strong frame, the base of which serves for it to stand on; and which is caused to revolve aud 
rise, carrying the load ou top of it, by turning a nut, or otherwise. 

Jack-rafters, or common rafters; small rafters laid on the purlins of a roof, Tor supporting the 
sbiugling laths, &c. 

Jag-spike ; a spike whose sides are jagged or notched, with the mistaken idea that its holding power 
is thereby much increased. If a spike or bolt is first put iuto its place loosely, aud then has melted 
lead run around it, the jaggiug does assist; but not when it is driven into wood. 

Jambs; the sides of an opening through a wall, &c; as door, window, and fireplace jambs. 

Jamb-linings ; the facing of woodwork with which jambs are covered and hidden. 

Jaw; an opening, often V-shaped, the inuer edges of which are for holding something in place. 

Jettie, or jetty ; a pier, mound, or mole projecting into the water; as a wharf-pier, Ac. 

Jib; the upper projecting member or arm of a crane, supported by the stay. 

Jig-saw; a very narrow thin saw worked vertically by machinery, and used for sawing curved 
ornameuts in boards. 

Joggle; a joint like that at 3 or 4, &c, p 613, of Trusses, for receiving the pressure of a strut at 
righ't'angles or nearly so. Also applied to squared blocks of stone sometimes inserted between 
courses of masonry to prevent sliding, &e. 

Joist ; binding joists are girders for sustaining common joists. The common ones are then called 
bridging joists. Ceding joists are small ones under roof trasses, or under girders, and for sustain¬ 
ing merely the plastered ceiling. 

Journal-box ; a fixture upon which a journal rests and revolves, instead of a plumroer-block. 

Journals ; the cylindrical supporting ends of a horizontal revolving shaft. Their iength is usually 
about 1 to 1}4 times their diam. In lines of shafting 4 diams. To find the dram, see Gudgeon. 

Jumper; a drill used for boring holes in stone by aid of blows of a sledge-hammer. 

"Keepers ; the pieces of metal or wood which keep a sliding bolt in its place, and guide it in sliding. 

Kerf; the opening or narrow slit made in sawing. 

Key-bolt. See Cotter-bolt, 

Keystone; the center stone of an arch. 

Kibble; the backet used for raising earth, stone, &c. from shafts or mines - .. 

King-post, kinq-rod ; the center post, vertical piece, or rod, in a truss; all those<m each side of it 
are queen-posts,"or queen-rods. Frequently called simply kings and queens. 

Knee ; a piece of metal or wood bent at an angle; to sene as a bracket, or as a means of uniting 

two surfaces which form with each other a similar angle. 

Lagging, or sheeting ; a covering of loose plank; as that placed upon centers, and supporting t 
archstones. Also, an outer wooden casing to locomotive boilers andotheis. 

Landing ; the resting-place at the end of a flight of stairs. 

u7ae*-* ifart p^oje&ng^ver like a shelf; a rock so projecting. A narrow strip of board nailed 

•«**** ** in a block 

Lighter ^ a'scow.aft) or otherVessel. used for unloading vessels out from the shore. 

wide span, and supporting heavy bnekwo:rk or t concealed within the thickness of the door, are 

JZdiSSX EWZhSZ-SS^^ewXgainsIthe faceof a door, rim locks., ft must he remem. 

ber rfSrT- atiXf vertMndow. frequently at the tops of roofs of depots, Ac, provided with hor- 
5SW! for various purposes, such as for 

Mallet! the wooden hammer used by stonecutter*. 





826 


GLOSSARY OF TERMS 


Mandrel; an iron rod used as a core around which a flat piece may be bent into a cylindrical shape 
.Also the shaft that carries the chuck of a lathe J ^ 


Manhole; an opening by which a man can enter a boiler, culvert, &c. to clean or repair it. 

Mattock; a kind of pick with broad edges for digging. 

Maul; a heavy wooden hammer. 

Mean, arithmetical: half the sum of two numbers. 

“ , geometrical; the sq rt of the product of two numbers. 

Mean-proportional; the same as the geometrical mean. 

Meridian ; a north and south line. Noon. 

Mitre-joint; a joint formed along the diagonal line where the ends of two pieces are united at an 
angle with each other. 

Mitre-sill; the sill against which the lock gates of a canal shut. 

Modulus : a datum serviug as a means of comparison. Same as constant or coefficient. 

Modulus of elasticity; see p i'Slb. Modulus of Rupture, plfcid. 

Moment; tendency of force acting with leverage. See p 335. 

Moment of inertia. See p 486. 

. Moment of rupture, or of bending; the tendency which any load or force exerts to break or bend a 
body by the aid of leverage. Its amount is fouud in foot-pounds by multiplying the force in tbs, by 
the length of leverage in feet between it and that part of the body upon which the tendency is exerted. 
Moment of stability. See Artfi), of Force iu Rigid Bodies. 

Monkey; the hammer or ram of a pile-driver. 

Monkey-wrench, or serew-wrench; a spanner, the gripping end of which can be adjusted by mean 
of a screw to tit objects of different siaes. 

Moorings: fixtures to which ships, Ac, can make fast. 

Mortise; a hole cut in one piece, for receiving the teuon which projects from another piece. 

Muck: soft surface soil containing much vegetable matter. 

Muntins. or mullions; the vertical pieces which separate the panes in a window-sash. 
Nailing-blocks; blocks of wood inserted in watls of stoue or brick, for nailing washboards, Ac. to. 
Nave; the main body of a building, having connecting wings or aisles on eaeb side of it. The hub 
of a wheel. 

Newel; the open space surrounded by a stairway. 

Newel-post; a vertical post sometimes used for sustaining the outer ends of steps. Also the lar 
baluster often placed at the foot of a stairway. 

Nippers ; pincers. An arrangement of two curved arms for catching hold of anything. 

Normal; perpendicular to According to rule, or to correct principles. 

Nosing ; the slight projection often given to the front edge of the tread ot a step ; usually rounded. 
Nut, or burr ; the short piece with a central female screw, used on the end of a screw-bolt, Ac, f 
keeping it in place. 

Ogee; a moulding in shape of an S. the same as a etma. 

Ordinate ; a line drawn at right angles from the axis of a curve, and extending to the curve. 
Oscillate; to swing backward and forward like a pendulum. 

Out of wind, pronounced wynd; perfectly straight or flat. 

Ovolo; a projecting convex moulding of quarter of a circle; when it is concave it is a eavetto, 
hollow. 

Packing ; the material placed in a stuffing-box, Ac, to prevent leaks. 

Packing-pieces ; short pieces inserted between two others which are to be riveted or bolted together, 
to prevent their coming in contact with each other. 

Pall, or pawl. See Ratchet. 

Parapet: a wall or any kind of fence or railing to prevent persons from falling off. 

Parcel; to wrap canvas or rags round a rope. 

Parge ; to make the inside of a flue smooth by plastering it. 

Patent hammer; a hammer with several parallel sharp edges for dressing stODe. 

Pay. To cover a surface with tar, pitch, Ac. A ship word. 

Pay out. To slacken, or let out rope. 

Pediment; the triangular space in the face of a wall that is included between the two sloping aid 
•f the roof and a line joining the eaves. 

Penstock. See Forebay. 

Pier; the support of two adjacent arches. The wall space between windows, Ac. A structure built 
•ut into the water. 

Pierre perdue; lost stone; random stone, or rough stones thrown into the water, and let find their 
own slope. 

Pilaster; a thin flat projection from the face of a wall, as a kind of ornamental substitute for a 
column. 


Pile-planks; planks driven like piles. 

Pillow-block, or phtmmer.block; a kind of metal chair or support, upon which the journals of hor¬ 
izontal shafts are generally made to rest, and on which they revolve. 

Pinion; n small cog-wheel which gives motion to a larger one. 

Pintle : a vertical projecting pin like that often placed at the tops of crane-posts, and over which 
the holding rings at the tops of the wooden guys &t. Also, such as is used for the hinges of rudders, 
or of window-shutters to turn around. 












GLOSSARY OF TERMS. 


827 


1 Pitch ; the elope ol a roof, Ac. The distance from center to center of the teeth of a cog wheel, or 
he threads of » screw. Boiled tar. Also the dist apart of rivets, &c. 

Pitman; a connecting-rod for transmitting motion from a prime mover to machinery at a distance, 
loved by it. 

Pt-saw ; a large saw worked vertically by two men, one of whom (the pitman; stands in a pit. 
Pivot; the lower end of a vertical revolving shaft, whether a part of the shaft Itself, or attached to 
,. It should be flat; and both it and the step or socket upon which it rests should be of hard steel, 
f a steel pivot has to revolve rapidly and continuously, it is well to proportion its diam, so as not to 
ave to sustain more than '250 lbs per sq inch ; otherwise it will wear quickly. Dust and grit should 
>r the same reason be carefully guarded against. Pivots which revolve but seldom, and slowly, as 
aose of a railroad turntable, may be trusted with half a ton. or even a whole ton per sq inch. As a 
ude rule, cast-iron pivots should not be loaded with more than half as much as steel ones. A steel 
ne may be welded to the foot of its cast iron shaft; or may be inserted part way into it; and the 
'hole strengthened by iron bands shrunk on. 

Planish; to polish metals by rubbing with a hard smooth tool, 
i Plant; the outfit of machinery, &c, necessary for carrying on any kind of work. 

Plaster-head; a small vertical strip of iron or wood nailed along projecting augles in rooms, to 
rotect the plaster at those parts. 

Platband; a plain, Hat, wide, slightly protecting strip, generally for ornament. When narrow, It 
3 called a fillet. 

1 Pliers; a kind of pincers. 

• Plinth ; the square lowest member of the base of a column or pillar, 
h Plug; a piece inserted to stop a hole. Screw-plug, a plug that is screwed into a hole. 

► Plumb; vertical. 

! Plummet, or plumb-bob; a weight at the lower end of a string, for testing verticality. 
j Plunger ; a kind of solid piston, or one without a valve. 

f Point; a kind of pointed chisel for dressing stone. To put a finish to masonry by touching up the 
i >uter mortar joints. To dress stone with a point and mallet. 

Pole-plate; a longitudinal timber resting on the ends of tie-beams of roofs; and for supporting 
be feet of the common or jack rafters, when such are used. 

Port; the opening or passage controlled by a valve. 

* Prime ; to put on the first coat of paint. Priming also is when water passes into a steam cylinder 
along with the steam. . 

Projection. If parallel straight lines be imagined to be drawn in any one given direction, from 
:very point in any surface s. whether flat, curved, or irregular, then if all these lines be supposed to 
ie intersected or'cut by a plane, either at right angles to their direction, or obliquely, the figure 
1 which their cross-section thus made would form upon said plane, is called the projection of the sur¬ 
face s If such lines be supposed to be drawn from a person's face, in a direction in front of him, 
ind to be cut by a plane at right angles to their direction, their projection on the plane would be the 
person's full-face portrait. If the lines he drawn sideways from his face, the projection will be his 
orofile The projection of a globe upon a flat plane, will evidently be a circle if the plaue cuts the 
lines at right angles ; and an ellipse if it cuts them obliquely. Shadows cast by the sun are projec- 

^Piddle; earth well rammed into a trench, Ac, to prevent leaking. A process for converting cast 
Iron into wrought by a puddling furnace. 

' Pug-mill; a mill for tempering clay for bricks or pottery. Ao. 

Pulley ; a oircular hoop which carries a belt in machinery. 

Puppet; in maohinerv, a small short pedestal or stand. Puppet-valve. See Valves. 

Purlins; the horizontal pieces placed on rafters, for supporting the roof covering. 

Pit-logs, or put-locks ; horizontal pieces supporting the floor of a scaffold; one end being inserted 
Into put log holes left for that purpose in the masonry. 



I U 3 Ill *411 »» 4 ’ O . _ - * _ I lUUIUCIib s/l II1V.I vie* 

Radius of gyration. See Center of gyration. Rad of gyr is — / - —— 

Ra bolt See Jag-spike. X weight of body 

Rag-wheel, or Sprocket wheel; one with teeth or pins which catch into the links of a chain. 

^Random stone; riprap, or rough stones thrown promiscuously into the water, to form a founda¬ 



tion, Ao. 








828 


GLOSSARY OF TERMS, 


Re-entering angle ; an angle or corner projecting inward. See Salient, below. «. 
Revetment; steep lacing of stone to tbe sides of a ditch or parapet in fortification. A retaining- 
Ritt: the curved pieces which form the arches of iron or wooden bridges, Ate. Also, those to wh 


the outer planking of a sailing vessel, Ac, are fastened. 

Ridge of a roof; its peak, or the sharp edge aloug its very top. Has various similar applicatioi 

Ridge pole, ridge-piece, or ridge-plate; the highest horizontal timber in a roof, extending trout 
to top ot the several pairs ol rafters of the trusses ; for supporting the heads of the jack-rafters. 

Right and left; a lock which in its proper position suits one Hap of a pair of folding doors, 
not suit if fastened to the other flap ; nor even to the same flap if required to open to the right 
stead of to the left, or vice versa, according to whether it is a right or a left hand lock. And so v 
many other things, as, for instance, certain arrangements for working railway switches, Ac. Hi 
and left boots and shoes are a familiar illustration ; also, right and left screws. Therefore, in oru 
ing several of anything, it is necessary to consider whether they may all be of the same pattern 
whether some must be light-hand, and others left-hand ones. 

Right shore of a river; that which is on the right hand when descending the river. 

Right-solid body ; one which has its axis at right angles to its base \ when not so, it is oblique. 

Ring-bolt; a bolt with an eye and a ring at one end. 

Rip rap. See Random stone. 

Roadstead; anchorage at some distance from shore. 


Rock-shaft; a shaft which only rocks or makes part of a revolution each way, instead of revolvi 
entirely around. 


Rockwork; squared masonry in which the face is left rough to give a rustic appearance. 

Rubble ; masonry of rough, undressed stones. Scabbled rubble has only the roughest irregularit 
knocked oil by a hammer. Ranged rubhle has the stones in each course rudely dressed to ueur 
uniform height. 1 

Randle , or round ; the step of a ladder. 

Rustic; much the same as rockwork. 

Saddle; the rollers and fixtures on top of the piers of a suspension bridge, to accommodate 

The top piece of a stone cornice of a pediment, lias uia 


pansion and contraction of the cables, 
other applications. 

Sag ; to bend downward. 

Salient; projecting outward. Sec Re-entering, above. 

Sandbag ; a bag filled with sand for stopping leaks. 

Scabble , to dress off the rougher projections of stones for rubble masonry, with a stone-axe 
scabbliug hammer. 

Scantling : the depth and breadth of pieces of timber; thus we sav, a scantling of 8 by 10 ins 

Scarf; the uniting of two pieces by a long joint, aided by bolts, Ac. 

Scarp; a steep slope, ft) fortification the inner slope of'a ditch. 

Scotia ; a receding mouldiug consisting of a semi-circle or semi-ellipse, or similar figure. 

Screeds ; long narrow strips of plaster put on horizontally along a wall, and carefully faced out 
wind, to serve as guides for afterward plastering the wide intervals between them. 

Screw-bolt, a bolt with a screw cut on oue end of it 

Screw-jack. See Jack. 

Screw-wrench. See wrench. 

Scribe ; to trim off the edge of a board, Ac, so as to make it fit closely at all points, to an irregu 
surface. Tliejower edges of an open caisson are scribed to fit the irregularities of a rockv river botto 

Scroll; an ornameutal form consisting of volutes or spirals arranged somewhat in the shape of 

Scupper nails ; nails with broad heads for nailing down canvas, Ac. 

Scuppers ; on shipboard, holes for allowing water to flow off from the deck into the sea 

Scuttle ; a small hatchway. To make holes in a vessel to cause sinking. 

Sea-wall; a wall built to prevent encroachment of the sea. 

Secret nailing; so nailing down a floor by nails aloug the edges of the boards, that the nail 
do not show. 




‘ 




Serve ; to wrap twine or yarn, Ac, closely round a rope to keep it from rubbing. 

Set screw, or tightening-screw: a screw for merely pressing one thing tightly against another 
will; such as that which confines the movable leg of a pair of dividers in its socket. 

Shackle, or clevis; a link in a chain shaped like a U, and so arranged that by drawing out a b 
or pin, which fits into two holes at the ends of the U, the chain can he separated at that point 
Shaft; a vertical pit like a well. The body of a column. A large axle 
Shank; the body of a bolt exclusive of its head. The long straight part of many things, as of 
micnor, «i Keyi <sc. 


Shears, or sheers; two tall timbers or poles, with their feet some distance apart, and their t< 
fastened together; and supporting hoisting tackle. 

Sheave; a wheel or round block with a groove around its circumference for guiding a rone 
Sheeting, or sheathing; covering a surface with boards, sheet iron, felt, Ac. 

Shingle; the pebbles on a seashore. 

Tbe wail she 


Shoes; certain fittings at the ends of pieces ; as the pointed iron shoes for piles 
into which the lower euds of iron rafters generally fit, Ac. 

Shore; a prop. 

Shot; the edge of a board is said to he shot when it is planed perfectly straight. 

" Ir ‘ii an iron hoop or band is first heated, and then at once placed upon tbe body whi 
It is intended to surround, it shrink* or contracts as it cools, and therefore clasps the body more firm 1 
This is called shrinking on the hoop. K - 

Shuttle ; a small gate for admitting water to a water-wheel, or out of a canal lock Ae 

; a sh ° rt p; e ° e of - I ; a !, lr : ,ad t™ek. parallel to the main one, to serve as a passing-piace. 

Silt ; soft fine mud deposited by rivers, Arc. v 1 

Siphon culvert; a culvert built' in shape of a U. for carrying a stream under an obstacle, and alio I 

outlie siphonYs no°t involved. 1 * l ° na ‘ Ura ' ‘ eVeL The t * rm 18 impPoper - inaRmuoh as the P™ci, ; 
Skewback ; the inclined stone from which an areh springs 

Skids; vertical fenders, on a ship’s sides. Two parallel t’lmhers for rolling things udod 
Skirting; narrow boards nailed along a wall, as the washboards in dwellings ’ 

Sledge; a heavy hammer. " 

Sleeper; any lower or foundation piece In contact with the ground. 

Sleeve; a hollow cylinder slid over two pieces to hold them together. 













GLOSSARY OF TERMS. 


829 


Slide-bars, or slides; bars for anything to slide aloug; as those for the cross-heads of piston-rods, 

I. Ofteu called guides. ' . . 

Slings ; pieces of rope or chain to be put around stones, &c, for raising them by. 

Slip'; the sliding down of the sides of earth-cuts or banks. A long narrow water space or dock 

,\veeu two wharf-piers. .. . 

Slope-wall; a wall, generally thin and of rubble stoue, used to preserve slopes from the action of 
ter in the banks of canals, rivers, reservoirs, &c ; or from the action of rain. 

Slot; a long narrow hole cut through anything. . 

Sluice; a water-chauuel of wood, masoury, &e; or a mere trench. The now is usually regulated 

a sluice-gate. , , . . , , 

Smoke box; in locomotives, that space in front of the boiler, through which the smoke passes to 

e chimney. 

Snag ; a lug with a hole through it, for a bolt. 

Socket; a cavity made in one piece for receiving a projection from, or the end of, another piece; as 
at into which the movable leg of a pair of dividers fits. 

Soffit; the lower or underneath surface of au arch, cornice, window, or door-opening, sc. 

Solder; a compouud of different metals, which when melted is med for uniting pieces of metal also 
ated. Soft solder is a compound of lead and tin, aud is used for uniting lead or tin. There are 
.rious hard solders, such as spelter solder, composed of copper and zinc, for uniting iron, copper, or 

6’ofe : that lining around a water-wheel which forms the bottoms of the buckets. 

Spandrel; the space, or the masonry, &c, between the back or extrados of an arch and the roadway. 
Spanner; a kiud of wrench, consisting of a handle or lever with a square eye at one end of it; much 
ted for tightening up the nuts upon screw-bolts, &c. The eye fits over or surrounds the nut. 

Spar; a beam; but generally applied to round pieces like masts, &c. 


The smaller end of a common cast-iron water or gas pipe. 


Spelter; zinc. 

Spigot ; the pin or stopper of a faucet. 

Spindle ; a thin delicate shaft or axle. , . . _ ... 

Splay; to widen or flare, like the jambs of a common fireplace, or those of many windows , or like 

le wing-walls of most culverts. 

Splice; to unite two pieces firmly together. 

Sormuer • the lowest stone of an arch. . , ... 

Sprocket wheel, or rag-wheel ; one with teeth or pins which catch in the links of a chain. 
Spur-wheel ;_ a common cog-wheel, in which the teeth radiate from a common cen, like those of a spur. 

f^ture-’Acud° 0 a U square terndnatfon like that upon which a watch-key fits for winding ; or that 
xm which the eye of the handle of a common grindstone fits for turning it, iic. 

Staging ; the temporary flooring of a scaffold, platform, &c. 

Standhig-bolt oTstod-fcoif: °a boltwitb a screw cut upon each end ; one end to be screwed perma- 
ent?y into something, and the other end to hold by means of a nut something else that may be le- 
uired to be removed at times. 

Sefa kinf e o7dou 9 ble pin in shape of a U; its two sharp points are driven into timber, and 
irved part is left projecting, to receive a hoop, pin, or hasp, &c. . 

Stnrlinns • the projecting up and down-stream ends or cutwaters of a bridge pier. 
f<«7Xiously P appTiedfo props, struts, and ties, for staying anything or keeping it in place. 
Stag-bolts; long bolts placed across the inside of a boiler, &c, to give it gieatei strength. 

Steam chest; the iron box in locomotive engines and others, through which the steam is admitted 

’ Steam nTf-'the one which leads steam from a boiler to the steam-chest. 

Step; a cavity in a piece for receiving the pivot of an upright shaft; or the end of any upright piece. 
Stiles • the flat vertical pieces between and at the sides of the panels, in door^. & . , 

IS-' .he .,‘ wHb handle, for turning it. in which the die. Tor the cutting of .crew, are held 

ws*jr»sr.s& t sz 

ther part of the hinge which is attached to the wall. 

IrnSr.-VlS.'orchil of' . wall. A fran.e for fetching an, 

k f ’*Lr a h?mde7on a roof represent inclined strata of rooks, then either the ridge or the eaves 
t^rKMSthe. Will represent their strike. The inclination is 
afled the dip of the strata: and the strike is always at right angles to it by compass. 

th. outer ends of ..op. in .U.r—. I. hide. 

"‘slZT'-ZS? X"”gtoSon*a2,1™ oVS^VrTaso^Wotinc a little bejond the other.! 

md often introduced for ornament. 

Stringer; an y longitudinal timber or beam, &Q. 





830 


GLOSSARY OF TERMS. 




Strut ; a prop. A piece that sustains compression, whether vertical or inclined. 

Strut-tie, or tie-strut; a piece adapted to sustain both tension and compression. v 
Stub-end; a bluut end. 

Stud; a short stout projecting pin. A prop. The vertical pieces in a stud partition. 

Stud-bolt. See Standiug-boit. 

Stuffing box; a small boxing on the end of a steam cylinder, and surrounding the piston-rod 
a collar ; or in other positions where a rod is required to move backward and forward or to revo 
iu an opening through auy kind of partition, without allowing the escape of steam, air’, or water 
as the ease may he. The box is lilled with greased hemp or other packing, which is kept pressed c 
around the moving rod by means of a top-piece or kind of cover called the gland which mav 
acre wed down more or less tightly upon jt at pleasure. The rod passes through the gland also 
Sumpt, or sump; a draining well into which rain or other water may be led by little ditches f 
different parts of a work to which it would do iujurv. 1 

Surbase; the inside horizontal mouldings just under a window-sill. Also those around the tOD 
pedestal, or of wainscoting, &c. ' 

Sivage, or swedge; a kind of hammer, on the face of which is a semi-cvliudrical. or other sha 
groove or indentation ; and which, being held upon a piece of hot iron and struck bv a heavy hamu 
leaves the shape of the indentation upon the iron. " ' 

Sivitcb; the movable tongue or rail by which a train is directed from one track to another 
Sunvels; devices for permitting one piece to turn readily in various directions upon auother wi 
out danger of entanglement or separation. At 13. p 583, of Trusses, is a tightening swivel • 
castors under the legs of heavy furniture are swivelled rollers. 

Synclinal axis; in geology, a valley axis, or one toward which the strata of rocks slope downv 
fr m opposite directions. The line of the gutter in a valley roof may represent such an axis 
t - J- p, ,r , . n, i taI that sha P e ' whether to serve as straps, or for other purposes. So also w 
L s. S s, W s, -ps, &c. See figs to Welded Iron Tubes, p 405. 

Tackle; a combination of ropes and pulleys. 

Talus; the same as batter. 

rmnroTf^Ki I'!- “ P wi t h S1 i? d or earth > & °> the remalt »<ler of the hole In which the powder has 
poured for blasting rock. To compact earth generally, as under cross-ties, &c. 

Tap; a kind of screw made of hard steel, and having a square head which mav be erasned h 
wrench for turning it around, and thus forcing it through a hole arouud the inside ofwhich if cuts 
interior screw. To strike with moderate force. To make an opening in the side of any vessel 
Tappet, a pin or short arm projecting from a revolving shaft; or from an alternating bar and 
tern led to come into contact with, or tap, something at each revolution or stroke. * 

Teeth; or cogs of wheels. 

Temper; to change the hardness of metals by first heating, and then plunging them into water r 

&c. ro mtx mortar, or to prepare clay for bricks, &c. ** b mem into water,c 

Templet; the outline of a moulding or other article, cut out of sheet metal or thin wood to set 
as a pattern for stonecutters, carpenters. &o. or min wood, to set 

" rllJ “ ned * 

"X-p: dr.* 'bbsssr sms 

Thionghstone; a stone that extends entirely through a wall. 

Throw; the radius, or distance to which a crank “thrown out" Us arm 

tt&vsttxsr ■— - VtJsr* Atfs&ss, ss 

Tie ; auy piece that sustains tension or pull. 

Ke-struf ; a pteoe adapted to sustain either tension or compression. 

Jhgbtnmg-ring. See 14. of Figs ‘ilH, of Trusses. V 

Tightning.screw. See Set-sorew. 

Tire; the iron ring placed around the outer ciroumferenoe of the felloe of a wheel 

'*“«* >»•“ .Torre, ponding g„„r„. „ 

T-idling ; dressing stone hv means of a tool and mallet’ tho tool hnlno. „ ... 

or I ro '* mono, «Me. Tooling I, genor.lly dnno In purnliel etripoa noro*. the stone * ° Utt ng 

Tramway; any two smooth parallel tracks upon which wheels without flanges may run. In ra 









GLOSSARY OF TERMS 


831 


nways the rails themselves have flanges; but in wide stone ones for common vehicles, none are 
tired. 

ransom : a beam across the opening for a door, &c. Also, a horizontal piece dividing a high 
dow into two stories, &c, &c. Also, an opening above a door, for ventilation or light. 
read ; the horizontal part of a step. 

readle ; a kind of foot-lever, for turning a lathe, griudstone, &c, by the foot. 
reenail; a long wooden pin. 

rimmer; a short cross-timber framed into two joist3 so as to sustain the ends of intermediate 
ts, to prevent the latter from entering a chimuey-liue, or interfering with a window, &c. 
rip-hammer, or tilt-hammer; a large hammer worked by camb machinery, and used for heavy 
» work, especially for hammering irregular masses into the shape of bars, &c. 
ruck; a kind of small wagon consisting of a platform on two or more low wheels. Also, those 
lies and wheels usually placed under railroad cars and engines, and which, by means of a pintle 
necting the two, allow them to vibrate or move laterally to some extent independently of each other. 
rundle, lantern-wheel , or wallower; used instead of a cog-wheel, and consisting of two parallel 
ular pieces some distance apart, and united by a central axis, and by cylindrical rods placed 
und and parallel to the axis, to serve instead of cogs or teeth. 

'runk; a long wooden boxing forming a water channel. 

'runnions ; cylindrical projections, as at the sides of a cannon, forming as it were an interrupted 
3 or shaft for supporting the cannon on its carriage; and allowing it to revolve vertically through 
ie distauce. 

'umbler; a kind of spring catch, which at the proper moment falls or tumbles into a notch or 
3 prepared for it in a piece; thus holding the piece in position until the tumbler is liliedoutoi th* 
ch. 


« ’umbling-bay; see “ waste-weir.” 

’umbling-shaft; in locomotives, a shaft used in the “ link motion." 

’urnbuckle ; variously applied, as to the ordinary fastenings at the outer face of a wall, for hold 
window-shutters back when opened; also, to the tightening swivel at 13, page 583, of Trusses, &c. 
’nrntable; the well-known arrangement for turning locomotives at rest. See page 791. 

Undermine; to excavate beneath anythiug. 

'nderpin; to add to the height of a wall already constructed, by excavating and building beneath 
| Also, to introduce additional support of any kind beneath anything already completed. 

See Stove-up. 

> 'aloes; various devices for permitting or stopping at pleasure the flow of water, steam, gas, Ac. 
afety vai.ve is one so balanced as to open of itself when the pressure becomes too great, for 
ity. A slide valve is one that slides backward and forward over the opening through which the 
/ takes place. A ball valve, or spherical valve, is a sphere, which in any position tits the open- 
'. When the pressure below it raises it off from its seat, it is prevented from rolling away by 
ms of a kind of open caging which surrounds it. A conical or puppet valve is a horizontal slice 
i cone, which fits into a corresponding conical seat made in the opening. In rising and falling it 
ept in position by a vertical valve-stem or spindle, which passes through its center, and which 
ys through guide-holes in bridge-pieces placed above and below the valve. A trap, clack, flap, 
>oor valve, is a plate with hinges like a door. When two such valves are used, with their hinged 
es adjacent to each other, so that in opening and shutting they flap like the wings of a butterfly, 
y constitute a butterfly valve. A throttle valve is one which when closed forms a partition 
oss a pipe; and opens by partially revolving upon an axis placed along its diameter. A rotary 
,ve works like a common stopcock. A snifting valve is one which lets out steam under water ; and 
jo called from the snifting noise thereby produced. The port valve is the sliding one which ad- 
J.s steam from the steam-chest into the cylinders. A double seat, or double beat valve is a pe- 
■f jar one with two seats, one above the other; and so arranged that the pressure of steam or water 
j .inst it when shut, does not oppose its being opened. A cup valve is in shape of au inverted 
(jindrical cup, with a length somewhat greater than its diameter. Its lower or open edge is ground 
(it the seat over which it rests. As this cup rises and falls, it is kept in place by a cylindrical 
ing closed at top, and having for its sides four or more vertical pieces, against the inner sides of 
ich the sides of the cup play. A check valve is any kind so placed as to check or prevent the 
urn of the fluid after its passage through the valve into the pipe or vessel beyond it. 

7 ault; an arch long in comparison with its span. The space covered by such an arch. 

7 eneer; a very thin sheet of ornamental wood glued over a more common variety. 

Yainscot; a wooden facing to walls in rooms, instead of plaster, or over a facing of plaster; usually 
, more than 3 or 4 feet high above the floor. 

V,Ues ; long longitudinal timbers in the sides of a ship, coffer-dam, caisson, Ac. 

JWalloio; a water-wheel, &c, is said to wallow when it does not revolve evenly on its journals. 

? Wallower. See Trundle. ... „ . 

.. VaU plate , or raising-plate ; a timber laid along the tops of walls for the roof trusses or rafters to 
Hi t on, so as to distribute their weight more equally upon the wall. 

1 Warped; twisted, as a board, or the face of a stone, &c, which is not perfectly flat. To warp; to 
al a vessel ahead by means of an anchor dropped some distance ahead. To flood an extent of 
>und with water for a short time to increase its fertility. . . 

Washboards; boards nailed around the walls of rooms at the floor, so as to prevent injury to the 
.ster when washing the floors. ... , .. . „ 

Washers ; broad pieces of metal surrounding a bolt, and placed between the faces of the timher 
•ough which the bolt passes, and the head and nut of the bolt, so as to distribute the pressure over 
arger surface, and prevent the timber from being crushed when the bolt is tightly screwed up. 
Waste-weir; an overfall provided along a canal, &c, at which the water may discharge itself in 

le of becoming too high by rain, &c. Sometimes called a tumbling-bay. . 

Watch-tackle; ropes running in different directions from a boat, and used in bringing it into a 
lired position. 











GLOSSARY OF TERMS. 


fTnfer-sAecf ; the sloping ground from which rain-water descends into a stream. 

Water-table; a slight projection of the lower masonry or brickwork on the outside of a wall, a 
reaching to a few feet above the ground surface, as a partial protection against rain, or as oruaniet 
jT'ijts ■ the inclined timbers along which a vessel glides when being launched. 

1 ^ther-boards: boards used instead of bricks or masonry for the outsides of a building, or bridp 
kc. I hey are nailed to vertical and inclined Indoor timbers; and may be either vertical or hr 
\ hen nor, they are so placed that the lower edge of one overlaps the upper edge of the one belo 
w hen vert, their edges should be tongued and grooved ; and narrow slips be nailed over the vert join 
to keep out ratn, Ac. 

Weir, or wier; a dam, or an overfall. 

Weld; to join two pieces of metal together by first softening them by heat, and then hammeri 
them in contact with each other. In this operation fluxes are used. 

Welt; see “ butt joint.” 


Wharf; a level space upon which vessels lying along its sides can discharge their cargoes ; or fro 
which they can receive them. 

Wheel-base; the distance from center to center from the extreme front wheels, to the extreme hit 
ones in a locomotive, car, Ac. 

Wicket; a small door or gate made in a larger one; as the shuttle or valve in a lock-gate, for lei 
out the water. 

H mch; a handle bent at right angles, and used for turning an axis ; that of a common grindston 

Hun 4. See Out of wind. 6 

Winders; those steps (often triangular) in a staircase by which we wind, or turn angles. 

Windlass; the wheel and axle, or winch and drum, as often used in common wells. Also, a hor 
lontal shaft on shipboard, by which the auchur is raised; the wiudlass being revolved bv means c 
wooden levers called handspikes. • 5 

. Wing-dani; a projection carried out part way across a shallow stream, so as to force all the w 
to flow deeper through the chanuel thus contracted. 

Wings ; applied in mauy ways to projections. The flanges which radiate out from a gudgeon • an 
by which it is fastened to the shatt. Small buiidiugs projecting from a main one. The wings i 
flaring wing-walls of a culvert or bridge. wings c 

Wing-waUs; the retaining-walls which flare out from the ends of bridges, culverts. Ac. 

Wiper. See Camb. 

Working-beam, or walking-beam; a beam vibrating vertically on a rock-shaft at its center, as see 
in some steam-engines ; one end of it having a connection with the piston-rod ; and the other end wit 
a crank, or with a pump-rod, &c. 

Worm; the so-called endless screw, which by revolving without advancing gives motion to a cos 
wheel (worm-wheel), the teeth of which catch in the thread of the screw. * 

Wrench; a long handle having nt one end an eye or jaw which msv catch hold of anvthing to b 
. OI \ turn . ed f arou nd. as a screw-nut. Ac. When it has a jaw'which by means of a screw i 
adaptable to nuts, Ac, of different sizes, it is a monkey-wrench, or screw-wreech. 







(1 


it: 


«> INDEX. 

1 .- 

tie numbers refer to the pages. In the alphabetical arrangement, minof 
r«ls, as “and,” “between,” “in,” “on,”“ through,” Ac, are neglected. 
, also Glossary, p 819, Ac, and Table of Contents, p xxiii. 


* : I 

Abrasion—Angle. 


A. 

* 

asion 

' cement, 678. 
y streams, 279/, 633, Ac. 

„ orbent bodies, specific gravity of, 
id 11. 

orbents for nitro-glycerine, 662, 664. 

“f sorption by bricks, 671. 

(, jtment, Abutments, 
i f arch, 697. 
atter of, 699. 

jurses in, inclination of, 359, 700. 
f dams, 285. 
mndations for, 633, Ac. 
ne of thrust in, 359, 700. 
lasonry, quantity in, 703. 
iers, 699. 

5 proportion, 697. 

ubble, dimensions, rule, 698. 

hickness of, rule, 697. 

;eleration 310. 

f gravity, 258, 311, 312, 362-364. 
n inclined planes, 363. 
d fumes, effect of, on roofs, 428. 
re. Acres, 

,rea of, 389. 

equired for railroads, 722. 
urveying, 168. 

! tion and re-action, 309. 
hesion 

>f cement, 677. 

>f glue, 466, 824. 

>f locomotives, 374a, 808, 809. 

)f mortar, 670. 

)f nails aud spikes, 425b, 762. 
justment 

)f box sextant, 195. 

)f clinometer, 206. 

jf compass, 195. 

jf error in survey, 168, Ac. 

Df hand level, 205. 

of level, 202. 

of plumb level, 206. 

of slope instrument, 206. 

of theodolite, 193. 

of transit", 191. 

ijutages, flow through, 259. 


Admiralty knot, 387. 

Air, 215. 

buoyancy of, 234. 
compressed, 215, 648, 658. 
breathing, 215, 648, Ac. 
in diving bells, 215. 
in foundations, 647. 
in rock-drills, 658. 
compressors, 658. 
lock, 648. 
in pipes, 297. 
pressure. 215. 

barometer, leveling by, 207. 
of compressed air, 215, 648, Ac. 
slacking, 669. 
in tunnels, 754. 
valves, 297. 

ventilation, quantity required fot 
215. 

vessel, 298. 
weight, 215, 381. 
wind, 216. 

Alcohol, weight, 381. 

Alk>th. 178,179. 

Alligation, 36. 

Altitude. See Height. 

Aluminium, weight, 381. 

Anchorages of suspension bridges, 620 
Aneroid barometer, 207. 

Angle, Angles, 54. 
acute, 54. 
adjacent, 54. 
alternate, 55. 

arc, angle subtended by, 141b.- 
to bisect, 56. 

blocks, of Howe truss, 594. 

in a circle, 56. 

chords subtending, 105. 

complement and supplement of, 56. 

contiguous, 54. 

co-secants of, 59. 

cosines of, 59, 60. 

co-versed sines of, 59. 

defined, 54. 

deflection, 726-731. 

degree, decimals of, mins Ac in, 5T. 

of direction, 615. 

to draw a given angle, 56. 

833 








834 


INDEX. 


Angle, Angles—continued, 
to draw a right angle, 55. 
exterior, interior, Ac, 55. 
of friction,318#,355, 371. 
frog, 781, 785, Ac. 
given, to draw, 56. 
interior, exterior, Ac, 55. 
iron, 441, 442, 525. 
limiting, of resistance, 355. 
of maximum pressure, 687. 
to measure 

with the hand, Ac, 58. 
with the sextant, 114. 
with the tape line, 114. 
with the two-foot rule, Ac, 58. 
minutes and seconds in decimals of a 
degree, 57. 
obtuse, 54. 
opposite, 55. 
in a parallelogram, 57. 
in pipes, 256. 
plates for rail-joints, 764. 
in polygons, 110. 
of reflexion, 255. 
of repose, .353. 

of resistance, limiting, 353, 355. 
right, to draw, 55. 
rule, 2-ft, to measure by, 58. 
secant, 59. 

seconds in decimals of a deg 57. 
in a segment, 56. 
in a semicircle, 56. 
sines. 59. 
table, 60. 

of sliding. 353, 355, 369. 

of slope, 724. 

on sloping ground, 113. 

subtended by arc, 111 6. 

supplement and complement of 56 

switch, 773. 

tangent of, 59, 60. 

tangential, tables of, 726-728. 

in triangles, 110, Ac. 

versed sines of, 59. 

vertical, 55. 

Angular velocity, 365. 

Animal power, 377. 

Anthracite 
heat from, 212. 
for locomotives, 809, Ac. 
weight of, 381, 389. 

Annual earnings, expenses, Ac. See Earn¬ 
ing^, Expenses, Ac. 

Anti-friction rollers, 374e, 792, Ac. 

Antimony 
strength, 464. 
weight, 381. 

Anti-resultant, 322, Ac. 

Apertures 

contiguous, flow through, 261. 
flow through. 257, Ac. 
shape of, effect on flow, 260. 
in thin partition, 260. 

Apothecaries’ weight, 387. 

Application of force, 308, 318 e, Ac. 
direction of, 308, 318 e, Ac. 
line of, 318 g. 
oblique, 318 e, Ac. 


force—continued 
perpendicular, 318 e, Ac. 
point of, 309,318/,Ac. 

Aqueduct, Aqueducts, 
flow in, 268. 

Kutter’s formula,271. 

Pittsburgh, 624. 

Arc, Arcs, 
circular, 141. 

angles subtended by, 14't b. 
center of gravity of, 351 a. 
chords of. 105, 141. 
co-secant, 59. 
cosine, 59. 
table, 60. 

co-versed sine, 59. 
graduated, 189. 
large, to draw, 141 b. 
lengths of, 141. 

ordinates of, 141 a, 726, 761, 786. 
radii of, 141 «. 
rise of, 141 a. 
secant, 59. 
sine. 69. 

table, 60. 
tables of, 143-145. 
tangent, 59. 

table, 60. 
versed sine, 59. 
elliptic, 149. 
circumference of, 149. 
ordinates of, 149. 
table, 150. 

tangent to, to draw, 150. 
parabolic, 152. 
semi-elliptic, 149. 
circumference of, 149. 
ordinates of, 149. 
table, 150. 

Arch, Arches, 693. 
abutments of, 697. 
back of, defined, 693. 
braced, strains in, 592, 698 
brick, 522, 709, 712. 
bridge, 693. 
cast-iron, 599. 
centers for, 711. 
concrete, 681, 695, 696. 
crown of, defined, 693. 
derangement of, 713. 
elliptic, 696, Ac. 

joint in, to draw, 150. 
existing, dimensions of, 695 
extrados, 693. 
face, 693. 
foot, 693. 
intrados, 693. 
keystone 
defined, 693. 

depth of, rules, 693, Ac, 695. 
lever, principle of, applied to, 34L 
line of pressure in, 359, Ac, 700. 
line of resistance in, 359, Ac, 700. 
line of thrust in, 359, Ac, 700. 
pressure in, 694. 
pressure, line of, 359, Ac, 700. 
radius, to find, 694. 
rise of, 693. 


Ang-le—Arch. 

Application of 











INDEX. 

Arch—Beam 


835 


Arch, Arches—continued, 
roof, 600. 
rubble, 681, 696. 
skf'wbtick, 693. 
soffit, 693. 
span,693. 
spring, 693. 
stability of, 359, Ac. 
stones, 693. 
chamfering, 714. 
pressure in, 694. 
pressure of, on centers, 713. 
i voussoir, defined, 693. 

Archimedes screw, 379. 

Area 

of a circle, to find, 123. 

tables, 125-140. 
crippling, of rivet, 471. 
sectional, of flange in beams, 518, 521, 
537. 

sectional, of tunnels, 754. 
sectional, of web, in beams, 518, 521, 
539, 540. 

of surfaces. See the surface in ques¬ 
tion. 

Arithmetic, 33-35, -fee. 

decimal, 34, 35. 

Arithmetical progression, 36. 

Artesian wells, 627. 

Artificial 
horizon, 195. 
islands, 651. 
stones, 466, 678, 681. 

Asc ent. See Grade, Height, Slope, 
effect of, 

on power of horse, 375. 
on power of locomotive, 808. 

Ash wood 

strength, 434 e, 436,463,493. 
weight, 381. 

Ashlar masonry, cost of, 667, 668. 
Asphaltum, weight, 381. 

Atlas powder, 664. 

Atmosphere, 215. See Air. 
buoyancy of, 234. 
in tunnels, 754. 
r ventilation, 

quantity required for, 215. 
weight of, 215, 381. 

Augers for earth and sand, 626. 
Automatic 
frogs, 784, 785. 
switch-stand, 775. 

Average pressure of steam, 809. 
Avoirdupois weight, 387. 

Axis. See the given surface or solid 
of buoyancy,235. 
of equilibrium, 235. 
of flotation, 235. 
neutral, 479, 485, 487. 

to find position of, 487. 
of symmetry, 235. 

Axle 
car. 813. 

driving, of locomotive, 808. 
friction, 374<i, 371c. 

standard dimensions, master mechan¬ 
ics’, 813. 


Back, Backs, 
of arch, defined, 693. 
of retaining wall, 683, Ac. 

Backing of walls, 683. 

Back-stays of suspension bridges, 616, 
<fcc. 

Bag-scoop or spoon, 632. 

Baggage cars, 811. 

Bailing by bucket, day’s work at, 378. 
Baldwin locomotives, 805-810. 

Ballast for railroads, 759. 
cost of, 804. 

Balloon, principle of, 234. 

Balls, weight of, 398, 400, 416. 

Bar, Bars, 
brass, 415. 
copper, 415. 
dredging, 63L 
iron, weight of, 400, 401. 
lead, 415. 
sand, 669. 

Barbed fence, 803. 

Barometer, 
aneroid, 207- 
effect of latitude on, 207. 
leveling by, 207. 
pressure of air, 215. 

Barrel, contents of, 390. 

Barrow. See Wheelbarrow. 

Base, wheel-, of locomotives, &c, 546,805, 
Ac. 

Batter 

of abutments, 699. 
of retaiuing walls, 685, Ac. 

Battered walls, 685, Ac. 

Beam, Beams, 
angle iron, 525. 
box,537. 

breaking loads, constants, 491-493. 

formula, 4SS. 
cast-iron 

modifications in sections of, 518. 
strength of, 519, Ac. 
channel, 521, Ac. 

as pillars, 441, 442, 456. 
closed, 485. 

concrete, strength of, 682. 
constant 

for breaking loads, 491-493. 
for deflection, 506, 507, 509. 
continuous, 515. 
curved beams, 484. 
curved flanges, 530. 
cylindrical, 492, 497,503, 511, 516. 
deck, 521. 

deflections of, 499, 505<x-513. 
constant for, 506-51©. 
dimensions for a given, 508, 510. 
load for a given, 605a, uUH, Ac., 510. 
rules for, 505/1-511. 
under sudden loading, 434/, 499. 
table, 499, 500. 
dimensions of 
to find, 497. 

for a given deflection, 508, 510. 
elastic limit in, 505. 
elasticity, modulus of, 434c. 







836 


INDEX. 

Beam—Bolts. 


V 


Beam, Beams—continued, 
flanges of 
curved, 530. 
oblique. 530. 
strains in, 529, 537. 
granite, 504. 

Jdodgkinson’s, 518. 
hollow, strength of, 516. 

I and channel, 521, Ac. 
brick arches with, 522. 
deflections of, 505 b, 522. 
as pillars, strength of, 441, 442, 454. 
for railroad bridges, 524. 
separators for, 523 b, d. 
steel, 523 c. 
inclined, 340, 496. 
iron 

angle, 525. 

breaking loads, table, 592. 
cast, 518, 519. 

flanges, strains in, 529, 537. 

loads, table of, 502. 

rolled, 521, A<\ See also Beams I, Ac. 

safe loads and deflections, 500. 

and steel, loads, table, 512,513,523, Ac. 

T, 525. 

wrought, 521, Ac. See Beams I. 
irregularly loaded, 498. 
lover, principle of. applied to, 339. 
limit of elasticity in, 505. 
loads on, 488. See also Beams, Strength, 
applied irregularly, 496. 
applied suddenly, 434 f, 499. 
constants for, 49L, 493. 
formula, 488. 

for a given deflection, 505 a, 508, 510. 
within limit of elasticity. 505.' 
longitudinal sections of 495. 
modulus of elasticity, 434 c. 
moments in, 478-489, 528, Ac., 537, Ac. 
©pen, 528. 
plate, 537. 
resistance of, 484. 
riveted, 537. 

rolled. See Beams I, Beams Channel, 
Ac. • 

shearing strains in, 532. 
of solid cross-section, 484. 
to splice, 610. 
square, on edge, 494. 
steel and iron, loads, table, 512, 523, Ac. 
stone, 493, 504. 
strains in flanges of, 529, 537. 
strains in, shearing or vertical, 532. 
strains in, vertical, 532. 
strength of, 478, Ac., 493. See also 
Beams, Loads on. 

affected by methods of supporting 
and loading, 494, 496. 
when loaded irregularly, 496. 
practical methods for finding, 491, 

Ac. 

suddenly loaded, 434,/j 499. 

T-iron, 525. 
with thin webs, 528. 
tie-, 551, Ac. 

timber. See Beams, Wooden, 
tubular, strength of, 516. 


Beam, Beams—continued, 
vertical strains in, 532. 
wooden 

deflections, table, 499. 
loads, tables. 499, 502, 512, 513. 
for railroad bridges, 514. 

Bearing piles, 641. 

Bearing power of soils, 634, 644. 
Bearing .and reverse bearing, 171. 

Beech-wood, strength, 434e, 436,463 193. 
Bell 

diving, pressure in, 215, 648, Ac. 
joint for pipes, 295. 

Belts, leather, strength of, 466. 

Bending 

of beams, rules for, 505-513. Se 
Beams, Deflections of. 
rails, ordinates for, 761. 

Bends in pipes, 255. 

Bents in trestles, 756. 

Bessemer steel ties, 760. 

Beton concrete, 678, 681. 

Beveled joints for rails, 763. 

Birch, strength, 434e,436, 463, 493. 
Birmingham gauges, 410, 411. 

Bismuth, 
strength, 464. 
weight, 381. 

Bitumen, weight, 381. 

Bituminous coal, 

for locomotives, 809, Ac. 
weight of, 381, 389. 

Blake stone-crusher, 680. 

Blasting, 651-666, 754. 

Blocks, concrete, 679, 681. 

Board measure, table, 420. 

Boat, canal, 
cost of, 376, 
traction of, 376. 

Bodies, Body. See the body in questio 
absorbent, specific gravity of, 381, 38 
center of gravity, 348, Ac. 
defined, 306. 

expansion of, by heat, 212. 
falling, 258, 362. 
floating. 234, Ac. 
mass of, 312. 

regular, volumes, Ac, of, 154. 
rigid, force in, 306. 
rotating, 365. 
specific gravity of, 380. 
strength of, 434-525. 
vibrating, 364. 
weight of, 380. 

Boiler, 

incrustation of, top of 218. 
iron, 402, 464. 
pressure, 809, Ac. 
thickness for, 233. 
tubes, 405. 

Boiling-point, 213, 217. 

leveling by, 209. 

Bollman truss, 586, 603. 

Bolster plates, 514, 524, 544. 

Bolts, 406. 

copper, strength of, 408. 
expansion, 407. 
iron, table, 408, 409. 
























INDEX 


837 


Boltless 

Boltless rail-joint, 767. 
luring 

Artesian wells, 627. 
augers for earth, 626, 627. 
test, 626, 633. 
wells, 626. 

Bor row-pits, to measure, 155,156. 
Bottom, 
heading, 754. 

of stream, scouring action on, 270/. 
velocity, 268. 

Bowstring 
; centers, 717. 

truss, strains in, 588, 597. 

Box 

beams, 537. 
cars, 811. 

center, of turntable, 792. 
drains, 707. 
girders, 537. 
sextant, 194. 

3raced arch, strains in, 592, 598. 

3 racing, 

counter, 549, 564, 714. 
lateral, 542, 610. 
sway, 543. 

Brad spikes, 762. 

Brake friction, 374, Ac. 

Branches in pipes, 296. 

3randt drill, 652. 

3rass 

balls, weight of, 416. 
bars, 415. 

compressibility of, 434 e. 
ductility of, 434 e. 

effect of cement, mortar, Ac, on, 670, 
673. 

effect of water on, 218. 
elastic limit of, 434 e, 
expansion by heat, 212. 
friction of, 373. 
modulus of elasticity, 434 e. 
pipes and tubes, seamless, 417. 
sheets, 415. 

strength of, 438, 464, 493. 

stretch of, 434 e. 

tubes, seamless, 417. 

weight of, 381, 398-400, 401, 410, 415. 

wire, 410, 412. 

Breaker, stone, 680. 

Breaking. See also Strength of Mate¬ 
rials, Ac. 

loads for beams, constants for, 491. See 
Beams. 

moment, 479-4S4. 

Breathing, 

air consumed in, 215. 
in diving-bells, 215, 648. 

Brick, Bricks, 670, 671. 
absorption of water by, 671. 
adhesion to cement, 677. 
adhesion to mortar, 670. 
arches, 522. 709, 712 
buildings, cost of, 668. 
cylinders, sinking of, 650. 
dust, 669. 
friction of, 373. 
laying, 671. 


-Brokerage. 

Brick, Bricks—continued, 
number of, in a sq ft of wall, 671. 
strength of, 437, 466, 493. 
weight of, 381, Ac. 
work, 671, Ac. 
cylinders, sinking of, 650. 
incrustations ou, 673, 678. 
rod of, English, 672. 
strength of, compressive, 437. 
water, to render impervious, 672. 
weight of (under Masonry), 383. 
Bridge, Bridges. See also Arch, Beam, 
Girder, Trestle, Truss, Ac. 
arch, 693. 
cast-iron, 599. 
centers for, 711. 
brick, 709, 712. 

centers for, 711. 

Brooklyn, foundations, 649. 
camber of, 607. 
cast-iron, 599. 

Chestnut St, Phila, 599. 

East River, foundations, 649. 
factor of safety, 607. 
false works, 608. 
floor girders for, 610. 
girder, 537-546. 
iron, cast, 599. 

iron, wrought. See Trusses, Ac. 

joints, 470, 611, 613. 

loads on, greatest probable, 606, 623. 

loads on. moving, 546, 564, 805, Ac. 

plate girders, 537-546. 

repairs, aunual, 815. 

riveted girders, 537-546. 

roadways, drainage of, 708. 

rollers, expansion, 614. 

safety, factor of, 607. 

Severn Valley, 599. 
stone, 693. See also Arch. 

centers for, 711. 
suspension, 615-625. 

cables of, 412, 413, 615, Ac. 
suspension links, 614. 
swing, 593. 

friction rollers for, 798. 
truss, 547. See Truss. 

Whipple, 599. 
widths of, 542, 609. 

Wissahickon, Phila, 720. 
wooden, 514. See also Truss, Trestle, 
Ac. 

British 

Imperial measure, 391. 
measures, to reduce U S measures to, 
anti vice versa, 390. 
railroads, miles, Ac, 818. 
rod of brickwork, 389, 672. 

Broach channeling, 658. 

Broken 

bubble-tube, to replace, 193. 
cross-hairs, to replace, 193. 
joints, 763. 
pipes, 296. 

stone (see also Rubble), 
foundations, 634. 
voids in, 380, 678, 751. 

Brokerage, 37. 










838 


INDEX. 

Broil ze—Chord 


V 


Bronze 
weight, 381. 

phosphor, wire, strength, 464. 
Brooklyn bridge, foundations, 649. 
Brunlee’s iron piles, 647. 

Bubble-tube, to replace, 193. 

Buckled plates, 409. 

Builder’s level, to adjust, 206. 
Building, Buildings, 
cost of, per cubic foot, 668. 
repairs, railroad, 815. 

Buoyancy 
of air, 234. 
of liquids, 234-236. 

Burleigh rock-drill, 657. 

Burnettizing, 425a. 

Burnham’s frost-proof tank, 801. 

Burr truss, 601. 

Bursting 

of pipes, 234, 298, 303. 
pressure in pipes, 239. 

Bushel, volume of, 390. 

Butt-joint, 469. 

Buttresses, 692. 

c. 

Cable, Cables, 
number of wires in, 412. 
stays, 616. 

of suspension bridges, 
dimensions of, 616. 
strains in, 616. 

Caisson, 636. 

Brooklyn bridge, 649. 

Calking, 295, &c. 

Camber of trusses, 607. 

Cambria nut-lock, 765. 

Canal, Canals, 
boats, cost of, 376. 
boats, traction of, 376. 
flow in, 268, &c. 

Rutter’s formula, 244, 271. 
leakage from, 222, 269. 
traction on, 375. 

Cantilevers, 479-482, 494-496, 535, 593. 
Caps, blasting, 665. 

Car, Cars, 811, 812. 
axles of, 812. 
derrick, 750. 
earthwork, 749. 
friction of, 374e, 808, 812. 
pile-driver, 642. 
repair, cost of, 815. 
resistance of, 374e, 808, 812. 
wheels, 765, 812. 
wrecking, 750. 

Carat, 385. 

Cart, Carts. 

earthwork, 742, <fcc, 747. 
excavating (wheeled scrapers), 747. 
road, repairs of, 743. 
rock, removing, 752. 
traction, 375. 

Cartridge, dynamite, 663. 

rack-a-rock, 664. 

Cast-iron. See Iron, Cast. 

Castelli’s quadrant, 269. 


Casting, 
rough, 674. 
safety, 770, 779. 

weight of, by size of pattern, 398. 

Cattle-cars, 811. 

Cedar, strength, 436, 463, 493. 

Ceiling, weight of, 553. 

Cellar walls, cost of, 668. 

Cement, 673. 
abrasion of, 678. 
brick-dust, 669. 
concrete, 678. 
expansion of, 678. 
and iron, pipes of, 294. 
for leaks, 429, 431. 
moisture, effect on, 673. 
mortar, 676. 
setting of, 674. 
in stone bridges, 696. 
strength of, 437, 466, 493. 
weight of, 382, 674. 

Center, Centers, 
for arches, 711. 
of buoyancy,235. 
of circle, to find, 123. 
of force, 347 1. 
of gravity, 348. 
of gyration, 440. 
of oscillation, 365. 
of percussion, 365. 
of pressure, 227, 235, 347 1. 700. 

Centigrade thermometer, 213. 

Centimetre, length of, 392. 
cubic, weights per, 381. 

Central forces, 368. 

Centrifugal force, 368. 

Centripetal force, 368. 

Chain, Chains, 

Gunter’s, 176. 

iron, 414. 

pump, 379. 

riveting, 470. 

surveying, 168, 176. 

of suspension bridges, 615. 

Chaining, slope, allowance for, 113, 176. 

Chair, railroad, 767. 

Chalk, 

strength, 437. 
weight, 381. 

Chamfering arch-stones, 714. 

Channel, Channels, 
flow in, 268. 

Rutter’s formula, 244, 271. 
iron, 521. 

as pillars, 441, 442, 456. 

Channeling in rock, 658. 

Charcoal, weight, 381. 

Cherry-wood, weight, 381. 

Chestnut St bridge, Pliila, 599, 636. 

Chestnut-wood, 
strength, 434 e, 436, 463, 493. 
weight, 382. 

Chord, Chords, 
of arcs, to find, 141. 
in circles, 124, 141. 
long, table, 729. 

natural (to radius 1), table, 105. 
of trusses, 550, 588, 610, 612, &c. 







839 


INDEX. 

Churn-drilling—Conical. 


Churn-drilling, 651. 

Circle, Circles, 123. See also Circular, 
angles in, 56. 
areas of, to find, 123-140. 
center of, to find, 123. 
center of gravity of, 348. 
chords in, 124, 141. 
circumference, to find. 123, 133, 140. 

tables of, 125-140. 
diameter, to find, 123, 133,140. 
to draw, 123. 
great, earth’s, 144. 
mensuration of, 123. 
radius of, 123. 141 o. 
tables, 125-140. 
tangents to, to draw, 124. 

Circular 

arc, 141. See also Arc. 

tables of, 143-145. 
curves for railroads, 726. 
inch, 389. 
lune, 146. 
motion, 365. 
ordinates, 141 a. 

tables, 726, 728, 761, 786. 
rings, 146, 167. 
sector, 146. 

center of gravity of, 351c. 
segment, 146. 

center of gravity of, 351 d. 
table of, 147. 

; spindle, 167. 
zone, 146. 

Circumference 

of circle, to find, 123, 133, 140. 

tables, 125-140. 
of ellipse, 149. 

Cisterns, 233, 800-803. 

City 

water-supply, 287. 

Civil time, 395. 

Clamp, Clamps, 

pouring, for pipe-joints, 294, 295. 
rod, switch, 771. 

Clay, 

effect on mortar, 670. 
in foundations, 634. 
loosening of, 742. 
swelling of, by absorption, 634. 
Clearing, cost of, 804. 

Clinometer, 206,724. 

Clock, 

to regulate by star, 395. 
time, 395. 

Close piles, 641. 

Cloth, tracing, 433. 

Coal 

cars, 811. 

consumption of, by steam-engines, 

805, etc. 

corrosive fumes from, 403. 418. 
for locomotives, 805-810. 
oil, weight (Petroleum), 383. 
ton of, volume of, 389. 
weight of, 381, 389. 

Cocks, corporation, 299. 

Coefficient, Coefficients. See Strength, &c. 
for beams, 491. 


Coefficient, Coefficients—continued, 
of contraction, 261. 
for deflection, 506, 507, 509. 
of friction, 371. 
for loads on beams, 491. 
for loads within elastic limit, 505. 
of resistance, 485. 
of roughness, 244, 272, 273. 
of rupture, 485. 

of safety. See Safety, Factor of. 
of torsion, 476. 
for transverse strength, 491. 
Coffer-dam, 636, 637, &c. 

Cog-wheels, 342. 

Cohesion, 463. 

Cohesive strength, 463. 

Coignet beton. 681. 

Coin, Coins, 386, 387. 

Coke, weight, 381. 

Cold, 

effect on cement, 675. 
on explosives, 661, 663. 
on iron, 466 763. 
on mortar, 672. 
on trusses, 614. 

Collision, 318 e. 

Color, Colors, 

of cement, 674, 678. 
draughtsmen’s, 433. 

Columns (pillars). See Pillars. 

water, 801. 

Combination, 36. 

Commercial 

measures, size of, by weight of water, 
391 

weight, 387. 

Commission, 37. 

Compass, 

to adjust, 195. 
variation of, 196,197. 

Compensating reservoir, 290. 
Compensation water, 290. 

Complement and supplement, 56. 
Component, 319, Ac. 

Composition of forces, 319, Ac. 
Compound 
interest, 37. 
levers, 342. 
proportion, 35. 

Compressed 

air, 215, 648, Ac, 658. 
gun-cotton, 664. 

Compressibility, 
of air, 215. 
of liquids, 236. 

Compressive strength, 436. Ac. 

Concrete, 678. 

beams of, strength of, 682. 
beton Coignet, 681. 
strength of, compressive, 437. 
under water, 680. 
weight, 681. 

Concretions in pipes, to prevent, 292. 
Cone, Cones, 160. 

center of gravity of, 351 e, g. 
frustum, 160 a. 

center of gravity of, 351 e, g. 

Conical rollers, 792. 









840 


INDEX. 

Conoid—Curved. 


Conoid, 
parabolic, 167. 
frustum of, 167. 

Consolidation locomotives, 546, 805-810. 
Constants. See Coefficieuts. 
Construction, 
railroad, 722. 
cost of, 804. 

Consumption 

of coal by steam-engines, 805, etc. 
of fuel, effect of grades on, 810. 
by locomotives, 808, Ac. 

Contiguous openings, flow through, 261. 
Continuous 
beams, 515. 
girders, 515. 

Contour lines, 197. 

Contracted vein, 258, 260. 

Contraction, 
coefficients of, 261. 
by cold, 212, 763 

and expansion in trusses, allowance 
for, 614. 

incomplete, 259, Ac, 263, Ac. 

on weirs, 263, Ac. 
of outflow, 258, 260. 
of rails, 763. 
of water-way, 703. 

Contractor’s profit, 743. 

Contrary flexure, point of, 515. 

Converse pipe-joint, 293. 

Copper 
bars, 415. 

balls, weight of, 416. 
compressibility of, 434e. 
cost of, 415-417. 
ductility of, 434 e. 

effect of cement, mortar, Ac, on, 670, 
673. ’ ’ 

of water on, 218. 
elastic limit of, 434 e. 
expansion by beat, 212. 
modulus of elasticity, 434 a. 
pipes, seamless, 417. 
roofs, 416. 
sheets, 415, 416. 

strength of, 408, 438, 464, 476, 477. 
stretch of, 434 e. 
tubes, seamless, 417. 
weight of. 382, 398, 399, 400, 401, 410, 
415,416. 

Cord (funicular machine), 325, 344. 

of wood, volume of, 389. 

Cork, weight, 382. 

Corporation cocks or stops, 299. 
Corrosion, 
by acid fumes, 428. 
by coal fumes, 403, 418. 
of timber, prevention, 425. 
by water, 218, 645. 

Corrugated sheet-iron, 403. 

Co-secants, 59. 

Cosines, 59, 60. 

Cost of articles. See article in question. 
Co-tangent, 59. 

Counter-bracing, 549, 564. 

of centers, 714. 

Counter-forts, 692. 


Counter-scarp revetment, 692. 
Counter-sloping revetment, 692. 

Couples,347 d. 

Couplings for pipes and tubes, 294, Ac 
405. 

Courses of masonry, inclination of, 683 
700. 

Cover in a butt-joint, 469. 

Cover-plate, 545, Ac. 

Co-versed sines, 59. 

Cracks in pipes, 296. 

Creeping of rails, 763, 764. 

Creosote, 425, 759. 

Crescent truss, strains in, 588. 

Crib, Cribs, 635. 
coffer-dam, 638. 
dams, cost of. 285. 
foundations, 635. 

Crippling, 
of beams, 478, Ac. 
of riveted joints, 471, Ac, 539. 
Cross-girts, turntables, 792. 

Cross-hairs, to replace, 193. 

Cross-section paper, 433. 

Cross-ties, 759. 
cost of, 804, 815. 

Crossings, railroad, repair, annual, 815. 
Crowds, weight of, 606. 623. 

Crown 

of arch, defined, 693. 

(coin) value of, 386'. 

Cruinlin viaduct, 756. 

Crusher, stone, 680. 

Crushing 
loads, 436, Ac. 
of stone, 680. 

Cube, Cubes, 41,154, 155. 
center of gravity of, 348, Ac. 
roots, 40. 

of decimals, to find, 53. 
of large numbers, to find, 52. 
tables, 40. 
tables, 41. 

Cubic 

centimetre, weights per. 3S1. 
foot, equivalents of, 389. 

weights per, 381. 
inch, equivalents of, 389. 
measure, 389. 
metric, 392. 

yards, earthwork, 732, Ac. 
yard, equivalents of, 389. 

Culvert, Culverts, 
arches for, 693. 
box, 707. 

foundations of, 707. 
lengths of, 702. 

quantity of masonry in. 702, Ac. 
Curvature of the earth, table, 115. 

Curve, Curves. See Arc, Circle, Ellipse* 
Parabola, Ac. 
in pipes, 255. 
railroad, 726. 
tables of, 726-731. 
in tunnels, 754. 
in turnouts, 786. 

Curved 
beams, 484. 







mi 


INDEX. 

Curved—Dodecagon. 


Curved—continued, 
fiauges, 530. 

profiles, retaining walls, 692. 
Curvilinear motion, 365. 

Cuttings, level, 732. 

Cycloid, 154. 

center of gravity of, 351 d. 

Cylinder, Cylinders, 156. 

brickwork, hollow, in foundations, 650. 
center of gravity of, 348, &c. 
contents, table, 157, 246, Ac., 390. 
in foundations, 647, 650. See also 
Foundations. 

frustum of, center of gravity of, 348, Ac. 
iron, foundations, 645. 

friction of, 644. 
of locomotives, 805, Ac. 

masonry, hollow, in foundations, 650. 
plenum process, 648'. 
pneumatic, 647. 
pressure in, 232. 
steam, 809, Ac. 

sinking, for foundations, Ac, 647-650. 
strength of, 232, 516. 
vacuum process, 647. 

Cylindrical 

beams, 492, 467, 503, 511, 516. 

riveted sheet-iron, 516. 
pillars, 441, 442, 443, Ac. 
ungula, 159. 

center of gravity of, 351 g. 

Cyma, to draw, 151. 

13. 

Dam, Dams, 279e, 282, 287. 
coffer, 636, 637, Ac. 
construction of, 282, 636. 
discharge over, 264. 
height of water on, 286. 
leakage through, 222, 288. 
pressure on, 222, Ac. 
stability of, 229. 
trembling in, 285. 
wall8, 229, Ac, 691. 

Day, 395. 

Dead 

load, cars, 811. 

bridges, 564, etc. 
oil, 425. 

Decagon, 110. 

Decimal, Decimals, 34, 35. See Metric, 
of a degree, mins and secs in, 57. 
of a foot, inches in, table, 388. 
fractions reduced to, 34. 
roots of, 53. 

Deck beams, 521. 

Deflection, Deflections, 
angle, 726-731, 785. 

of beams, 505 a, Ac. See Beams, Defls of, 

distances, tables, 726-728. 

of shafting, 510. 

of suspension bridges, 615. 

of trusses (camber), 607. 

of turnouts, 785. 

of turntables, 791, Ac. 

Degree 

of latitude, length of, 144, 388. 


Degree—continued, 
of longitude, length of, 387, 388. 
mins and secs in decimals of, 57. 
Dekametre, 392. 

chord of 2 dekams, curve, table, 728. 
Demi-revetment, 692. 

Density, 314. 

Departures and latitudes, 168. 

Depot, railroad, cost of, 804. 

Depth, Depths, 
on dams, 286. 

flow at different, 237, &c, 268. 
hydraulic mean, 272. 
of keystone, 693, Ac. 
pressures at different, 223, Ac, 286. 
of rain-fall, 220. 

Derrick car, 750. 

De Vout’s switch-stand, 776. 

Dew-point, 215. 

Diagonal 

bracing, 542, &c, 608. 
of parallelogram, 57, 119. 
of trapezoid, Ac, 120. 
of truss, 547, 608. 

Diagram, Diagrams, 
of forces, 320, &c. 
for Kutter’s formula, 278. 
of loads, moving, 546. 
of moments, 482, 483. 
of pressure of water, 240. 
of trusses, 551, Ac. 

Dial, to make, 397. 

Diameter, Diameters, 
of bolts, 406. 

of circle, to find, 123,133, 134, 140. 
of pipes, 245, &c, 248, 405, 416. 
of rivets, for safety, 471, Ac, 539. 
square roots of, 247. 
of wire, 410, Ac. 

Diamond drill, 652. 

Diffusion of force, 224, 227, 231a. 
Dimensions. See the article in question. 
Direction, angle of, 615. 

Discharge 

through adjutages, 259. 
through apertures, 257, Ac. 
through channels, 2.68, Ac. 
through contiguous openings, 261. 
over dams, 264, Ac. 
head for a given, to find, 248. 
through notches, 267 b. 
through pipes, 236, &c. 
through sewers, 279c. 
through short tubes, 259. 
through thin partition, 260. 
over weirs, 264, &c. 

Discount, 37. 

Distance, Distances, 
frog, 785, Ac- 
polar, of North star, 177. 
by sound, 211. 

Distributing reservoirs, 290. 

Distribution of pressure 
in plane surfaces, 231a. 

Diving-bell, pressure in, 215. 

Diving dress, 651. 

Docks, concrete for, 679. 

Dodecagon, 110. 








842 


INDEX. 

I>o<lecabettron—End-wheels. 


Dodecahedron, 154. 

Dollar, 387. 

U S, weight, Ac, of, 387. 

Talue of, iu different countries. 386. 
Double 
float, 269. 
riveting, 468. 
rule of three, 35. 
shear, 470, 476. 

Draft 

of horse, 375, 377, Ac. 
of locomotive, 808. 
of vessels, 236. 

Drag scrapers, 747. 

Drain, Drains, 
area drained by, 279c. 
box drains, 707. 
foundations of, 707. 
pipe, 279d. 

Drainage 

of roadways of bridges, Ac, 708. 
sewers, 279e. 
of tunnels, 754. 

Draw-bridge, strains in, 593. 

Drawing materials, 423. 

Drawn pipes and tubes. 417. 

Dredge, 631. 

land, 750. 

Dredging, 631. 

by screw-pan, 647. 

Dress, diving, 651. 

Dressing of stone, 667. 

Drill. Drills, 
churn, 651. 
jumper, 651. 
machine, 652. 
rock, 651, &e. 
steam, 652. 

Drilling, 

Artesian wells, 627. 
rock, 651-658. 
tunnel, 754. 

Driving 
axles, 808. 
tires, 81)6. 
wheels, 805. Ac. 
weights on, 546, 564, 806, Ac. 

Drop limbers, 284. 

Dry 

drains, 707. 
measure, 390. 
rot, 425. 

Dualin, 664. 

Ducat, value of, 386. 

Ductility, 434 a, Ac. 

Dump-cars, 811. 

Duodecimals, 35. 

Dynamic rock-drill, 657. 

Dynamics, 306. 

Dynamite, 662. 

E. 

E and W line, to run, 171. 

Earnings, railroad, 814, Ac. 

Earth, 

augers for, 626, Ac. 
bearing power of, 634, 


Earth—continued, 
blasting of, 663. 
boring of, 626. 
cars (dump-cars). 811. 
curvature of, table, 115. 
triction of, 375, 692. 
hanling of, 743. 
heat of, 215. 

leakage through, 222, 288. 
leveling of, 743. 
loosening of, 742. 
natural slope of, 6S7, 690. 
pressure of, 687. 
shoveling, 742. 
shrinkage of, 741. 
slope of, natural, 687, 690. 
spreading of, 743. 
supporting power of, 634. 
weight of, 382. 
work, 732-753. 
cost of, 742, 804. 
cubic yards of, 732. 
in tunnels, 754. 
volume of, 732. 

East Kiver bridge, foundations of, 649. 
East and west line, to run, 171. 
Easting, 168. 

Eclipse rock-drill, 654. 

Economizer rock-drill, 656. 

Edge Moor turntable, 793. 

Effective cross-section, 638. 
Efflorescence, 673, 678. 

Elastic limit, 434 d, 504. 

Elastic ratio, 434 d. 

Elasticity, 
limit of, 434 d, 505. 

in beams, 505. 
modulus of, 4346. 

Electric blasting machines, 665. 
Elevation of outer rail in curves, 729 
Ellipse, 149, 150. 
area of, 150. 

center of gravity of, 348, Ac. 
to draw, 150. 
false, to draw, 151. 
ordinates of, 149. 
tangent to, to draw, 150. 

Ellipsoid, 166. 

center of gravity of, 348, Ac. 

Elliptic 
arc, 149. 

ordinates of, 149. 
table of, 150. 
arch, 696. 

joints in, to draw, 150. 
ordinates, 149. 

Elm wood, 

strength, 434e, 436,463, 493. 
weight, 382. 

Elongation 
by heat, 212. 
of North star, 177, 178. 
under tension, 434 a, Ac. 
Embankment, 732-753. 
cost of, 742, 804. 
shrinkage of, 741. 
volume of, 732. 

End-wheels of turntables, 792. 





INDEX. 843 

Energy—Flow. 


Energy, 318 a. 
kinetic, 318 a. 
potential, 318 d. 

Engine 

locomotive, 805-810. 
dimensions, 805, Ac. 
weight, 805, &c. 
performance, 808. 
pumping, 801. 

English cement, 673. 

English rod of brick-work, 389, 672. 

Enlargement in pipes, effect of, 257. 

Entry head, 237. 

Equality of moments, 338. 

Equation of payments, 37. 

Equilibrium 

of floating bodies, axis of, 235. 

of forces, 338. 

indifferent, 235, 348. 

of moments, 338. 

stable, 235, 348. 

unstable, 235, 348. 

vertical of, 235. 

Equipment, railroad, cost of, 804, 814, Ac. 

Erection of trusses, 608. 

Establishment of a port, 219. 

Europe, railroad, miles of, 818. 

' Evaporation, 222, 269. 
by locomotives, 803, 809. 

Even joints, 763. 

Excavating carts (wheeled scrapers), 747. 
; Excavation, 732-753. 
cost of, 742, 804. 
cubic yards, 732. 
in tunnels, 754. 

! volume of, 732. 

Excavator, steam (land dredge), 750. 

Expansion 
bolts, 407. 
of cements, 678. 
by heat, 212. See Heat, 
of rails, 763. 

in trusses, allowance for, 614. 
steam, 809. 
i Expense, Expenses, 
fuel, 815. 

1 locomotive, 810. 
railroad, 814, Ac. 
telegraph, 815. 
train, 815. 

Exploders for blasting, 665. 

Explosive, Explosives, 660, Ac. 
foreign, 664. 
freezing of, 661, 663. 
gelatine, 66\ 
modern, 661. 

Express cars, 811. 

Extrados, 693. 

Eye bars and pins, 612. 


F. 

Face 

of arch, 693. 
wall, 683. 

Facing switch, 770. 

Factor of safety. See Safety, factor of. 
for piles, 644. 


Factor of safety—continued, 
for pillars, 442, 446. 
for truss bridges, 607. 

Fahrenheit thermometer, 213. 

Fall, Falls, 
rain, 220. 

required for a given discharge, 274,279b, 
in sewers, 279c, Ac. 

Falling bodies, 258, 362. 

Falling water, horse-power of, 280. 

False ellipse, to draw, 151. 

False works, 608. 

Fascines, 650. 

Fatigue of materials, 435. 

Faucet in pipe-joint, 295. 

Fellowship (partnership), 37. 

Fence, 803, 815. 

Ferrule for water-nipe, 299. 

Fifth 

powers, 251. 

square roots of, 253. 
roots, 251. 

Figure, Figures, 110, Ac. 
center of gravity of, 348, Ac. 
defined, 54. 
to draw, 121, Ac. 
to enlarge, 122. 

irregular, area of, to find, 122. 

Filling, spandrel, 693. 

Filtration, 222. 

Finish, hard, 426. 

Fink truss, 574, 578-580, 584, G03. 

Fir, strength, 460, 463. 

Fire, Fires, 
heat of, 212. 

hydrant (fire-plug), 304. 

-proof floors, 522. 

Firing, simultaneous, of blasts, 665. 
Fish-plates, 764. 

Fisher rail-joints, 766, 767. 

Fittings for pipes, 293, Ac, 405. 

Flagging, strength, constants for, 493. 
Flange, Flanges, 
curved, 530. 
oblique, 530. 
strains in, 529, 537. 
strains in, in riveted girders, 537. 
Flexible joints for pipes, 296. 

Flexure, contrary, point of, 515. 

Floating 
bodies, 234, Ac. 
mills, 280. 

Floats, 268, 269. 

Floor 

beams. See Beam, 
buckled plates for, 409. 
fire-proof, 522. 
girders, 545, Ac, 610. 
glass, 432. 
loads on, 606, 623. 
weight of, 553. 

Florin, value of, 386 
Flotation, 234, Ac. 

Flow 

through adjutages, 259. 
through apertures, 257, Ac. 
in channels, 268, Ac. 
through contiguous openings, 261. 








844 


INDEX. 

Flow—Fractions. 


Flow—continued, 
full, 259. 

Kutter’s formula, 244, 271. 
obstructions to, 279e, Ac. 
in pipes, 236, Ac. 
in sewers, 279c. 
in streams, 268, Ac. 
through short tubes, 259. 
in syphons, 241, Ac. 
through thin partition, 260. 
in troughs, 263. 

Fluid, Fluids, 
friction of, 374c. 

Follower, in pile driving. 645. 

Foot, 

cubic, equivalents of, 389. 

of substances, weight of, 381. 
decimals of, inches in, table, 388. 
pound, 316. 

spherical, equivalents of, 389. 

Force, Forces, 

application of, point of, 309, 318/, Ac. 

center of, 227. 235, 347 1 , 700. 

centrifugal, 368. 

centripetal, 368. 

composition of, 319, Ac. 

defined, 308. 

in different planes, 332, Ac. 
diffusion of, 231 a. 

through liquids, 227. 
equilibrium of, 338. 
imparted, 318 e. 
gradually, 311. 

on inclined planes, 352, Ac., 363. 

living, 318 a. 

measure of, 310, 314. 

momentum, 318 c. 

obliquely applied, 318 e. 

parallel, 347. 

parallelogram of, 320, Ac. 

parallelopiped of, 333. 

point of application of, 3 )9, 318 f, Ac. 

polygon of, 329. 

resolution of. 319. Ac. 

in rigid bodies, 306. 

‘‘single,” “ triple,” Ac., caps, 665. 
in trusses, 551. 
work, 316. 

Forcite, 664. 

Foreign coins, 386. 

Foreign explosives, 664. 

Formula. See also the given problem. 
Gordon’s, 439. 

Kutter’s, 244, 271. 
prismoidal, 161. 

Foundations, 633. 
of arches, 693. 
artificial islands, 651. 
brick cylinders, 650. 

Brooklyn bridge, 649. 
caissons, 636. 
car pile-driver, 642. 
for centers, 711. 
in clay, 634. 
close piles, 641. 
coffer-dams, 636, 637, Ac. 
concrete, 680, 681. 
crib, 635. 


Foundations—continued, 
of culverts, 707. 
cylinders, 645-647. 
brick, 650. 

plenum process, 648. 
vacuum process, 647. 
with concrete, 679. 
friction of, 644, 645. 
masonry, 650. 
with piles inside, 651. 
diving dress, 651. 
of drains, 707. 

East River bridge, 649. 
fascines, 650. 
on gravel, 634. 
grillage, 641. 

iron cylinders, 645-647. See Found® 
tions, Cylinders, 
iron piles, 645. 
islands, artificial, 651. 
leveling by concrete, 6S0. 
loads for, 634. 
masonry cylinders, 650. 

Nasmyth pile-driver, 642. 

Pierie perdue, 634. 
pile, piles, 626, 640. 
adhesion of ice to, 645. 
bearing, 641. 
blasting of, 663. 
in cylinders, 651. 
drivers, 379, 641, 642. 
car, 642. 
gunpowder, 641. 

Nasmyth, 642. 
driving, by jets, 646. 
factor of safety, 644. 
friction of, 644. 
grillage, 641. 
heads for, 645. 
hollow, 647. 
ice, adhesion to, 645. 
iron, 645. 
loads for, 643. 

Mitchell’s screw, 645. 
sand, 626, 650. 
screw, 645. 
sheet, 641. 
shoes for, 644. 
sustaining power of, 643. 
water-jet for driving, 646. 
withdrawal of, 645. 
plenum process, 648. 
pneumatic, 647, Ac. 
random stone, 634. 
of retaining walls, 692. 
rip-rap, 634. 
on sand, 634. 
sand piles, 626,650. 
sand pump, 650. 
screw piles, 645. 
sheet piles, 641. 
sustaining power, 634, 643, 
for trestles, 756. 
for turntables, 791. 
vacuum process, 647. 

Four-way stop-valve, 302. 

Fractions, 33. 
addition, Ac, of, 33. 












845 


INDEX. 

Fractions—Glass. 


Fractions—continued, 
decimal, 34. 

greatest common divisor, 33. 
lowest terms, to reduce to, 33. 

Franc, value of, 386. 

Franklin Inst standard dimensions of 
bolts, Ac., 406. 

Freezing 
of cement, 675. 
of dynamite, 663. 
i of explosives, 661, Ac. 
of mercury. 213. 
of mortar, 672. 
of nitro-glycerine, 661. 
in pipes, 293 

behind retaining-walls, 684. 
in stand-pipes, 298. 
in track tank, how prevented, 802. 
of water, 217. 

Freight 
cars, 811, 812. 
earnings, 814. 
expenses, 814, Ac. 
locomotives, 805-810. 
ton-mile, 809, Ac, 814, Ac. 
train expense, 815. 

French 

measures. 391, 393. 
weights, 393. 

Freyburg suspension bridge, 622. 
Friction, 353, 370. 
angle of, 318 < 7 , 355, 371. 
axle, 374d. 
of cars, 374e, 808. 
coefficients of, 371, Ac. 
of earth, 375, 692. 
at feet of rafters, 355. 
head, 237, 248. 

on inclined planes, 352-361, 364. 
of iion cylinders, 644. 
journal, 374d. 
kinetic, 370. 
launching, 374c. 

Morin’s laws, 372. 

of masonry, 373, 375, 692. 

of piles, 644. 

in pipes, 257. 

of pivots, 371. 

in pumping mains, 257. 

rollers, 374e, 792, Ac. 

rolling, 3746. 

static, 370. 

of walls, 688 . 

of water, 257, 374c. 

Frictional stability, 352-361. 

Fritzsohe turntable, 794. 

Frog, Frogs, 780-786. 
angle of, 781, 785, Ac. 
distance, 785. 
length, 781. 
number, 781, 786, Ac. 
point., 781. 

Frost-jacket in fire-hydrant, 304. 
Frost-proof tank, 801. 

Frozen. See Freezing. 

Frustum, 

center of gravity of, 348, Ac. 
of cone, 160 a. 


Frustum—continued, 
parabolic, 152. 
of paraboloid, 167. 
of prism, 155. 
of pyramid, 160 a. 

Fuel, 

coal, 809, Ac. 

consumption by locomotive, 809. 

effect of grade on, 810, Ac. 
expense, of locomotives, 810, 815. 
grades, effect on consumption, 810, Ac. 
wood, 810, Ac. 

Fulcrum, 335. 

Full flow, 259. 

Fumes, 

acid, effect on roofs, 428. 
coal, effect on iron, 403, 418. 

Funicular machine, 325, 344. 

G. 

Gallon, 390, 391. 

Galton’s experiments, 374. 

Galvanic action in water-pipes, 293. 
Galvanized 
iron, 403. 
pipes, 299. 

Gas, weight of, 381, Ac. 

Gasket, 295, Ac. 

to prevent washing into pipe, 295, 296. 
Gates for water-pipes, 301. 

Gauge, Gauges, 

Birmingham, 410, 411. 
narrow 

cars for, 811. 
car-wheels, 812. 
locomotives, 805, Ac. 
statistics, 818. 
railroad, 773, 814. 

Stub’s, 411. 
stuff, 426. 
wire, 410, 411, 412. 

Gauging of streams, 268, Ac. 

Gautliey’s pressure plate, 269. 

Gearing, 342. 

Gelatine, explosive, 665. 

Genesee viaducts, 756, 757. 

Geographical mile, 387. 

Geometrical 
progression, 36. 
similarity, 54. 

Geometry, 54. 

German cement, 673. 

Giant powder, 664. 

Gibbon boltless rail-joint, 767. 

Gin. 378. 

Girder, Girders. See also Beam, 
box and plate, 537. 
continuous, 515. 
floor, 545, Ac, 610. 
riveted, 537. 
transverse, 545, Ac, 610. 
of turntables, 791, Ac. 

Girts, cross-, of turntables, 792. 

Glass, 431. 

compressibility, 434 e. 
dimensions, Ac, 431. 
ductility, 434 e. 











846 INDEX. 

Glass—Hemisphere. 


Glass—con t i n ued. 
elastic limit, 434 1 . 
expansion by heat, 212. 
friction of, 373, Ac. 
modulus of elasticity, 434e. 
prices of, 432. 

strength, 432, 437, 466, 493. 
stretch of, 434 e. 
weight, 431. 

Glazing, 431. 

Globe, 162, 163. 

Glossary of terms, 819. 

Glue, adhesion of, 466. 

Glycerine, nitro-, 661. 

Gneiss, weight, 382. 

Gold, 

strength. 464. 
value, 387. 
weight, 382, 3S7. 

Gondola cars, 811. 

Gordon’s formula, 439. 

Grade, Grades, 

allowance for, in chaining, 176. 
contour lines, 197. 
effect on fuel consumption, 810. 
on horse, 375. 
on locomotive, 808. 
hydraulic, 240. 
resistance of, 808. 
of roads, 375,723. 
of sewers, 279c, d. 
tables of, 176, 354, 723, 724, 725. 
traction on, 375, 808. 
in tunnels, 754. 
on turnpikes, 723. 
of water-pipes, 290. 

Grading, cost of, 742, Ac, 804. 

Granite, 
beams, 504. 
cost of blocks, 667. 
expansion by heat, 212. 
rubble, cost of, 668. 
strength of, 437,493, 504. 
weight, 383. 

Granular bodies, specific grav of, 381,384. 
Gravel, 

boring in. 626. 
dredging in, 631. 
for foundations, 634. 
natural slope of, 690. 
weight, 382. 

Gravity, 

acceleration of, 258, 311, 312, 362, 363. 

center of, 348, Ac. 

on inclined planes, 363. 

line of, 350. 

plane of, 350. 

specific, 380. 

Great Bear, 179. 

Great Britain, railroads in, miles of, 818. 
Greenleaf turntable, 795. 

Greenwood switch-stand, 772. 

Grillage, 641. 

Ground lever, 772. 

Grout, 670. 

Grubbing, cost of, 804. 

Guard rails, 774, 779,781. 

Gudgeon 374 d, 824. 


Guide-rails, 774, 779, 781. 
Gun-cotton, compressed, 664. 

Gun metal, strength,464. 
Gunpowder, 660. 
pile-driver, 641. 

weight of (under Powder), 384. 
Gunter’s chain, 176, 387. 
Gutta-percha 
pipe, 294. 
weight of, 382. 

Gypsum process, 425a. 

weight, 382. 

Gyration, 
center of, 440. 
radius of, 366, 440, 538, 540. 


H. 

Hairs, cross, to replace, 193. 

stadia, 190. 

Hand level. 205. 

Hard finish, 426. 

Haul, mean, 743. 

Hauling, 375, 377, Ac, 743, 747. 

Head, Heads, 
block, 771. 
of holts, 406. 
entry, 237. 
friction, 237, 248. 

for a given discharge, to find, 248. 
for a given velocity, to find, 245, 248 
for piles, 644. 
plate, 771. 
pressure, 239. 

required for bends, Ac, 255. 

theoretical, 258. 

tripod, 189. 

velocity, 237. 

virtual, 258,280. 

of water, 223, 237, Ac. 

for water supply, 290, Ac. 

Heading in tunuel, 754. 

Headway iu bridges, 609. 

Heat, 

of the air, 215. 
conduction of, by air, 215. 
effect of, on cement, 675. 
expansion of air by, 215. 
of rails by, 212, 763. 
of solids by, 212. 
of surveying chains ,by, 168. 
of trusses by, 614. 
of fires, 212. 
subterranean. 215. 
thermometer, 213. 

Hecla powder, 664. 

Heel 

of frog, 781. 

- of switch, 771, 774, 785. 

Height, 

effect on temperature, 215. 
effect on weight, 312, 362. 
to find, by barometer, 207. 
by boiling point, 209. 
by trigonometry, 113. 
of locomotive smoke-stack, 805, Ac. 
of water. See Head. 

Hemisphere, center of gravity of, 348, A 













INDEX. 

He m lock—I r on 


847 


Hemlock, 

strength, 436, 460, 476, 493. 
weight, 382. 

Hemp ropes, 414. 

Heptagon, 110. 

Hercules powder, 664. 

Hexagon, 110,121. 

Hickory, 

strength, 436, 463, 493. 
weight, 382. 

High explosives, 661. 

Hodgkinson beams, 518. 

Holes 

for blasting, 651. 
boring, in earth, 626. 
boring, in rock, 651. 
for rivets, 470. 

Hook-head spikes, 762. 

Horizon, artificial, 195. 

Horizontal 
defined,115. 

Horse, Horses, 
power of,'375, 801. 

-power, 318, 377. 

of falling water, 280. 
of running streams, 280. 
pumping, day’s work, 801. 

-walk, diameter of, 377. 
weight, 377. 

House, engine, cost, 799. 

Howe truss, 594. 

Hydrant, fire (fire-plug), 304. 

Hydraulic, Hydraulics, 236. See also 
Water, Flow, Velocity, Discharge, 
&c. 

cements, 673. See also Cement. 

dams, 229, 264, 279e, 282. 

grade-line, 240. 

lime, 673. 

mean depth, 272. 

radius, 244, 272. 

ram, 280. 

weirs, 229, 264, 279e, 282. 

Hydrogen, specific gravity of, 382. 
Hydrometric pendulum, 269. 

Hydrostatic, Hydrostatics, 222. 
press, 227. 

I. 

I beams, 521, Ac. 
as pillars, 441, 442, 454. 
see Beams, I. 

Ice, 217, Ac. 
adhesion to piles, 645. 
blasting of, 663. 
in staud-pipes, 298. 
streugth of, compressive, 437. 
weight, 217, 383. 

Icosahedron, 164. 

Impact, 3L8e. 

Impartation 
of force, 308, 318 e. 
of velocity, gradual, 311. 

Imperial measure, British, 391. 

Impulse, 313. 

Inch, Inches, 
circular, 389. 


Inch, Inches—continued, 
cubic, equivalents of, 389. 
in decimals of a foot, 388. 
fractions of, common and decimal, 34. 
of rain, 221. 

spherical, equivalents of, 389. 
Inclination. See Grade. 

of courses in masonry, 683, 700. 
tables of, 176, 354, 723-725. 
in tunnels, 751. 

Inclined 

beams, 340, 480, 496. 
plane, 352, &c, 363. 
descent on, 363. 
ropes for, 413, 414. 
stability on, 352-361. 
tables, 176, 354, 723-725. 
velocity on, 363. 

Incomplete contraction, 263. 

Incrustation 
of boilers, top of 218. 
of walls, 673, 678. 

India-rubber, weight, 383. 

Indifferent equilibrium, 235, 348. 

Inertia, 314. 

moment of, 365, 486, 4S7. 

Ingersoll rock-drill, 654. 

Initial pressure of steam, 809. 

Instability, 235, 348, 356. 

Insurance,37. 

Interest, 37. 

Intrados, defined, 693. 

Iron 

angle, 525. 
beams of, 525. 
pillars of, 441-442. 
strength of, transverse, 525. 
balls, weight, 416. 
bars, weight, 400, 401. 
beams. See Beams, Iron, 
blasting of. 663. 
bolts, 406-409. 
bridges, cast, 599. 
bridges. See Trusses, Bridge, &c. 
buckled plates, 409. 
cars, 811. 
cast 

beams of, modification iu sections of, 
518. 

beams of, strength of, 519. 
bridges, 599. 
cohesive streugth, 464. 
compressive strength, 438. 
crushing strength, 438. 
expansion by heat, 212. 
friction of, 373, &c. 
pillars, 439, &c. 
pipes, flow in, 243, &c. 
pipes, weight of, 293, 297, 399. 
salt water, effect on, 218, 645. 
shearing strength, 476. 

Sterling’s toughened, 520. 
strength, 438,464, 476, 477, 493. 
tensile strength, 464. 
torsional strength, 477. 
transverse strength, constants for, 
493. 

turntables, 792. 







848 


INDEX. 


Iron—Keystone. 


Iron—continued, 
cast 

water, salt, effect on, 218, 645. 
weight, 382, 398, 401. 
casting, weight of, by size of pattern, 
398. 

and cement, pipes of, 294. 
chains, 414. 
channel, 521. 

as pillars, 441, 442, 456. 
cohesive strength of, 409, 464. 
cold, effect on, 168, 466, 763. 
columns, 439, Ac. See also Pillars, Iron, 
compressibility of, 434 e. 
compressive strength of, 438. 
contraction of, by cold, 168, 763. 
corrosion of, by coal fumes, 403. 
corrugated sheet, 403. 
cost of, 402. 

crushing strength, 438. 
cylinders, bursting pressure in, 232,233. 
cylinders, in foundation, 644, Ac. See 
also Foundations, 
ductility of, 434 e. 
effect of cement on, 673. 
of cold on, 168, 466, 763. 
of heat on, 168, 212, 763. 
of mortar on, 670, 673. 
of water on, 218, 645. 
elastic limit, 434 «. 
expansion of, by heat, 168, 212, 763. 
-frame cars, 811. 
friction of, 373, Ac. 
galvanized, 403. 
heat, effect on, 168, 212, 763, 
limit of elasticity, 434 «. 
modulus of elasticity, 434 e. 
net, 470. 

paints for preserving, 430. 
piles, 645 See also Foundations, 
pillars, 439, Ac. See also Pillars, 
pipes, 

cast, weight, 293. 
fittings for, 405. 
flow in, 243, Ac. 
galvanized, 299. 
joints for, 293, 295. 
kalameined, 293. 
thickness of, 233, 293. 
wrought, 293. 
plates, 

buckled, 409. 
prices of, 402. 
porosity of, 233. 
prices of, 402. 
re-rolled, 402, 464. 
rolled. See Iron, Wrought, 
roofs, 403, Ac, 582, Ac. See Roofs, 
salt water, effect on, 218, 645. 
shearing strength, 476. 
sheet, 402, 403. 
corrugated, 403. 
galvanized, 403. 
specific gravity of, 382. 
spikes, 762. 

strength. 409, 438, 464, 476, 477, 493. 
stretch of, 434 <•. 

T, 441, 442, 525. 


Iron—continued, 
tensile strength, 409, 461. 
torsional strength, 477. 
transverse strength, 477, 493. 

trestles, 756. 
tubes, 405. 

water, effect on, 218, 645. 
weight of, 3S2, 400, Ac. See also Iron, 
Cast; Iron, Wrought, 
wire, 412. 
rope, 413. 

-wood (Canadian), strength, 493. 
wrought 

bars, weight, 400, 401. 
beams, 521, Ac. See Beams, 
cohesive strength, 464. 
compressive strength, 438. 
crushing strength, 438. 
expansion by heat, 168, 212, 763. 
friction of, 373, Ac. 
pillars, 439, Ac. See also Pillars, 
pipes, 293, 294. 
fittings for, 405. 
joints for, 293, 295. 
weights, prices, Ac, 293, 405. 
prices, 402. 

shearing strength, 476. 
strength, 438, 464, 476, 477, 493. 
tensile strength, 464. 
torsional strength, 477. 
transverse strength, 493. 
tubes, weight, 405. 
water-pipes, 293. 
weight, 383, 400, 401. 

Islands, artificial, for foundations 651. 

J. 

Jag-spikes, 762. 

Jet, pile-driving, 646. 

Joint, Joints, 
in arches, 709. 
bell, for pipes, 295. 
butt, lap, 469. 

in chimneys, Ac, cement for, 429, 431. 
Converse, for wrought iron pipes, 293. 
distribution of pressure in, 231«. 
flexible, for pipes, 296. 
iron, 583. 
lap, butt, 469. 

masonry, inclination of, 683, 700. 

distribution of pressure in, 231a. 
net, 470. 

for pipes, 293, 295, Ac, 405. 
pressure in, distribution of, 231a. 
rail, 763. 
riveted, 468, 539. 
roof, 418, 427, 429. 
timber, 610, 612. 
in trusses, 610, 612, Ac. 

Journal friction, 374d. 

Judson dynamite, Ac, 664. 

Jumper drill, 651. 

K. 

Keyed frog, 782. 

Keystone, 693, 695. 
pressures on, 330, 359, 694. 















INDEX. 

Kieselgahr-Link. 


Kieselguhr, 662. 

Kinetic energy, 318 a. 

friction, 37U. 

King 

-post, 554. 

-rod, 553, &c. 
truss, 551, &c, 578. 

Kinzua viaduct, 758. 

Knees in pipes, 256. 

Knife-edge, strength of, 438. 

Knot (nautical), length of, 387. 

Kutter’s formula, 244, 271. 

Kyanizing, 425a. 

L. 

jagging for centers, 711,712, 719. 

jaitance, 681. 

jand 

dredge, 750. 
measure, 389. 
metric, 392. 

required for railroads, 722. 
section of, 
area of, 389. 
surveying, 168. 

! ties, 692. 
jap-joint, 469. 

jap-welded boiler-tubes, 405. 

water-pipe, 293. 
jard 

as a lubricant, 374c. 
weight of, 383. 
jateral bracing, 542, 610. 
jaths, 426, 427. 

.atitude. Latitudes, 
degree of, length of, 387. 
and departures, 168. 
effect of, on barometer, 207, 209. 

on gravity, 312, 362. 
secants of, 177. 

.attice truss, 596. 
jaunching, friction of, 374c. 
jaying 
bricks, 670. 
out of turnouts, 785. 
pipe, cost of, 297. 
track, cost of, 804. 

Lead 

balls, weight of, 416. 
bars, 415. 

compressibility, 434 e. 
ductility, 434 e. 
elastic limit, 434 e. 

effect of cement, mortar, &c, on, 670, 
673. 

expansion by heat, 212. 
in masonry joints, 438. 
modulus of elasticity, 434 e. 
paint, 429. 
pencils, 433. 
pipe, 234, 416. 

for pipe-joints, 293, 295, 297. 
roofs, 415, 416. 
sheets, 415. 416. 
strongth, 438, 464. 
stretch of, 434 e. 
tensile strength of, 464. 


Lead—continued, 
weight, 383, 398-401, 410, 415, 416. 
white, cement, 
for leaks, 429. 
white, paint, 429. 

Leaded tin, 418. 

Leak in roof, to stop, 429, 431, 
Leakage, 222, 269, 282, 288. 

Leather, 
friction, 374c. 
strength, 466. 

Length, 
frog, 781. 
switch-rail, 776. 
switch, 786, &c. 

Lengthening scarfs, 610. 

Level, 201. 

builder’s, to adjust, 206. 
cuttings, 732. 
hand, Locke’s, 205. 
lines, defined, 115. 
note-book, form of, 204. 

Y, 201. 

Leveling 

by barometer, 207. 
by boiling-point, 209. 
by concrete, 680. 
of earth, on embankments, 743. 
screws, 189, 202. 

Lever, Levers, 335. 
compound, 342. 

principle of, applied to arches, 341. 

beams, 339. 
switch, 772, 776, 779. 
tumbling, 772, 776, 779. 

Leverage, 335. 

Life, average, 
of cars, 811. 
of locomotives, 810. 
of rails, 760. 
of shingles, 429. 
of ties, 759. 
of wire ropes, 414. 

Lignum vitae, 
strength, 463, 493. 
weight, 383. 

Lime, 669. 

effect on cement, 680. 
hydraulic, 673. 
paste, 670. 

to preserve timber, 425a. 
weight, 383. 

Limestone, 383, 437. 

Limit of elasticity, 434 d, 505. 

in beams, 505. 

Limnoria, 425. 

Line, Lines, 54. 
center of gravity of, 351 a. 
contour, 197. 
of gravity, 350. 
hydraulic grade, 239. 
of no variation, 197. 
parallel, to draw, 56. 
of pressure, 359, 700. 
of resistance, 359,700. 
of thrust, 359, 700. 

Lining of tunnels, 754. 

Link. See Chaiu. 




850 


INDEX. 


Liquid-Members. 


Liquid, Liquids. See Water, 
buoyancy of, 234, 235. 
compressibility, 217, 236. 
flow of, 236, Ac. 
friction of, 374c. 
measu re, 390. 

pressure of, 222, Ac, 239, Ac. 
pressure, transmission through, 227. 
specific gravity of, 381. 

Lithofracteur, 664. 

Little Giant rock-drill, 656. 

Live load, 546, 564, Ac, 805-807. 

Living force, 318 a/ 

Load. See also Beam, Truss, Pillar, 
Bridge, Ac. 

on beams, constants for, 491. 

on bridges, greatest probable, 606, 623. 

cart load, of earth, Ac. 742, Ac. 

on driving-wheels, 546, 564, 805, Ac. 

on earth, safe, 634. 

for a given deflection, 505 a, b, 510. 

for I beams, 521. 

live, 546, 564, 805-807. 

for locomotives, 808. 

moving, 546, 564, 805-807. 

for piles, 643. 

on roof, 216, 221, 580. 

of sand, 427. 

suddenly applied, 434/. 

on turntables, 794, 795. 

Lock, air, 648. 

Lock-nut, 408, 765, 768. 

Lock-Ken viaduct, 599, 647. 

Locke level, 205. 

Locomotive, Locomotives, 805-810. 
“adhesion” of, 374a, 808, 809. 
driving-wheels, 805, Ac. 

weights on, 546, 564, 805-807. 
evaporation by, 803. 
house, cost, 799. 
repairs, 810, 815. 
statistics, 814, Ac. 
turntables for, 791. 
water for, 218, 800. 
weights, 546, 805, Ac. 

Locust, strength, 436, 463, 493. 

Logarithms, 38, 39. 
to find roots by, 39. 

Long 

chords, table, 729. 
measure, 387, 392. 

Longitude, degree of, length of, 387, 388. 

Lorenz switch, 775. 

Lowering of centers, 711-713, 720. 

Lubricants, 374c. 

Lubrication of turntables, 792, Ac. 

Lumber, 420-425. See also Wood, Timber. 

Lune, circular, 146. 


Machine, 
drill, 652,754. 
funicular, 325, 344. 
riveting, 471. 

for tapping pipes, 294, 299. 
Magneto-electric blasting, 665. 
Mahogany, strength, 434e, 436, 463, 493. 
weight, 383. 


Mail 

cars, 811. 
earnings, 814. 

Maintenance of road and real estate, 815 
Man, 

power of, 378. 
weight, 606, 623. 

Manilla rope, 414. 

Map, to reduce or enlarge, 122. 
Maple-wood, 
strength, 463, 493. 
weight, 383. 

Marble, 
cost, 668. 

expansion by heat, 212 . 
strength, 437, 493. 
weight, 383. 

Masonry, 

in abutments, quantity, 703. 
adhesion of cement to, 677. 

of mortar to, 670. 
in arch bridges, quantity, 702-708. 
backed by concrete, 679. 
cost, 667. 

courses, inclination of, 683, 700. 

lead between, 438. 
friction of, 373, Ac, 683, 700. 
incrustation of, 673, 678. 
in piers, quantity, 708. 
joints. See Joints, 
quantity in arches, 702-708. 
in piers, 708. 
in retaining walls, 690. 
in walls of wells, 158. 
in wing-walls, 704. 
railroad, cost, 804. 
in retaining walls, 683, Ac. 
strength of, compressive, 437 . 
weight, 229, 381, Ac. 

Mass, 312. 

Materials, 
fatigue of, 435. 
strength of, 434. 
weight of, 381. 

Matter, defined, 306. 

Maximum 


pressure, angle of, 687. 
prism of, 687. 
slope of, 687. 
velocity, 268. 

Mean 


depth, hydraulic, 244, 272. 
haul, 743. 
radius, 244, 272. 
velocity. 243, 268, Ac. 
Measure, Measures, 385. 


commercial 

391. 


, size of, by' weight of 


cubic, 389. 

French, 391-393. 
long, 387. 
metric, 391, &c. 

Russian, 394. 

Spanish, 394. 
square, 389. 

Mechanics, 306. 

Melting points, 212. 

Members, web, of trusses, 529. 

























INDEX. 


851 


Mercury 

' Mercury, 

barometer, 207, 215. 
freezing-point, 213. 

i thermometer, 213, 215. 
weight, 383. 

Meridian, to find by North star, 177. 
longitude, 387, 388. 
variation of compass, 193, 196. 

Metacenter, 235. 

Metal, Metals. See also the names of the 
several metals, 
blasting of, 663. 
cohesive strength, 464. 
compressibility, 434 1 . 
compressive strength, 438. 

i ductility, 434 e. 
effect of cement on, 670, 673. 
of heat on, 212. 
of lime on, 670, 673. 
of mortar on, 670. 
of w ater on, 218, 645. 
elastic limits of, 434 «. 
expansion by heat, 212, 763. 
friction of, 370, Ac. 

! limit of elasticity, 434 «. 
modulus of elasticity, 434 e. 
shearing strength, 476. 
sheet, 402-404, 410, 415, 416, 418, 419. 
strength, 434, 438, 464, 476, 477, 493. 
stretch of, 434 «. 
tensile strength, 464. 
torsional strength, 477. 
transverse strength, 493. 
weight of, 381, Ac. 

Meters, wheel, 270; Venturi, 260. 

Metre, Metres, 
length of, 391. 
radii, &c, of curves in. 728. 

Metric 

measures, 391. 
railroad curves, table, 728. 
weights, 381, Ac, 393. 

Mica, weight, 383. 

Middle ordinates,141 a,726,728,730,761,786. 

Mile, Miles, 

freight-ton-mile, 809, Ac, 814, &c. 
geographical, 387. 
land and sea, 387. 
nautical, 387. 

I passenger, 814, Ac. 
scale of, 187. 
sea-mile, 387. 
square (section), 389. 
ton-mile, 809, Ac, 814, Ac. 
train-mile, 809, Ac. 

Mills, floating, 280. 

Miner’s friend powder, 664. 

Minutes in decimals of a degree, 57. 
of time, 395. 

Mitchell's screw pile, 645. 

Mitred joints for rails, 763. 

Modern explosives. 661. 

Modulus. See Coefficient, Strength, Ac. 
of elasticity, 434 b. 
of flow', 259, 266. 
of friction, 371. 
of resilience, 434/. 
of resistance (coef of res), 485. 
of rupture, 485. 


-Natural. 

Mogul locomotives, 805, Ac. 

Moisture, effect on cement, 673. 
on sound. 211. 
on zinc, 419. 

Moment, Moments, 335. 
in beams, 478-489, 528, 537, Ac. 
breaking, 479-484. 
equilibrium of, 338. 
of inertia, 365, 486, 487. 
of resistance, 484, 486, 488. 
of rupture, 479-484. 
of stability, 229, 235, 337, 357, 688. 

Momentum, 314, 318 c. 
of water, 234. 

Money, 386, 387. 

Monkey-switch, 772. 

Mont Cenis tunnel, 754. 

Morin’s laws of friction, 372. 

Mortar, 

adhesion of, 670. 
in arches, 696, 7,09, 713. 
bricks, Ac, 669. 
cement, 676. 
clay, effect on, 670. 
effect on iron, 620, 670, 673. 

on wood, 670. 
frozen, 675. 
grout, 670. 
pointing, 674. 
in retaining walls, 684. 
rubble, cost of, 668. 

weight of, 383. 
salt, effect on, 670, 678. 
sand for, 669, 677. 
strength of, tensile, 466, 676,678. 
in water tanks, 80L. 
weight, 383, 670. 

Moseley roof, 600. 

Motion, 307. 
accelerated. 307. 
circular, 365. 
defined. 307. 
quantity of, 314. 
retarded, 307. 
uniform, 307. 

Moving 
force, 308. 

load, 540, 564, 805-807. 

Mud 

penetrability, 644. 
in reservoirs, 288. 
weight of, 383, 632. 

Muskrats, 288. 

N. 

Nails, 4255. 
shingling, 429. 
slating, 425 5, 428. 

Narrow'-gauge railroad 
cars, 811, 812. 
locomotives, 806, Ac. 
statistics, 818. 

Nasmyth pile-driver, 642. 

Natural 
chords, 105. 

Portland cement, 673. 
sines, Ac, 59, 60. 
slope, 684, 686, 690. 

Nautical mile, 387. 








852 


INDEX. 

Sfeedle—Paste. 


Needle, compass, 190, 196. 

Net 

earnings of railroads, 814, Ac. 
ivou, net plate, net joint, 470, 538. 
Neutral axis, 479, 485, 487. 

Niagara suspension bridge, 622. 
Nitro-glyeerine, 661. 

NoiMgon, 110. 

Nortlr and south line, to find, 177. 
North star, 177. 

Northing, 168. 

Note-book, level, form of, 204. 
Number 

of frog, 781, 786. 
by wire-gauge, 410, 411, 412. 

Nuts, 406. 

Nut-locks, 408, 765, 768. 

Cambria, 765. 


o. 

Oak, 

strength, 434 «, 436,463, 476, 493. 
weight, 383. 

Oblique, Obliques, 
beams, 340, 480, 496. 
flanges, 530. 
lines, 54. 
pillars, 457. 

pressure, 225, 318 e, 352, 687. 
in trusses, best inclination for, 548. 
length of, to find, 122, 608. 
Obstacles in surveying, to pass, 175. 
Obstruction, Obstructions, 
to flow, 279d, Ac. 
by piers, 279d, Ac. 
in pipes, to prevent, 292. 

Octagon, 
area, 110. 
to draw, 121. 

Octahedron,154. 

Offset, 683. 

Oil, Oils, 

coal, weight (Petroleum), 383. 
dead, 425. 

for locomotives, cost, 810, 815. 
olive, 374c. 
weight, 383. 

wells, nitro-glycerine in, 661. 

Olive oil, lubricating power of, 374c. 
Open channels, flow in, 268, Ac. 
Openings, 

contiguous, flow through, 261. 
flow through, 257, Ac. 
with short tubes, How through, 259. 
in thin partition, flow through, 260. 
Ordinate, Ordinates, 
elliptic, 149. 
to find, 141 a. 

middle, 141 a 726-731, 761, 786. 

parabolic, 152. 

tables, 726-731, 761, 786. 

Oscillation, center of, 365. 

Osgood excavator, 750. 

Otis excavator, 751. 

Outer rail, elevation of, 729. 

Outflow, velocity of, theoretical, 258. 
Outlet valves, 290. 


Oval, to draw, 151. 

Overfall, 
dams, 282, 
discharge over, 264. 
for reservoir, 289. 

P. 

Packing piece, 471, 545, Ac. 

Paint, Paints, 429. 
brushes, to clean, 430. 
for iron, 430. 
on zinc, 403. 

Painting, 429. 

Panel, 

defined, length of, 548. 
diagonal of, to find length of, 122, 60 
Paper, 433. 
car-wheels, 812. 
pipes, 294. 
wheels, 812. 

Parabola, 152, 153. See Parabolic, 
arc of, 152. 

center of gravity of, 351 d. 
to draw, 153. 
frustum of, 152. 
ordinates, 152. 

semi-, center of gravity of, 351 d. 
tangent to, to draw, 153. 
zone of, 162. 

Parabolic 
arc, 152, 153. 
conoid, 167. 

frustum of, 167. 
curve, 152, 153. 
frustum, 152. 
ordinates, 152. 
zone, 152. 

Paraboloid, 167. 
center of gravity of, 351 h. 
frustum of, 167. 

Parallel 
forces, 235, 847. 
lines, to draw, 56. 
plates, 189. 

Parallelogram, Parallelograms, 57, 119 

angles in, 57. 

center of gravity of, 348, Ac. 
of forces, 320, Ac. 1 

Parallelepiped, 155. 
center of gravity of, 348. <feo. 

of forces, 333. 

Parapets of suspension bridges, 621, A< 
Parlor cars, 811, 812. 

Partial 

contraction, 259, Ac, 263, Ac. 
on weirs, 263, Ac. 

payments (Equation of Payments), J 
Partition, thin, flow through, 260. 
Partnership, 37. 

Passage-way in tunnel, 754. 

Passenger 
cars, 811, 812. 

earnings and expenses, 811, 814, Ac. 
locomotives, 805-810. 
mile, 814, Ac. 
train expenses, 815. 

Paste, lime, 670. 













INDEX. 853 

Patterns—Pipe. 


’atterns, weight of castings by size of, 
398. 

’aving, 

Belgian, 668. 
brick, 671. 

’ayments, 
equation of, 37. 

partial (Equation of Payments), 37. 

5 encils, lead, 433. 

’endulums, 364. 
hydrometric, 269. 
seconds, 385. 

jr’ennsylvania R R, 
bridges 

of I beams, 524. 
of riveted girders, 543, Ac, 
standard rolling loads, 546. 
locomotives, 546, 806, 807. 
track-tank, 802. 

I Pentagon, 110. 

Perch 389. 

Percussion, 
center of, 365. 
drills, 653. 

Perimeter. See also Circumference, 
wet, 244, 271. 

Permanent way, 759. 

Permutation, 36. 

Perpendicular, to draw, 55. 

Persian wheel, 379. 

Petroleum, weight, 383. 

Philadelphia, 

Chestnut St bridge, 599. 

foundations, 636. 

South St bridge, 648. 

Wissahickon bridge, 720. 

Phoenix segment-columns,441,442,443,449 
dimensions, weights, Ac, 449. 
strength, 442, 443, 449. 
in trestles, 758. 

Phosphor-bronze wire, strength, 464. 

Picks, wear of, 743. 

Pier, Piers, 
abutment, 699. 
foundations for, 633, Ac. 
masonry, quantity in, 708. 
obstructions by, 2794, &c. 
of suspension bridges, 618, Ac. 

Pierre perdue, 634. 

,Piezometer, 239. 

Piles, foundation, 640. See also Founda¬ 
tions. 

Pillar, Pillars, 439, Ac. 
of angle-iron, 440-442. 
capitals of, shapes of, 457. 
of channel-iron, 441, 442, 456. 
ends of, shapes of, 439, 457. 
factor of safety, 442, 446. 

Gordon’s formula, 439. 
hinged ends, 439. 
of I beams, 441, 442, 454. 
iron, 439, Ac. 

coefficients of safety, 442, 446. 
strains usually allowed, 457. 
strength of, 439, Ac. 
masonry, strength of, 437. 
oblique, 457. 

Phoenix segment, 441-443, 449, 758. 


Pillar, Pillars—continued, 
pin-ended, 439. 
radius of gyration, 440. 
with rounded ends, 439. 
safety, factor of, 442, 446. 
segment, Phoenix, 
dimensions, weights, Ac, 449. 
strength, 441, 442, 443, 449. 
in trestles, 758. 
steel, 442, 458. 
strength of, 439, Ac. 

T and -f iron, 441, 442. 
wooden, 458, Ac. 

Pin, Pins 

and eye-bars, 439, 612. 
surveying, 176. 

Pine, 

pillars of, 459. 

strength. 434e. 436. 458, 465, 476, 493. 

weight, 383, 384. 

Pinions and wheels, 342. 

Pipe, Pipes, 
air-valves for, 297. 
angles in, 256. 

areas and contents of, 157, 247. 
bends in, 255. 
branches in, 296. 
brass, seamless, 417. 
bursting of, 234, 298, 303. 

thickness required to prevent, 232- 
234, 293. 

bursting pressure in, 239. 
cast-iron, 290-293, 297, 399. 

weight, 293, 297, 399. 
cement and iron, 294. 
concretions in, to prevent, 292. 
contents and areas of, 157, 247. 
copper, seamless, 417. 
cost of, 293, 297. 

of laying, 297. 
couplings for, 293, 295, 405. 
cracks in, 296. 
curves in, 255. 

diameters of, 245, 290, 291, 293, 297. 
square roots of, 247. 
for water-supply, 290, Ac. 
discharge from, 
formulae, 243. 
principles of, 236, Ac. 
drain, 279d. 

terra-cotta, 279cL 
drawn brass and copper, 417. 
enlargements in, 257. 
ferrules for, 299. 
flexible joints for, 296. 
flow in, 236, Ac. 

Kutter’s formula, 244, 271. 
friction in, 257. 
galvanic action in, 293. 
galvanized, 299. 
gates for, 301. 
gutta-percha, 294. 
iron, 

cast, 290-293, 297, 399. 

weight, 293, 297, 399. 
and cement, 294. 
fittings for, 293, Ac, 405. 
joints for, 293, 295, 407. 






854 


INDEX. 

PI pe—Pole n t ia 1 


Pipe, pipes, iron—continued, 
thicknesses of, 233, 293. 
wrought, 293, 405. 
weight, 293, 405. 
joints for, 293, 295, 407. 

flexible, 296. 
kalameined, 293. 
knees in, 256. 
laying, 297. 

lead for, 293,295, 297. 
leaden, 234, 299.416. 

thicknesses of, 234. 
long, pressure of water in, 257. 
material of, effect on velocity, 244. 
to mend, 296. 

obstructions in, to prevent, 292. 
paper, 294. 

pressure of water in, 232, Ac, 239. 

resistance to pumping, 257. 

seamless, 417. 

service, 294, 299, 416. 

sleeves for, 296. 

stand, 298. 

steam, 405. 

stop-valves for, 301. 

street, 290. 

swellings in, effect of, 257. 
tapping of, 294, 299. 
terra-cotta, 279d. 
thickness required, 232-234, 293. 
valves for, 301. 

of varying diameter,disch through, 254. 
velocity in, 243, 245, Ac. 
water, 290, Ac, 293, 297, 399. 
weight, 293, 297, 399,405. 
wooden, 294. 

wrought-iron, 293, 294, 405. 

Pit of turntable, 791. 

Pitch 

of rivets, 472, Ac, 539. 
of roofs, 428, 581. 
weight of, 384. 

Pitot’s tube, 269. 

Pittsburgh suspension bridge, 624. 

Pivot 

of turntables, 792, Ac. 

Plan, to reduce or enlarge, 122. 

Plane, Planes, 110. 
of flotation, 235. 
forces in different, 332, Ac. 
forces in one, 319, Ac. 
of gravity, 350. 

inclined, 352, Ac, 363. See also In¬ 
clined Plane, 
surfaces, 110. 
trigonometry, 112. 

Plank, 

board measure, table, 420. 
in foundations, 633. 
sheet piling, 641. 
thickness for a given pres, 637. 

Plant, railroad, 814, Ac. 

Plaster, 426. 
of Paris 

effect on metals, 673. 
price of, 673. 
strength, 437, 466. 
weight, 384. 


Plate, Plates. See Sheet, 
angle, 764. 
beams, 537. 
bolster, 524, 544, 614. 
buckled, 409. 
fish, 764. 
frog, 783. 
girders, 537. 
glass, 432. 
iron 

prices, 402. 
turntables, 793. 
net, 470. 

openings in, flow through, 257, Ac, 

parallel, 189. 

steel, tinned, 418. 

terne, 418. 

tin, 418. 

tinned steel, 418. 
wall, 524, 544, 614. 

Platform 
cars, 811. 
revolving, 799. 

Platinum, 384, 464. 

blasting caps, 665. 

Plenum process, 648. 

Plug, fire (fire-hydrant), 304. 

Plumb level, to adjust, 206. 

Plumbago, as a lubricant, 374c. 
Pneumatic 
foundations, 647, Ac. 

Pocket sextant, 194. 

Point, Points, 

of application of force, 309, 318 f Ac 
boiling, 210, 217. 

leveling by, 209. 
of contrary flexure, 515. 
freezing, 212, 217. 
frog, 781-789. 
melting, 212. 
position of, to find, 118. 
switch, 774-776. 

Pointing mortar, 674. 

Pole star, 177. 

Polygon, Polygons, 110. 
center of gravity of, 348, Ac. 
of forces, 329. 

irregular, to find area of, 122. 
to reduce to a triangle, 121. 
regular, to draw, 121. 

Polyhedron, Polyhedrons, regular, 154 
Pond, discharge of, time required for 2 
Poplar, strength, 436, 463, 493. 

Pound (coin), value of, 386. 
weight, 387. 

Porous bodies, specific gravity of, 381,3 
Port, establishment of, 219. 

Portage viaduct, 756, 757. 

Portland cement, 673. 

Post, Posts. See also Pillars, 
fence, 803. 
king, 554. 

pivot, in turntables, 792, Ac. 
queen, 555. 

Samson, 630. 
in trestles, 756. 
in trusses, 547,549. 

Potential energy, 318 d. 















855 


INDEX. 

Pouring-—Rail. 


Pouring-clamps for pipe-joints, 294, 295. 
Powder, 384, 660. 

[Power, Powers. See Steam, Water, Wind, 
Ac. 

animal, 377. 
defined, 318. 
fifth, 251. 

square roots of, 253. 
gain of, 336. 
of horse, 375, 377, 801. 
of locomotives, 808. 
man, 378. 

second and third, tables, 41. 
tractive, 375. See also Traction. 

Pratt truss, 595. 

Prejudicial work, 318. 

Press, Presses, 
hydrostatic, 227. 

Pressed brick, 670. 

Pressure. See Load, 
of air, 215, 648, Ac. 

barometer, leveling by, 207. 
in arches, 342,359, 694. 

, boiler, 809. 

center of, 227, 235, 347 l , 700. 
on centers for arches, 713. 
cylinder, of steam, 809, &c. 
in dams, 286. 
distribution of, 
in plane surfaces, 231a. 
of earth, 683, Ac, 687. 
effect of, on friction, 371. 
on foundations, 634. 
on inclined plates, 352, Ac, 363. 
initial, of steam, 809. 
line of, 359, 700. 
maximum, angle of, 687. 
prism of, 687. 
slope of, 687. 
in pipes, 232, 239. 
plate, Gauthey’s, 269. 
in reservoirs, 288. 
on retaining walls, 683, Ac. 
of running streams, 279/, Ac. 
of running water in pipes, 239. 
steam, 809. 

strength, compressive, 436. 
transmission of, through liquids, 227. 
of water, 222, Ac, 239, Ac. 
in cylinders, 232. 
in long pipes, 257. 
in pipes, 232, Ac, 239. 
running, 239, 279/, Ac. 
still, 222. 

walls to resist, 229, Ac. 
of wind, 216. 

Prices. See the article in question. 
Prism, Prisms, 155. 
center of gravity of, 351 e, /. 
frustums of, 155. 

center of gravity of, 351 e, g. 
of maximum pressure, 687. 

Prismoid, 160 6. 

Prismoidal formula, 160 6. 

Profile, Profiles, 199, Ac. 
curved, 692. 
paper, 433. 

transformation of, 691. 


Progression, 36. 
arithmetical, 36. 
geometrical, 36. 

Proportion, 35. 
compound, 35. 
simple, 35. 

Protracting by chords, 105. 

Puddle-walls, 288. 

Pulley, 342. 

Pump, Pumps, 801. 
chain, 379. 

day’s work at, 378, 801. 
sand, 626, 650. 

Pumping. See Pump, 
engine, 801. 
mains, friction in, 257. 

Purlins, 551, 582. 

Pyramid, Pyramids, 160. 
center of gravity of, 348, Ac. 
frustum of, 160 a. 
center of gravity of, 348, Ac. 

Q* 

Quadrant, Castelli’s, 269. 

center of gravity of, 351 c. 

Quarrying, 651-667. 

Quartz, weight, 384. 

Queen truss, 555, 578. 

Quicklime, 669. 
to preserve timber, 425a. 
weight of, 383. 

R. 

Rack-a-rook, 664. 

Radii, Radius, 
to find, 123, 141 a. 
of gyration, 366, 367, 439, 440. 

square of, 440, 538, 540. 
mean, 244, 272. 

of railroad curves, tables, 726-728. 
of turnouts, 786. 

Rafter, Rafters, 551, Ac, 582. 

feet of, friction at, 355. 

Rail, Rails, 760, 763. 
bending, ordinates for, 761. 
creeping of. 763, 764. 
elevation of outer, 729. 
expansion by heat, 212, 763. 
fence, 803. 
frog, 781. 

guard or guide, 774, 779, 781. 
joints, 763. 

ordinates for bending, 761. 
outer, elevation of, 729. 
renewals, 815. 
roads, 722. 
ballast, 759. 

bridges. See Bridge, Truss, Arch, Ac 

cars, 811, 812, 814, Ac. 

construction, 722, 804. 

cost of, 804. 

cross-ties, 759. 

gauge, in U. S., 814. 

pile-driver, 642. 

resistance on, 374e. 

roadway, 759. 







856 


INDEX. 

Rail—Rock 


Rail, Rails—continued, 
roads, 

shops, cost of, 799. 
spikes, 7(32. 
statistics, 814 to 818. 
switch, 770. 
ties, 759. 

time, standard, 396. 
track-tank, 802. 
traction on, 377, 808, 810. 
turnout, 770. 
water stations, 800. 
safety, 779. 
stock, 774. 

switch-rail, length of, 776. 
way, see Rail-road. 

Rain, 220. 

reaching sewer, rate of, 279c. 
water, 218, 385. 

Ram, hydraulic, 28C. 

water, 234, 298, 303. 
Ramming 


of cement and concrete. 675, 680. 
Ramsbottom’s track tank, 802. 

Rand rock-drill, 656. 

Random stone, 634. 

Range of stress, 435. 

Ratio, 35. 

elastic, 434 d 
Reuc ion, 308, Ac. 
of soils, elastic, 644. 

Real estate, maintenance, railroad, 815. 
Reaumur thermometer, 213. 

Re-burning of cement, 674. 

Rectangle, Rectangles, ir9. 

center of gravity of. 348, Ac. 
Reflection, to measure heights by, 117. 
Reflexion, angle of, 255. 

Refraction and curvature, table, 115. 
Regular figures, 110. 

Regular solids, 154. 

Repair, Repairs, 
of bubble-tube, 193. 
of cars, 811, 815. 
of cross-hairs, 193. 
of locomotives, 810, 815. 
of pipes, 296. 
in reservoirs, 289. 
of road, 743. 

of road-bed, railroad, 815. 
of rolling-stock, 810,811, 815. 
of track, 815. 

Repose, angle of, 355, 371. 

Reservoir, Reservoirs, 287, Ac. 
discharge from and into, 262. 
evaporation from, 222. 
for railroads, 801. 

Resilience, 434/. 

Resistance. See also Loads, Strength Ac 
angle of, limiting, 355, 371. 
of beams, 484. 
of cars, 374e,. 808. 
coefficient of, 485. 
to flow, 244, 255, 257, 271, Ac. 
of friction, 370. 
on grades, 808. 
limiting angle of, 355, 371. 
line of, 359, Ac, 700. 


Resistance—continued, 
modulus of, 485. 
moment of, 484, 486. 4S8. 
on railroads, 374e. 

Resolution of forces, 319, Ac. 
Resultants, 319, Ac. 

Retaining walls, 683. 
clay backing, 634. 
curved profiles, 692. 
masonry in, quantity of, 690, 692. 
surcharged, 685, Ac. 
theory of, 686. 

transformation of profile, 691. 
Retarded velocity, 3u7. 

Reverse bearing, 171. 

Revetment, 692. 

Revolving bodies 365. 

Rhomb, 119, 155. 

Rhombic prism, 155. 

Rhombohedron, 155. 

Rhomboid, 119. 

center of gravity of, 348, Ac. 
Rhombus, 119. 

center of gravity of, 348 Ac. 
Rhumb-line, 171 
Right angle, to draw, 55. 

Rigid bodies, force in, 306. 

Ring, Rings. See Circle, Ellipse, Ac. 
circular. 146, 167. 
joint, 768. 
tightening, 583. 

Rip-rap, 634. 

Rise of arch, 693. 

of roof, effect on weight, 581. 
Rivers. See Water, Rain, Ac. 
dams, 282. 
flow in, 268, Ac. 
reservoirs, 287. 
scour of, 279 f. 

Rivet, Rivets, 468, 539. 

Riveted 
beams, 537. 
girders, 537. 
joints, 468, 539. 

Road, Roads, 

-bed, repairs, 815. 
cart, repairs, 743. 
grade, 375, 723. 

tables, 176, 354, 723-725. 
maintenance, 743, 815. 
rail-. See Railroad, 
traction on, 375. 

-way, acres required for, 722. 
drainage of, in arches, 708. 
railroad, items of, 759. 
width of, in bridges, 604. 

Rock, Rocks, 
blasting, 660. 

broken, voids in, 380, 678,751. 
channeling, 658. 
drill, 651, Ac. 
band, 658. 
machine, 652. 
steam, 652. 
removal, 751, Ac. 
strength of, 434, Ac. 
weight of, 381, Ac. 
work in tunnels, 754. 














INDEX. S57 

Rockers—Sand. 


Rockers, expansion, 614. 

Rod, Rods, 

of brickwork, 389, 672. 
iron, 402. 
i king, 553. <fcc. 
queen, 555. 
suspending, 620. 

) tie, 551, &c, 572. 
upset, 408. 

Rolled 

j iron. See Iron, Wrought. 

Roller, Rollers, 
anti-friction, 374e. 792, &c. 
expansion, 614. 
friction, 374e, 792, &c. 

Rolling 
friction, 3745. 
load, 546, 561, 805-807. 
stock, 805, &c, 814, 815, etc. 

repairs, 815. 
resistance of, 374e. 

Roof, Roofs, 
arched, 600. 
copper, 416. 
cost of, 580. 

coverings, weight of, 551, &c, 581. 
details, 582. 

effect of acid fumes on, 418, 428. 

of rise on weight of, 581. 

Fink, 574, 578-580. 
frost-proof, Burnham’s, 801. 
iron for, 403. 
arch, 600. 
details, 582, 583. 
lead, 415, 416. 
leak in, to stop, 429, 431. 

Moseley, 600. 

painting of, 403,430. 

pitch of, 428,581. 

purlin, 551, 582. 

rafter, 355, 551, &c, 582. 

rise of, effect on weight of, 581. 

sheet-iron, 403. 

shingle, 429. 

slate, weight of, 428. 

snow on, 221. 

to stop leak, 429, 431. 

tin, 418. 

truss. See Truss. 

weight of, affected by rise, 581. 

with load, 580. 
wind on, 216. 
wooden, details, 613. 
zinc. 418. 

Root, Roots, 40. 
cube and square, tables, 40. 
of decimals, to find, 53. 
fifth, 251. 

of large numbers, to calculate, 52. 
square and cube, tables, 40. 
square, of diameters, 247. 
of fifth powers, 253. 

Rope, Ropes, 414. 
strength of, 414, 466. 
weight of, 414. 
wire, 413. 

Rosendale cement, 673. 

Rosin, weight, 384. 


Rot of timber, 425. 

Rotary drills, 652. 

Rotary motion, 365. 

Rotating bodies, 365. 

Rough-casting, 431, 674. 

Roughness, 

coefficient of, 244, 272, 273. 

Rubble, 

adhesion to mortar, 676, 677. 
arches, 684, 696. 
cost, 668. 
foundations, 634. 
proportion of mortar in, 383. 
quarry, loose, 380, 751. 
retaining walls, 690u 
strength, 437. 
voids in, 669, 741- 
weight of, 383. 

Rule, Rules, 
of three, 35. 

two-fbot, to measure angles by, 58. 
Rupture. See Strength- 
coefficient of, 485. 
constant of, 485. 
modulus of, 485. 
moment of, 479-484. 

Russian weights and measures, 394. 

s. 

Safety, 

allowance for. See Safety, Factor of. 
castings, 770, 779. 

coefficient of. See Safety, Factor of. 
factor of, 

for beams, 499, 521, 540. 
for piles, 644. 
for pillars, 442, 446. 
for retaining walls, 685. 
for suspension bridges, 617. 
for truss bridges, 607. 
rail, 779. 

switch, 770, 775, 778. 

Salmon brick, 671. 

Salt, 

effect on mortar, 670, 678. 
water, effect on iron, 218, 645. 

weight of, 217. 
weight of, 384. 

Sauison 
joint, 765. 
post, 630. 

Sand # 

augers, 626, See. 
bar sand, 669. 
blasting of, 663. 
in cement, 674, 676, 678, 679. 
for centers, striking, 713. 
in concrete, 678. 
cost of, 669. 
dredging, 63L 

effect on cement, 674, 676,678. 
excavating in, 742, &c. 
for foundations, 634. 
load of, 427. 

for mortar, 669, 677, 678. 
natural slope of, 690. 
penetrability of, 644. 
piles, 626, 650. 









858 


INDEX. 


Sand-Single. 


Sand-—con tinned, 
in plaster, 426. 
pressure of, 683, Ac. 
price of, 669. 
pump, 626, 650. 
retaining walls for, 683, Ac. 
slope, natural, 690. 
specific gravity of, 381, 384. 
-stone, 

expansion by heat, 212. 
strength of, 437, 466, 493. 
weight, 384. 

sustaining power of, 634,644. 
voids in, 384,677. 
weight of, 381, 384. 

Sap of timber, 425. 

Saylor’s Portland cement, 673. 
Scales, track, 803. 

Scarfs, lengthening, 610. 

Scarp revetment, 692. 

Scoop, tender, 802. 

Scour of streams, 279/. 

Scrapers, earthwork, 147. » 

Screeding, 426. 

Screw, Screws, 542. 

Archimedes, 379. 
for centers, striking, 713. 
cylinders, 645. 
leveling, 189, 202. 
piles, 645. 

standard dimensions, 406. 
Seamless pipes and tubes, 417 
Sea 

mile, 387. 
tides, 219. 

water, 217, 219, 645. 
worms, 425. 

Secant, 59. 
of latitudes, 177. 

Seconds it) decimals of a degree, 57 
Seconds pendulum, 385. 

Section 


effective, in riveted girders, 538 
of land, area of, 389. 

Sector, circular, 146. 
center of gravity of, 351 e. 
spherical, center of gravity, 351,/. 

Sediment in reservoirs, 2>8. 

> Segment, Segments, 

circular, 146, 147, 351 d. 
columns, Phtenix, 441, 412. 449, 758 
spherical, 166. 
center of gravity of, 351 e,f. 

Self-acting frog, 785. J 

switch stand, 775. 

Sellers standard dimensions of bolts A 
406. 

Sellers turntable, 792. 

Semi-circle, center of gravity, 351 a c, 

8emi-parabola, center of gravity of 351 

Separators for beams, 5236, d. ' 

Service pipe, 294, 299, 416. 
insertion of, 294, 299. 

Setting of cement, 674. 

Settlement 
of backing, 684. 
of centers, 713, Ac, 720. 
ol embankment, 741. 


Sewer, Sewers, 
flow in, 279c. 

Kutter’s formula, 244, 271. 
grade of, 279c. 

rain-water, rate of reaching, 279c. 
velocities in, 279c. 

Sextant, 

angles measured by, 114. 
box or pocket, 194. 
center of gravity of, 351 c. 

Shaft of tunnel, 754. 

Shafting, 
deflection of, 510. 
friction of, 374d. 
strength of, 477. 

Shale, weight, 384. 

Sharpening tools, cost, 743. 

Shear, 47 6, 532. 

double and single, 470, 476. 

Shearing 
of beams, 532. 
of nails, 4256. 
of rivets. 470, Ac. 
strains, 632. 
strength, 476. 

Sheet, Sheets, 
brass, 415. 
copper, 415, 416. 
iron, 403. 
corrugated, 403. 
galvanized, 103. 
roof, 403. 
lead, 415, 416. 

metals, thickness of, 410, 411. 
piles, 641. 
zinc, 418. 

Sheeting of centers, 711, 719. 

-lime, 670. 
spherical, 166. 
weight of, 398, 400. 

Shilling, value of, 386. 

Shingles, 429. 

Shoes for piles, 644. 

Shops, railroad, cost, 799. 

Shoveling earth, 742. 

Shovels, wear of, 743. 

Shrinkage of embankment, 741. 
Siemens’ electrical blasting machine. 665 
Sieves for cement, 678. 

Signal target, 772, Ac, 775, 779. 

Silver, 

coins, Ac, 387. 
strength, 466. 
weight, 384, 387. 

Similarity, geometrical, 54. 

Simple 

proportion, 35. 
interest, 37. 

Simultaneous firing of blasts, 665. 

Sine, Sines, 59. 
natural, defined, 59. 
table, 60. 

of polar distances of Polaris, 177. 
Single 

riveting, 468. 

rule of three, 35. , 

shear, 470. 
















INDEX. 859 

Siphon—Statics. 


Siphon, 241. 

Skew-back, 693. 

Skidding of w heels, 374a. 

Slacking of lime, 669, 670. 

Slate, 427. 

compressibility, 434 e. 
expansion by heat, 212. 
roofs, weight of, 428. 
strength, 

compressive, 437. 
tensile, 466. 
transverse, 493. 
weight, 384. 

Slating, 427. 

Sleeping cars, 811, 812. 

Sleeves for pipes, 296. 

Sliding, 370. 
angle of, 355, 371. 
friction, 370, &c. 
of retaining walls, 692. 

Slope, Slopes. See Grade. 

allowance for, in chaining, 176. 
angle of, 176, 354, 723, 724. 
hydraulic, 244, 272. 
instrument, 206, 724. 
of maximum pressure, 687. 
natural, 684, 686, 690 
tables, 176, 354, 723-725. 
in tunnels, 754. 

Slugger rock-drill, 656. 

Sluices in dams, 285. 

Snow, 221, 384. 

Soakage, loss by, 269. 

Soap as a lubricant, 374c. 

Soapstone, weight, 384. 

Soap-wash for walls, 672. 

Soffit, defined, 693. 

Soil, Soils, 

boring in, 626. 
dredging, 631. 
excavation of, 742. 
leakage through, 222. 
penetrability of, 644. 
pressure of, 683, &c. 
reaction of. elastic, 644. 
scour of, 279/. 

sustaining power of, 634, 644. 
weight, 382, under *• earth.” 

Solid, Solids, 154. 

center of gravity of, 351 f. 
defined, 54. 

expansion by heat, 212. 
measure, 389. 

metric, 392. 
mensuration of, 154. 
specific gravity of, 380, &c. 

Sound, 211. 

South St bridge, Pnila, 648- 

Southing,168. 

Sovereign, 386. 

Span, defined, 693. 

Spandrel, 693. 
walls, 693, 698. 

Spanish weights and measures, 394. 

Specific gravity, 380. 

Speed, Speeds. See Velocity, 
of locomotives, 809. 
of teams, 743, 747. 


Spelter. See Zinc. 

Sphere, Sphere*, 162,163. See Spherical, 
center of gravity of, 348, &c. 
volume of, 389. 

Splierical 

sector, 

center of gravity of, 351/. 
segment, 166. 

center of gravity of, 351 e,f. 
shell, 166. 

weight of. 398, 400. 
zone, 166. 

center of gravity of, 351/ 

Spheroid, 166. 

center of gravity of, 351 e,f. 

Spigot in pipe-joint, 295. 

Spindle 

circular, 167. 
torsional strain in, 477. 

Spikes, 762. 

Splices, timber. 610, 612. 

Split switch, 774-776. 

Spreading of earth, 743. 

Spring, Springs, 
of arch, 693. 
in foundations, 634. 
frog, 784. 

Spruce, 

strength, 434 e, 436, 463, 476, 493. 
weight, 384. 

Spudding, 628. 

Square, Squares, 
area, 119. 

equivalents of, in circles, 123. 
center of gravity of, 351 a. 
measure, 389. 

metric, 392. 
mensuration of, 119. 
of numbers, table, 41. 
of radius of gyration, 440, 538, 540. 
roots, 40. 

of decimals, to find, 53. 
of diameters, 247. 
of fifth powers, 253. 
of large numbers, to calculate, 52. 
tables, 40. 
sides of, 119, 123. 
tables of, 41. 

Stability, 235, 356. 

of arches, 358, &c, 700. 
frictional, 352-361. 
on inclined planes, 352, &c. 
moment of, 357. 
of retaining walls, 683. 

Stable eqtiilibrium, 235, 348, 358. 
Stadia hairs, 190. 

Stand-pipes, 298. 

for railroad water-stations, 801. 
for water-works, 298. 

Stand, switch, 772. 

Standard railway time, 396. 

Star, Stars, 

Aliotli, 177. 

iron, sizes and prices, 402. 

North, 177. 

Pole, 177 . 

to regulate a watch, &c, by, 395. 
Statics, 306. 








S60 


INDEX. 

Station—Strength. 


Station, Stations, 
expense, 815. 
in surveys, 197, Ac, 204. 
water, 800. 
way, cost, 803. 

Statistics, 
arches. 095. 
railroad, 814-818. 
rainfall, 220. 

Stays, cable, 616. 

Steam, 

average pressure, 809. 
boiler pressure, 809. 
cylinder pressure, 809. 
dredges, 031. 
engine, 

locomotive, 805-810. 
pumping, 801. 
excavator, 750. 
expansion of, 809. 
initial pressure of. 809. 
pile drivers, 641, 642. 
pipes, 405. 
pressure, 809. 
rock-drill, 652. 

warming by, surface required, 399. 
Steel ’ 

beams, 493, 512, 523 c, d. 
buckled plates, 409. 
chains, strength, 415. 
cohesive strength of, 464, 465. 
columns, 442, 458. 
compressibility, 434 e. 
compressive strength, 438. 
cost of, 402. 
ductility, 434 e. 
elastic limit, 434 e. 
expansion by heat, 212. 
friction of, 373, Ac. 
modulus of elasticity, 434 e. 
nails, 4256. 
pillars, 442, 458. 
plates, 

buckled, 409. 
tinned, 418. 
price, 402. 
rails, 760. 

frogs of, 781. 
rope, 413. 

shearing strength, 476. 

strength, 438, 464, 465, 476, 477, 493 

stretch of, 434 e. 

tensile strength, 464, 465. 

ties, 700. 

tires, 807, 812, 813. 
torsional strength, 477. 
transverse strength, 493. 
weight, 384, 400, 401. 
wire, 412. 
rope, 413. 

Sterling’s toughened cast-iron, 520. 
Stiffeners for plate-girders, 539. 

Stock rails, 774. 

Stone, Stones, 
adhesion to cement, 677. 

mortar, 070. 
arch-, 693. 

arches, quantity in, 702. 


Stone, Stones—continued, 
artificial, 466, 678, 681. 
ballast, 759, 804. 
beams, 493, 504. 

-breaker, 680. 

bridges, 693. See also Arch. 

centers for, 711. 
broken, voids in, 380, 678, 751. 
buildings, cost of, C68. 
cohesive strength, 466. 
compressive strength, 437. 
for concrete, 678, Ac. 
crusher, 680. 

-cutter, day’s work, 667. 
dams, 229, 231. 
dressing, 667. 
drilling, 651, Ac. 
excavating, 751. 
expansion by heat, 212. 
friction of, 373, Ac. 
key-, 693. 

quantity in arches, Ac, 702, Ac. 
quarrying, 651-667. 
random, 634. 

Kan some’s, 466. 
strength, 437, 466, 493. 
tensile strength, 466. 
transverse strength, 493. 
weight, 381, Ac. 

-work, 651, 751. 
strength, 437. 
weight, 229, 383. 

Stop, Stops, 

corporation, for pipes, 294, 299. 
leak in roof, 429, 431. 

-valves for water-pipee, 301. 

Storage reservoirs, 289. 

Strain, Strains, 318 h, Ac., 434, Ac. 
in beams, vertical, 532. 
flange, 529, 537. 
repeated, 435. 
shearing, 532. 

in suspension bridges, 616, Ac. 
in trusses, 551, Ac. See also Tmeses, 
vertical, in beams, 532. 

Stream, Streams, 
abrasion by, 279/. 
flow in, 268. 
to gauge, 268. 
horse-power of, 280. 
pressure of running, 279 f. Ac. 
scour of*279/. 
virtual head, 280. 

Street pipes, 290, Ac. See also Pipes. 

Strength, Strengths. See also the article 
in question, 
of arches, 693, Ac. 
of beams, 478,493. 
of bridges. See Arch, Truss, Ac. 
cohesive, 463. 
compressive, 436, Ac. 
of cylinders, 232, 516. 
of materials, 434. 
of piles, 643. 
of pillars, 439, Ac. 
of retaining walls, 683, Ac. 
of riveted joints, 468, Ac, 472. 
of shafting, 477. 










861 


INDEX. 

Strength—Tie. 


Strength, Strengths—continued, 
shearing, 476. 
tensile, 463. 
torsional, 476. 
transverse, 478. 

Stress,defined,318 h .See Strain, Strength, 
range of, 435. 
repeated, 435. 

Stretch 

ut materials, 434 e. 

Striking of centers, 711, 713, 720. 
Stringers, track, 545, Ac. 

Strut, Struts, 
defined, 325, 547, 554. 

-tie, defined, 325, 547. 

Stub’s gauge, 411. 

Stub switch, 770, 771. 

Stucco, 426, 674. 

Stumps, blasting of, 663. 

Sub-delivery, cost, 804. 

Subterranean temperature, 215. 
Suddenly applied loads, 
deflection under. 434 > /‘. 

Sulphur, weight, 3S4._ 

Sun-dial, to make, 397. 

Supplement of angle, 56. 

Supported joints, 763. 

Surcharge, 685, Ac. 

Surface velocity, 268. 

Surveying, 168. 

Suspended joints, 763. 

Suspenders of suspension bridges, 620. 
Suspension 

bridges, 615, Ac. 

cables of, 412, 615, Ac. 
links, 614. 
trusses, 548. 

.Sway-bracing, 543. 

Swing bridges, strains in, 593. 

Switch, Switches, 770, Ae. 

Swivel. 583. 

Sycamore, 

strength, 434e, 436, 463, 493. 
weight, 384. 

Symmetry, axis of, 235. 

Syphon, 241, Ac. 

System, metric, 391. 

Systems 
aucien, 393. 
usuel, 393. 


T. * 

T iron, 442, 525. 

rails, 760, 763. 

Table, turning, 790. 

Tables. See the article in question. 
Tallow, 

as a lubricant, 374c. 
weight of, 384. 

Talus, 692. 

Tamping, 661, Ac. 

Tangential 

angle, table, 726-728. 
distance, table. 726-728. 

Tangent, Tangents. 59, 60. 
to circles, to draw, 124. 
to an ellipse, to draw, 150. 


Tangent, Tangents—continued, 
natural, 59. 
table, 60. 

to a parabola, to draw, 153. 
screw, 190, 202. 

Tank, 

frost-proof, 801. 

of tender, capacity of, 805, Ac, 808. 
thickness, 227, 803. 
track, 802. 
water, 800, Ac. 

Tapping of pipes, 294, 299. 

Tar, weight, 384. 

Target, signal, 772, Ac, 775, 779. 
Tarpaulin, 680. 

Taxes, railroad, 815. 

Teams, speed of, 743, 747. 

Telegraph expense, annual, 815. 
Temperature, 212. See Heat, 
of air, 215. 

altitude, effect on, 215. 
effect on 
cement, 674, 675. 
evaporation, 222. 
metals, Ac, 212. 
rails, 212, 763. 
rainfall, 220. 
strength of iron, 466. 
surveying chains, 168. 
trusses, 614. 
velocity of sonnd, 211. 
weight of water, 217, 385. 
subterranean. 215 
thermometers 213. 

Tender, Tenders, 546, 805-810. 

-scoop, 802. 
weights, 546, 805. Ac. 

Tensile strength, 463. 

of idveted joints, 472. 

Teredo, 425. 

Terne plates, 418. 

Terra-cotta pipes, 279d. 

Test 

borings, 626, 633. 
of cements, 674, 675. 
of instruments, 191-206. 
Tetrahedron. 154. 

Theodolite, 193. 

Thermometers, 213. 

Thilmany process, 425a. 

Thin partition, flow through. 260. 
Three, rules of, 35. 

Three throw switch. 776. 

Three-way valves, 302. 

Throat of frog, 781. 

Throw of switch, 773. 

Thrust, line of, 359, Ac, 700. 

Tides, 219. 

Tie, Ties, 

-beam, 551, Ac. 
cross, 759, 804, 815. 
defined, 325, 547. 
land-, 692. 
rod, 551, Ac. 

raised, 572. 
steel, 760. 

-strut, defined, 325, 547, 

and strut, to distinguish, 325, 590. 






862 


INDEX. 

Timber—Truss. 


Timber. See also Wood, Wooden, 
beams, deflections, 499. 
loads, 499, 502, 512, 513. 
for railroad bridges, 514. 
board measure, table, 420. 
cohesive strength, 463. 
compressibility, 434 e. 
compressive strength, 436. 
cost, 425. 
creosoting, 425. 
crushing strength, 436. 
dams, 282. 
ductility, 434 e. 
durability, 425. 
elastic limit, 434 e. 
friction of, 373, Ac. 
joints, 610, 612. 
limit, elastic, 434 e. 
modulus of elasticity, 434 e. 
preservation of, 425. 
shearing strength, 476. 
splices, 610, 612. 
strength, 436, 463, 476, 477, 493. 
stretch of, 434 «. 
tensile strength, 463. 
for ties, 759. 
torsional strength, 477. 
transverse strength, 493. 
trestles, 755. 
turntables, 797. 
weight, 381, Ac. 

Time, 395. 

-piece, to regulate by star, 395. 
standard railway, 396. 

Tin, 418. 

compressibility, 434 e. 
ductility, 434 a. 
elastic limit, 434 e. 
expansion by heat, 212. 
leaded, 418. 

modulus of elasticity, 434 e. 
roofing, 418. 
strength, 438, 466. 
stretch of, 434 e. 
weight, 385, 400. 

Tinned steel plates, 418. 

Tire, 

car-wheel, 811, 812. 
locomotive, 807. 
wagon, 380. 

Toe of switch, 771, 774, 785. 

Ton, x, 387. 

of coal, volume of, 389. 

-mile, S09, Ac, 814, Ac. 

Tongue 
of frog, 780. 

-switch, 774. 

Tonite, 664. 

Tools, wear of, 743. 

Top heading, 754. 

Torpedoes, nitro-glycerine, 661. 
Torsion, 476. 

Toughened cast-iron, Sterling’s, 520. 
Towers of suspension bridges, 618,620 
valve towers, 289. 

Town’s truss, 596. 

Tracing-cloth and paper, 433. 

Track. See Kail, 
gauge of, 773 814. 


Track—con t i nued. 
laying, cost, 804. 
repairs, 815. 

-scales, 803. 
stringers, 545, Ac. 
tank, 802. 
trough, 802. 

Traction, 375. 
of cars. 808. 
on grades, 808. 
of horses, 375, 377. 
of locomotives, 808, 809. 

Trailing switch, 770. 

Train, 

earthwork, 749. 
expenses, 815. 

-mile, 809, Ac. 
weight, 546, 564. 
of wheels, 342. 

Transformation of profile, 691. 

Transit, the engineer’s, 188. 
Transmission of pressure in liquids, 227. 
Transverse 
girders. 545, Ac. 
strength, 478. 

Trap rock, weight, 384. 

Trapezoid, 120. 

center of gravity of, 348, Ac. 
Trapezium, 120. 

center of gravity of, 348, Ac. 

Traverse table, 180-187. 

Tread, 765. 
of car-wheel, 765. 

-wheel, 378, 641. 
of wheel, 380,765. 

Trees, blasting of, 663. 

Trembling of dams, 285. 

Tremie, 680. 

Trenton wire gauge, 412. 

Trestles, 755. 

Triangle, Triangles, 110. 
center of gravity of, 348, Ac. 
in or about a circle, 123. 
of forces, 330, 588. 
mensuration of, 110. 
right-angled, 112. 

Triangular truss, 558. 

Trigonometry, plane, 112. 

Tripod, 189. 

Trough, 

flow through, 263. 
track, 802. 

Troy weight, 385. 

Trunnion, friction of, 374d 
Truss, Trusses, 547. 

Boll man, strains in, 5S6. 
bow-string, strains in, 588, 597. 
braced arch, strains in, 592, 598 
bracing, lateral, 610. 
bridge, arrangement of, 603, Ac. 

Burr, 601. 
camber, 607. 
cantilever, 593. 
for centers, 716. Ac. 
chords of, 550, 612. 
contraction by cold, 614. 
cost, 580. 

counter-bracing, 564. 
crescent, strains in, 588. 










863 


INDEX. 

Truss—Velocity. 


Truss, Trusses—continued, 
diagrams, 551, Ac. 
dimensions, for bridges, 595-605. 
distance apart in bridges, 609. 
erection of, 608. 

expansion by heat, allowance for, 
614. 

rollers, 614. 
eye-bars and pins, 612. 
factor of safety, 607. 
false-works, 608. 

Fink 

bridge, arrangement of, 603. 
strains in, 584. 

roof compared with king and queen, 
578. 

strains in, 574. 
floor girders, 610. 
forces acting on, 551. 
headway, 609. 

horizontal bracing, 542, 610. 

horizontal strains in, 562, 591, Ac. 

Howe, 594. 

joints, 583, 610, 612. 

king and queen, 551, Ac. 

compared with Fink, 578. 
lateral bracing, 610. 
lattice, 596. 

loads on, greatest probable, for bridges, 
606, 623. 

moving, 546, 564, 805-807. 

Moseley, 600. 

moving loads on, 546, 564, 805-807. 
obliques, 

best inclination for, 548. 
to find length of, 122, 608. 
overturning tendency, 609. 
panel, defined, 548. 

length of, to find, 548. 
pins and eye-bars, 612. 
posts in, 549. 

Pratt, 595. 

purlins, 551, 582. 

queen compared with Fink, 578. 

rafters of, 355, 551, Ac, 582. 

raising of, 608. 

rise of, effect on weight, 581. 

rollers, expansion, 614. 

roof, details, 582. 

forms of, 551, Ac, 570, &c. 
strains in, 551, &c, 570, &c. 
safety, factor of, 607. 
splices, 610, 612. 
strains in, 551, Ac. 
suspension, 548. 
in suspension bridges, 615. 
suspension links, 614. 
of swing bridges, strains in, 593. 
tendency to overturn, 609. 
tie-beams in, 551, &c. 

Town’s lattice, 596. 
triangular, 558. 
verticals in, 549. 

Warren, 558. 

counter-bracing, 568. 
weight of, affected by rise, 581. 
weights of, for bridges, 605. 
for roofs, 573, 578-580. 


Tube, Tubes. See also Pipes, 
boiler, 405. 

brass, seamless drawn, 417. 
bubble, to replace, 193. 
copper, seamless drawn, 417. 
flow in, 236, &c. See Pipes, Flow, Ac. 
iron, 405. 

Pitot’s, 269. 

pressure of water in, 232, Ac, 239. 
seamless, 417. 
short, flow through, 259. 
welded, 405. 

Tumbling lever, 772, 776, 779. 

Tunnel, 754. 

Turf, weight, 385. 

Turn-buckle, 583. 

Turnouts, 770,785. 

Turnpike, grades on, 723. 

Turntables, 790. 
cast-iron, 792. 
wooden, 797. 
wrought-iron, 793. 

Turpentine, 430. 

Tyler switch, 770. 

Tympan, 379. 

u. 

Cndecagon, 110. 

Ungula. cylindric, 159, 351 g. 

Uniform velocity, 307. 

Unit 

of rate of work, 318. 
of work, 316. 

United States 
cements, 673, 679. 

measures, to reduce to British, and 
vice versa, 390. 
railroad statistics, 814, Ac. 
standard dimensions of bolts, Ac, 406. 
Unstable equilibrium, 235, 348. 

Upright switch-stand, 772. 

Upset rods, 408. 

Useful work, 318. 

V. 

Vacuum process for sinking cylinders, 
647. 

Valve, Valves, 
air, 297. 
four-way, 302. 
outlet, 290. 
stop, 301. 
three-way, 302. 

-tower, 289. 

for water-pipes, 301. 

Variation 
of compass, 196. 
line of no, 197. 
vernier, 193. 

Vegetation in reservoirs, 289. 

Vehicles, friction of, 3745. 

Vein, contracted, 258, 260. 

Velocity, Velocities, 
of abrasion, 279/. 
affected by material of pipe, 244. 
accelerated, 307. 
through adjutages, 259. 
angular, 365. 








ot>4 INDEX. 

Velocity—Water. 


Velocity, Velocities—continued, 
through apertures, 257, &c. 
in channels, 268, Ac. 

Kutter’s formula, 271. 
defined, 307. 
effect of, on friction, 374. 

of material of pipe on, 244. 
of falling bodies, 258, 362. 
head, 237. 

for a given velocity, to find, 248. 
imparted gradually, 311. 
on inclined planes, 363. 

Kutter’s formula, 244, 271. 

material of pipe, effect on, 244. 

mean, 243, 268, Ac. 

of outflow, 258. 

in pipes, 243, Ac, 245, Ac. 

retarded, 307. 

in rivers, 268, Ac. 

in sewers, 279c. 

in short tubes, 259. 

of sound, 211. 

theoretical, of outflow, 258. 

of trains, 809. 

uniform, 307. 

virtual, 339. 

of water, 236, Ac. 

of wind, 216. 

Vena contracta, 258, 260. 

Ventilation. 

air, quantity required, 215. 
of tunnels, 754. 

Venturi meter, 260. 

Vernier, 190. 

variation, 193. 

Verrugas viaduct, 758. 

Versed sines, 59. 

Vertical, Verticals, 
defined, 54. 
of buoyancy, 235. 
of equilibrium, 235. 
of flotation, 235. 
strains in beams, 532. 
in a truss, 547. 

Vessel, Vessels, 
air, 298. 

contents of, 154, Ac, 390, Ac. 
floating, 235, 236. 

metallic, effect of water on, 218, 419 
Viaduct, 

Crumlin, 756. 

Genesee, 756,757. 

Kinzua, 758. 

Lock-Ken, 599, 647. 

Portage, 756. 757. 

Verrugas, 758, 

Vibrating bodies, 364. 

Vibration, 364. 

Virtual 

head, 258, 280. 
velocities, 339. 

Vis viva, 318 a. 

Voids 

in broken stone, 380, 678, 751. 
in concrete, 678. . 
in rubble, 669, 741. 
in sand, 384, 677. 

Voussoir, 693. 


w. 

Wages, locomotive, 810, 815. 

Wagons, friction of, 3745. 

Wall, Walls, 
backing of, 683. 
battered, 685, Ac. 

bricks, number iu a sq ft of, 669-671. 

cost, 667, 668. 

dam, 229, &c, 282, 287. 

face, 683. 

foundations for, 633, Ac. 

incrustation of, 673, 678. 

offset, 683. 

plates, 524, 544, 614. 

of reservoirs, 287. 

retaining, 683. 

soap-wash for, 672. 

spandrel, 693, Ac. 

stability of, 229, 686. 

surcharged, 685, Ac. 

water, to render impervious, 672. 

to resist pressure of, 229, Ac. 
wharf, 236, 691. 
wing, 704. 

Walnut, 

strength. 436, 463, 493. 
weight, 3S5. 

Ward’s flexible pipe-joint, 296. 

V arming by steam, surface required, 399. 
Warren truss, 558, 568. 

Washers, 406. 
lock-nut, 408. 

Washes for walls, 430, Ac, 672. 

W ashington monument, concrete, 679. 
Waste, 

for locomotives, cost, 810,815. 
of water, in cities, 287. 
weir for reservoirs, 289. 

Watch, to regulate by star, 395. 

Water, 217. 

for boilers, top of 218. 

boiling, to measure heights by, 209. 

buoyancy, 234, Ac. 

brick-work, to render impervious, 672, 

in cement, quantity required. 675, 678. 

cisterns, 233, 800-803. 

column, 801. 

compensation, 290. 

composition of, 217, 

compressibility of, 217. 

concrete under, 680. 

corrosion by, 218, 645. 

dams for, 229, 282, 

discharge. See Discharge. 

effect 

on cement, 673, Ac. 
on dynamite, 663. 
on iron, 218, 646. 
on lime, 669, 670. 
of zinc on, 219, 419. 
evaporation, 222. 
flow of. See Flow, 
foundations in, 634, Ac. 
freezing of, 217, 219. 
friction of, in pumping mains, 267. 
gates, 301. 
horse-power of, 280. 








INDEX. 

W ater—W ood 


865 


Water—continued, 
jet, for pile-driving, 646. 
leakage, 222, 269, 282, 288. 
for locomotives, 218, 801-810, 815. 
masonry, to render impervious, 672. 
meters, 270. 
momentum of, 234. 
pipes, 290, &c. See also Pipes, 
in pipes. See Pipes, Velocity, Flow, 
Discharge, Pressure, &c. 
power, 280. 
pressure, 222, &c, 239. 
in cylinders, 232. 
in pipes, 232, <tc, 239, 257. 
running, 279/, Ac. 
still, 222. 

walls to resist, 229, &c. 
quantity required in cities, 287. 
rain, 218, 385. 
ram, 234, 298, 303. 
reservoirs for, 287, &c. 
resistance to moving bodies, 280. 
running, pressure of, 279/, &c. 
salt, effect on metals, 218, 645. 
scouring action, 279/. 
size of commercial measures by weight 
of, 391. 
stations, 800. 
storage of, 288. 
supply, 287, &c. 
tank, thicknesses, 803. 
traction on, 375. 

in tubes, flow of, 236, &c. See also Flow. 

velocity. See Velocity. 

walls, to render impervious, 672. 

to resist pressure of, 229, 236. 
way, contraction of, 703. 
weight of, 217, 385. 

in pipes one foot long. 246. 
size of commercial measures by, 391. 
wheel, 280. 

W a Y 

weight, 385. 

Way, 

permanent, 759. 
station, cost, 803. 

Wear 

of cars, 811. 
of locomotives, 811. 
of rails, 760. 
of ropes, 414. 
of ties, 759. 
of tools, 743. 
of wheels, 807, 812. 
of wire ropes, 414. 

Web 

of beams, 529. 
members of truss, 547. 
of riveted girders, 539, 540. 

Wedge, Wedges, 

mensuration of, 161. 
striking, for centers, 711, 712, 720. 
Weight, Weights, 381, &c. See also the 
article in question, 
of centers for arches, 719. 
on driving wheels, 546, 664, 805, &c. 
French, old, 393. 
and measures, 385, &c. 


Weight, Weights—continued, 
metric, 381, &c, 393. 

Russian, 394. 

Spanish, 394. 

of substances, table, 381. See also the 
article in question. 

Weir, 264, &c. 

Well, Wells, 
artesian, 627. 
boring, 626. 
contents, 157. 

masonry, quantity in walls of, 158. 
Wellhouse process, 425a. 

Westing, 168. 

Westinghouse experiments, 374. 

Wet 

perimeter, 244. 
rot, 425. 

Wharf 
spikes. 762. 
walls, 236, 691. 

Wharton switch, 778. 

Wheel, Wheels, 
and axle, 339. 

barrows, loads of, 745, &c, 752. 

base, 805, &c. 

of car, 812. 

cog, 342. 

driving, 805, &c. 

loads on, 546, 564, 805-807. 
of locomotives, 805, &c. 

loads on, 546, 564, 805-807. 
meters, 270. 

Persian, 379. 
and pinions, 342. 
skidding of, 374a. 
tire of, locomotive, 805, &c. 
of cars, 812. 
of wagon, 380. 
train of, 342. 
tread of, 380, 765. 
tread-, 378, 641. 
water-, 280. 

Wheeled scrapers, 747. 

White effervescence on walls, 673, 678. 
lead paint, 429. 

-wash, 431. 

Whitworth standard screw thread, &c, 
406. 

Widths of bridges, 542, 608. 

Winch, 339, 378. 

Wind, 216. 

effect on suspension bridges, 616. 
mills. 801. 

pressure on roofs, 216, 580, 581. 

W r ing 

of frog, 780. 

-walls, 704. 

Wire, 410-412. 
brass, copper, 411. 
fence, 803. 
gauges, 410-412. 
iron, steel, 412. 
rope, 413. 
strength, 464. 

Wohler’s law, 435. 

Wood. See Timber, Wooden, 
board measure, table, 420. 









866 


INDEX. 

Wood—Zone, 


Wood—con tinned, 
cohesive strength, 463. 
compressibility, 434 e. 
compressive strength, 436. 
creosoting, 425. 
crushing strength, 436. 

De Yolson, rock-drill. 657. 
ductility, 434 «. 
durability, 425. 
effect, on cement, 678. 
of lime on, 670. 
of mortar on, 670. 
elastic limit of, 434 e. 
expansion by heat, 212. 
friction of, 373, Ac. 
fuel, 810, Ac. 
limit of elasticity, 434 e. 
modulus of elasticity, 434e. 
preservation of, 425. 
shearing strength, 476. 
shingles, 429. 
specific gravity of, 381, Ac. 
strength, 436, 463, 476, 477, 493. 
stretch of, 434 e. 
tensile strength, 463. 
torsional strength, 477. 
transverse strength, constants for, 493. 
weight, 381, Ac. 

Wood’s frog, 785. 

Wooden 

beams, 493, Ac. See Beams, Wooden, 
bridges, 514. See also Truss, Trestle, 
Bridge, Ac. 
dams, 282. 
pillars, 458, Ac. 
pipes, 294. 
trestles, 755. 
turntables, 797. 

Work, 316. 
of friction, 374/. 


Work—continued, 
rate of, unit of, 318. 
unit of, 316. 
useful, 318. 

World, railroad, miles in, 818. 

Worm fence, 803. 

Worms, sea, 425. 

Wrecking car, 750. 

Wrought-iron. See Iron, Wrought. 

Y. 

Yard, Yards, 385, 387. 
cubic, of earthwork, 732. 
equivalents of, 389. 

Yielding of centers for arches, 713, Ac, 
720. 

z. 

Zigzag riveting, 470. 

Zinc, 418. 

effect of cement, mortar, Ac, on, 670, 
673. 

of, on water, 419. 
of water on, 219. 
expansion by heat, 212. 
paint, 429. 
paint on, 403. 
price, 419. 
roofing, 418. 
sheets, 418. 

strength, compressive, 438. 
weight of, 385, 398, 400, 401, 410. 

Zone, Zones, 
circular, 146. 
of circular spindle, 167. 
parabolic, 152. 
spherical, 166. 

center of gravity of, 351 f. 


1 


THE END. 


f 


3 * 177-2 


f 

1 


























































